Research ArticleSingularities for One-Parameter Null Hypersurfaces ofAnti-de Sitter Spacelike Curves in Semi-Euclidean Space

Yongqiao Wang1 Donghe Pei1 and Ruimei Gao2

1School of Mathematics and Statistics Northeast Normal University Changchun 130024 China2Department of Science Changchun University of Science and Technology Changchun 130022 China

Correspondence should be addressed to Donghe Pei peidh340nenueducn

Received 19 April 2016 Revised 22 June 2016 Accepted 27 June 2016

Academic Editor Ajda Fosner

Copyright copy 2016 Yongqiao Wang et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider one-parameter null hypersurfaces associated with spacelike curves The spacelike curves are in anti-de Sitter 3-spacewhile one-parameter null hypersurfaces lie in 4-dimensional semi-Euclidean spacewith index 2We classify the generic singularitiesof the hypersurfaces which are cuspidal edges and swallowtails And we reveal the geometric meanings of the singularities of suchhypersurfaces by the singularity theory

1 Introduction

Semi-Euclidean space is a vector space with pseudoscalarproduct which is different from Euclidean space The studyof semi-Euclidean space has produced fruitful results pleasesee [1ndash5] It is well known that there exist spacelike sub-manifolds timelike submanifolds and null submanifolds insemi-Euclidean space Null submanifolds appear in manyphysics papers For example the null submanifolds are ofinterest because they provide models of different horizontypes such as event horizons of Kerr black holes Cauchyhorizons isolated horizons Kruskal horizons and Killinghorizons [6ndash12] Null submanifolds are also studied in thetheory of electromagnetism

Anti-de Sitter space is a maximally symmetric semi-Riemannian manifold with constant negative scalar curva-ture This space is a very important subject in physics it isalso one of the vacuum solutions of Einsteinrsquos field equationin the theory of relativity There is a conjecture in physicsthat the classical gravitation theory on anti-de Sitter spaceis equivalent to the conformal field theory on the idealboundary of anti-de Sitter space It is called the AdSCFTcorrespondence or the holographic principle by E Witten Inmathematics this conjecture is that the extrinsic geometricproperties on submanifolds in anti-de Sitter space havecorresponding Gauge theoretic geometric properties in its

ideal boundary Therefore it is necessary to investigate thesubmanifolds in anti-de Sitter space During the last fourdecades singularity theory has enjoyed rapid developmentThe French mathematician R Thom (Fields medallist) firstput forward the philosophical idea of applying singularitytheory to the study of differential geometry Porteous appliedthe thoughts of Thom to the study of Euclidean geometry[13] The first attempts to apply the singularity theory tonon-Euclidean geometry were undertaken by S Izumiya thesecond author and T Sano et al

Recently there appear several results on submanifolds inanti-de Sitter space from the viewpoint of singularity theoryThe timelike hypersurfaces are studied in the anti-de Sitterspace from the viewpoint of Lagrangian singularity theory[14] In the study of submanifolds the null submanifoldshappen to be the most interesting subjects both from theviewpoint of singularity theory and the theory of relativity[15 16] Fusho and Izumiya have discussed the spacelikecurves in de Sitter 3-space [17] they define the null surfaces ofspacelike curves The spacelike curves have degenerate con-tact with null cones at the singularities of the null surfaces In[18] L Chen Q Han the second author andW Sun considernull ruled surfaces along spacelike curves in anti-de Sitter 3-spaceThey give the classifications of singularities of the ruledsurfaces which are the codimensional two submanifolds insemi-Euclidean space with index 2 Null surfaces have been

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 2319741 8 pageshttpdxdoiorg10115520162319741

2 Journal of Function Spaces

studied in preceding literature As we all know the horizon ofthe black hole is a null hypersurface or a part However to thebest of the authorsrsquo knowledge no literature exists regardingthe singularities of one-parameter null hypersurfaces as theyrelate to spacelike curves in anti-de Sitter 3-space Thusthe current study hopes to serve such a need Therefore inthis paper we stick to the one-parameter null hypersurfaceswhich are generated by spacelike curves in anti-de Sitter 3-space When the parameter is fixed the sections of one-parameter null hypersurfaces are null surfaces Moreoverthe null ruled surfaces in [18] are the sections of one-parameter null hypersurfaces And the one-parameter nullhypersurfaces can be taken as the most elementary case forthe study of the lowest codimensional submanifolds in semi-Euclidean space with index 2

A singularity is a point at which a function blows up Itis a point at which a function is at a maximumminimumor a submanifold is no longer smooth and regular In [19]we have discussed the singularities of normal hypersurfaceassociated with a timelike curve In this paper we first con-sider spacelike curves in anti-de Sitter 3-space and then definethe one-parameter null hypersurfaces which are bundlesalong spacelike curves whose fibres are null lines or timelikecurves We also define the one-parameter height functionson spacelike curves and apply the versal unfolding theoryof functions to discuss them the functions can be used toinvestigate the geometric properties of one-parameter nullhypersurfaces In fact one-parameter null hypersurfaces arethe discriminant sets of these functions (the discriminantsets of one-parameter height functions are precisely thewavefronts of spacelike curves) the singularities of nullhypersurfaces are119860119896-singularities (119896 ge 2) of these functionsThe main result in this paper is Theorem 5 This theoremcharacterizes the contact of spacelike curves with null conesin semi-Euclidean space with index 2

A brief description of the organization of this paper is asfollows In Section 2 we review the concepts of submanifoldsin semi-Euclidean space with index 2 In Section 3 wegive one-parameter height functions of a spacelike curveby which we can obtain several geometric invariants ofthe spacelike curve We also get the singularities of one-parameter null hypersurfaces and the geometric meaning ofTheorem 5 is described in this section The preparations forthe proof of Theorem 5 are in Section 4 We give the proof ofTheorem 5 in Section 5 In Section 6 we give an example toillustrate the results of Theorem 5

We will assume throughout the whole paper that allmanifolds and maps are 119862infin unless the contrary is stated

2 Preliminaries

Let R4 be a 4-dimensional vector space For any two vectorsx = (1199091 1199092 1199093 1199094) and y = (1199101 1199102 1199103 1199104) in R4 theirpseudoscalar product is defined by

⟨x y⟩ = minus11990911199101 minus 11990921199102 + 11990931199103 + 11990941199104 (1)

The space (R4 ⟨ ⟩) is called semi-Euclidean 4-space withindex 2 and denoted by R4

2

For three vectors x = (1199091 1199092 1199093 1199094) y = (1199101 1199102 1199103 1199104)and z = (1199111 1199112 1199113 1199114) isin R4

2 we define a vector x and y and z by

x and y and z =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minuse1 minuse2 e3 e41199091 1199092 1199093 1199094

1199101 1199102 1199103 1199104

1199111 1199112 1199113 1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(2)

where e1 e2 e3 e4 is the canonical basis of R42 We have

⟨x0 xandyandz⟩ = det(x0 x y z) so xandyandz is pseudoorthogonalto x y and z A nonzero vector x isin R4

2is called spacelike null

or timelike if ⟨x x⟩ gt 0 ⟨x x⟩ = 0 or ⟨x x⟩ lt 0 respectivelyThe norm of x isin R4

2is defined by x = (sign(x)⟨x x⟩)12

where sign(x) denotes the signature of x which is given bysign(x) = 1 0 or minus1 if x is a spacelike null or timelike vectorrespectively

Let 120574 119868 rarr R42be a regular curve in R4

2(ie (119905) =

119889120574119889119905 = 0) where 119868 is an open interval For any 119905 isin 119868 thecurve 120574 is called spacelike null or timelike if ⟨(119905) (119905)⟩ gt 0⟨(119905) (119905)⟩ = 0 or ⟨(119905) (119905)⟩ lt 0 respectively We call 120574a nonnull curve if 120574 is a spacelike or timelike curve The arc-length of a nonnull curve 120574 measured from 120574(1199050)(1199050 isin 119868) is119904(119905) = int

119905

1199050(119905)119889119905

The parameter 119904 is determined such that 1205741015840(119904) = 1 forthe nonnull curve where 1205741015840(119904) = 119889120574119889119904 is the unit tangentvector of 120574 Some submanifolds in R4

2are as follows

The anti-de Sitter space is defined by

1198673

1= x isin R

4

2| ⟨x x⟩ = minus1 (3)

and the lightlike cone by

LC119901 = x isin R4

2| ⟨x minus p x minus p⟩ = 0 (4)

We also define the one-parameter anti-de Sitter space by

1198673

1(minussinh2120579) = x isin R

4

2| ⟨x x⟩ = minussinh2120579 (5)

Let 120574 119868 rarr 1198673

1be a spacelike regular curve that is 120574

satisfies ⟨(119905) (119905)⟩ gt 0 119905 isin 119868 Since the curve 120574 is spacelikewe can reparametrize it by the arc-length 119904 Then we havethe tangent vector t(119904) = 120574

1015840(119904) obviously t(119904) = 1 When

⟨t1015840(119904) t1015840(119904)⟩ = minus1 we define a unit vector

n (119904) = t1015840 (119904) minus 120574 (119904)1003817100381710038171003817t1015840 (119904) minus 120574 (119904)

1003817100381710038171003817

(6)

Let e(119904) = 120574(119904)andt(119904)andn(119904) thenwehave a pseudoorthonormalframe 120574(119904) t(119904)n(119904) e(119904) of R4

2along 120574 By direct calculat-

ing the following Frenet-Serret type is displayed under theassumption that ⟨t1015840(119904) t1015840(119904)⟩ = minus1

1205741015840(119904) = t (119904)

t1015840 (119904) = 120574 (119904) + 120581119892 (119904)n (119904)

n1015840 (119904) = minus120575120581119892 (119904) t (119904) + 120575120591119892 (119904) e (119904)

e1015840 (119904) = 120575120591119892 (119904)n (119904)

(7)

Journal of Function Spaces 3

Here 120581119892(119904) = t1015840(119904) minus 120574(119904) is the geodesic curvature

120591119892 (119904) = minus120581minus2

119892(119904) det (120574 (119904) 1205741015840 (119904) 12057410158401015840 (119904) 120574101584010158401015840 (119904)) (8)

is the geodesic torsion and 120575 = sign(119899(119904)) If ⟨t1015840(119904) t1015840(119904)⟩ = minus1we can obtain 120581119892(119904) = 0 it means that 120574(119904) is a geodesic curvein 1198673

1 We consider ⟨t1015840(119904) t1015840(119904)⟩ = minus1 (ie 120581119892(119904) = 0) in the

following sectionsLet 120574 119868 rarr 119867

3

1be a unit speed spacelike curve we write

120574120579= sinh 120579120574 and define 119871plusmn

120579 119868 timesR rarr 119867

3

1(minussinh2120579) by

119871plusmn

120579(119904 120583) = sinh 120579120574 (119904) + 120583 (n (119904) plusmn e (119904))

= 120574120579+ 120583 (n (119904) plusmn e (119904)) 120579 isin (0 +infin)

(9)

We call 119871plusmn120579(119904 120583) the one-parameter null hypersurfaces associ-

ated with 120574(119904) We also define the following model surfaceFor any k0 isin 119867

3

1(minussinh2120579)

LCV0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (10)

On the other hand let 119865 1198673

1rarr R be a submersion and

let 120574 119868 rarr 1198673

1be a spacelike curve We say that 120574 and 119865minus1(0)

have 119896-point contact at 119905 = 1199050 if the function 119892(119905) = 119865 ∘ 120574(119905)

satisfies 119892(1199050) = 1198921015840(1199050) = sdot sdot sdot = 119892

(119896minus1)(1199050) = 0 119892(119896)(1199050) = 0

We also have that 120574 and 119865minus1(0) have at least 119896-point contact

at 119905 = 1199050 if the function119892(119905) = 119865∘120574(119905) satisfies119892(1199050) = 1198921015840(1199050) =

sdot sdot sdot = 119892(119896minus1)

(1199050) = 0

3 One-Parameter HeightFunctions and the Singularities ofOne-Parameter Null Surfaces

In this section we discuss a kind of Lorentzian invariantfunctions on a spacelike curve in119867

3

1 It is useful to study the

null hypersurfaces of the spacelike curve Let 120574 119868 rarr 1198673

1be

a unit spacelike curve We now define a function

119867 119868 times 1198673

1(minussinh2120579) times (0 +infin) 997888rarr R (11)

by 119867(119904 k 120579) = ⟨120574120579(119904) k⟩ + sinh2120579 we call 119867 one-parameter

height function on the spacelike curve 120574 We denote thatℎV120579(119904) = 119867(119904 k 120579) (k 120579) isin 119867

3

1(minussinh2120579) times (0 +infin) Then

we have the following proposition

Proposition 1 Let 120574 119868 rarr 1198673

1be a unit spacelike curve and

(k 120579) isin 1198673

1(minussinh2120579) times (0 +infin) Then one has the following

(1) ℎV120579(119904) = ℎ1015840

V120579(119904) = 0 if and only if there exists 120583 isin R

such that k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904))

(2) ℎV120579(119904) = ℎ1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = 0 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (12)

(3) ℎV120579(119904) = ℎ1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = ℎ101584010158401015840

V120579(119904) = 0 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (13)

and 120590 = 0(4) ℎV120579(119904) = ℎ

1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = ℎ101584010158401015840

V120579(119904) = ℎ(4)

V120579(119904) = 0 if andonly if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (14)

and 120590 = 1205901015840= 0

Proof (1) Since k isin 1198673

1(minussinh2120579) we can find that 120578 120572 120583 120573 isin

R with minus1205782 + 1205722+ 1205751205832minus 1205751205732= minussinh2120579 such that k = 120578120574(119904) +

120572t(119904) + 120583n(119904) + 120573e(119904) Because

ℎV120579 (119904) = ⟨sinh 120579120574 (119904) k⟩ + sinh2120579 = 0 (15)

we can get 120578 = sinh 120579 when ℎ1015840

V120579(119904) = 0 it means that 120572 = 0

and 120573 = plusmn120583 Therefore k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904)) theconverse direction also holds

(2) By (1) an easy computation shows that

sinh 120579 ⟨t1015840 (119904) k⟩ = sinh 120579 ⟨120574 (119904) + 120581119892 (119904)n (119904) k⟩

= sinh 120579 ⟨120574 (119904) + 120581119892 (119904)n (119904) sinh 120579120574 (119904)

+ 120583 (n (119904) plusmn e (119904))⟩ = 0

(16)

we get 120583 = sinh 120579120575120581119892(119904) therefore

k = sinh 120579120574 (119904) + (sinh 120579

(120575120581119892 (119904))

) (n (119904) plusmn e (119904)) (17)

(3) Under the assumption that

ℎV120579 (119904) = ℎ1015840

V120579 (119904) = ℎ10158401015840

V120579 (119904) = 0

ℎ101584010158401015840

V120579 (119904)

sinh 120579= ⟨(1 minus 120575120581

2

119892(119904)) t (119904) + 120581

1015840

119892(119904)n (119904)

+ 120575120581119892 (119904) 120591119892 (119904) k⟩

(18)

we can get 120590(119904) = 1205811015840

119892(119904)∓120581119892(119904)120591119892(119904) = 0 assertion (3) follows

(4) Based on the assumption that

ℎV120579 (119904) = ℎ1015840

V120579 (119904) = ℎ10158401015840

V120579 (119904) = ℎ101584010158401015840

V120579 (119904) = 0 (19)

the relation

ℎ(4)

V120579 (119904)

sinh 120579= ⟨(1 minus 120575120581

2

119892(119904)) t (119904) + 120581

1015840

119892(119904)n (119904)

minus 120575120581119892 (119904) 120591119892 (119904) e (119904) k⟩1015840

(20)

follows the fact that ℎ(4)V120579(119904) = 0 is equivalent to

12058110158401015840

119892(119904) ∓ 120581

1015840

119892(119904) 120591119892 (119904) ∓ 120581119892 (119904) 120591

1015840

119892(s) = 0 (21)

so 1205901015840 = 0 This proves assertion (4)

4 Journal of Function Spaces

Now we do research on some properties of one-parameter null hypersurfaces of the spacelike curve in 119867

3

1

As we can know the functions 120581119892(119904) 120591119892(119904) and 120590(119904) haveparticular meanings Here we consider the case when theone-parameter null hypersurfaces have the most degeneratesingularities We have the following proposition

Proposition 2 Let 120574 119868 rarr 1198673

1be a unit spacelike curveThen

one has the following

(1) The set 119871plusmn120579(119904 120583) | 120583 = sinh 120579120575120581119892(119904) is the singulari-

ties of one-parameter null hypersurfaces 119871plusmn120579(119904 120583)

(2) If k0 = 119871plusmn

120579(119904 sinh 120579120575120581119892(119904)) is a constant vector one

has 120574120579(119904) isin 119871119862V0(120579) for any 119904 isin 119868 at the same time

120590(119904) = 0

Proof By calculations we have

120597119871plusmn

120579(119904 120583)

120597120579= minus cosh 120579120574 (119904) (22)

120597119871plusmn

120579(119904 120583)

120597120583= e (119904) plusmn n (119904) (23)

120597119871plusmn

120579(119904 120583)

120597119904

= sinh 120579t (119904)

+ 120583 (minus120575120581119892 (119904) t (119904) + 120575120591119892 (119904) e (119904) plusmn 120575120591119892 (119904)n (119904))

= (sinh 120579 minus 120575120583120581119892 (119904)) t (119904)

+ 120575120583120591119892 (119904) (e (119904) plusmn n (119904))

(24)

(1) If the above three vectors are linearly dependent wecan get the singularities of 119871plusmn

120579(119904 120583) if and only if sinh 120579 minus

120575120583120581119892(119904) = 0 Therefore assertion (1) holds(2) For any fixed 120579 isin (0 +infin) if

119891 (119904) = sinh 120579120574 (119904) + 120583 (119904) (n (119904) plusmn e (119904)) (25)

is a constant then

119889119891

119889119904= (sinh 120579 minus 120575120583 (119904) 120581119892 (119904)) t (119904)

+ (1205831015840(119904) minus 120575120583 (119904) 120591119892 (119904)) (n (119904) plusmn e (119904)) = 0

(26)

Since

120583 (119904) =sinh 120579120575120581119892 (119904)

1205831015840(119904) minus 120575120583 (119904) 120591119892 (119904) = 0 (27)

then

120590 = 1205811015840

119892(119904) ∓ 120581119892 (119904) 120591119892 (119904) = 0 (28)

We have

⟨120574120579(119904) k0⟩

= ⟨120574120579(119904) sinh 120579120574 (119904) + sinh 120579

120575120581119892 (119904)(n (119904) plusmn e (119904))⟩

= minussinh2120579

(29)

This completes the proof

4 Unfoldings of One-ParameterHeight Functions

In this section we classify singularities of the one-parameternull hypersurfaces along 120574 as an application of the unfoldingtheory of functions

Let 119865 (R times R119903 (1199040 x0)) rarr R be a function germ119891(119904) = 1198651199090

(119904 x0) We call 119865 an 119903-parameter unfolding of 119891If 119891(119901)(1199040) = 0 for all 1 le 119901 le 119896 and 119891

(119896+1)(1199040) = 0 we say

that 119891 has 119860119896-singularity at 1199040 We also say that 119891 has 119860ge119896-singularity at 1199040 if 119891

(119901)(1199040) = 0 for all 1 le 119901 le 119896 Let 119865 be an

119903-parameter unfolding of 119891 and 119891 has 119860119896-singularity (119896 ge 1)

at 1199040 we define the (119896 minus 1)-jet of the partial derivative 120597119865120597119909119894at 1199040 as

119895(119896minus1) 120597119865

120597119909119894

(119904 x0) (1199040) =119896+1

sum

119895=1

120572119895119894 (119904 minus 1199040)119895

(119894 = 1 119903)

(30)

If the rank of 119896 times 119903 matrix (1205720119894 120572119895119894) is 119896 (119896 le 119903) then 119865 iscalled a versal unfolding of 119891 where 1205720119894 = (120597119865120597119909119894)(1199040 x0)The discriminant set of 119865 is defined by

119863119865 = x isin R119903| exist119904 isin R 119865 (119904 x) = 120597119865

120597119904(119904 x) = 0 (31)

There has been the following famous result [20]

Theorem 3 Let 119865 (R timesR119903 (1199040 x0)) rarr R be an 119903-parameterunfolding of 119891(119904) which has 119860119896-singularity at 1199040 suppose that119865 is a V119890119903119904119886119897 119906119899119891119900119897119889119894119899119892 of 119891 Then one has the following

(a) If 119896 = 1 then119863119865 is locally diffeomorphic to 0 timesR119903minus1(b) If 119896 = 2 then119863119865 is locally diffeomorphic to 119862 timesR119903minus2(c) If 119896 = 3 then119863119865 is locally diffeomorphic to 119878119882timesR119903minus3

By Proposition 1 the discriminant set of the timelikeheight function119867(119904 k 120579) is given by

119863119867 = sinh 120579120574 (119904) + 120583 (n (119904) plusmn e (119904)) | 119904 120583 isin 119868 120579

isin (0 +infin)

(32)

Proposition 4 Let119867(119904 k 120579) be a one-parameter height func-tion on the spacelike curve 120574 k isin 119863119867 If ℎV has 119860119896-singularityat s (119896 = 1 2 3 4) then119867 is a versal unfolding of ℎV

Journal of Function Spaces 5

Proof Let 120574(119904) = (1199091(119904) 1199092(119904) 1199093(119904) 1199094(119904)) isin 1198673

1and k =

(V1 V2 V3 V4) isin 1198673

1(minussinh2120579)

Then119867(119904 k 120579)

= sinh 120579 (minus1199091V1 minus 1199092V2 + 1199093V3 + 1199094V4 + sinh 120579)

120579 isin (0 +infin)

(33)

Let V1 = plusmnradicminusV22+ V23+ V24+ sinh2120579 so

120597119867

120597V2(119904 k) = sinh 120579 (minus1199092 minus

V2V11199091)

1205972119867

120597119904120597V2(119904 k) = sinh 120579 (minus1199091015840

2minusV2V11199091015840

1)

1205973119867

1205972119904120597V2(119904 k) = sinh 120579 (minus11990910158401015840

2minusV2V111990910158401015840

1)

120597119867

120597V119894(119904 k) = sinh 120579 (119909119894 +

V119894V11199091)

1205972119867

120597119904120597V119894(119904 k) = sinh 120579 (1199091015840

119894+

V119894V11199091015840

1)

1205973119867

1205972119904120597V119894(119904 k) = sinh 120579 (11990910158401015840

119894+

V119894V111990910158401015840

1) (119894 = 3 4)

120597119867

120597120579(119904 k) = 120593120579 (119904)

1205972119867

120597119904120597120579(119904 k) = 120593

1015840

120579(119904)

1205973119867

1205972119904120597120579(119904 k) = 120593

10158401015840

120579(119904)

(34)

The 2-119895119890119905 of (120597119867120597V2)(119904 k) at 1199040 is given by

sinh 120579 (minus11990910158402119904 minus

1

211990910158401015840

21199042) +

V2V1

(minus1199091015840

1119904 minus

1

211990910158401015840

11199042) (35)

the 2-119895119890119905 of (120597119867120597V119894)(119904 k) at 1199040 is given by

sinh 120579 (1199091015840119894119904 +

1

211990910158401015840

1198941199042) +

V119894V1

(1199091015840

1119904 +

1

211990910158401015840

11199042)

(119894 = 3 4)

(36)

(1) By Proposition 1 ℎV has the 1198601-singularity at 1199040 if andonly if k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904)) Since the curve 120574(119904) isregular the rank of

(minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579) (37)

is 1(2) We can get that ℎ has the 1198602-singularity at 1199040 if and

only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (38)

and 120590 = 0 When ℎ has the 119860ge2-singularity at 1199040 we requirethe 2 times 4matrix

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091 1199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

) (39)

to have rank 2 which follows from the proof of the next case(3) It also follows from Proposition 1 that ℎ has the 119860ge3-

singularity at 1199040 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (40)

and 120590 = 0 1205901015840 = 0We require the 3 times 4matrix

(

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

12059310158401015840

120579

sinh 120579

)

)

(41)

to have rank 3Let 3 times 3matrix

119860 = (

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

) (42)

We denote

119860 (119894 119895 119896) = det(

119909119894 119909119895 119909119896

1199091015840

1198941199091015840

1198951199091015840

119896

11990910158401015840

11989411990910158401015840

11989511990910158401015840

119896

) (43)

thendet119860

= minus119860 (2 3 4) minusV2V1119860 (1 3 4) minus

V3V1119860 (2 1 4)

minusV4V1119860 (2 3 1)

= plusmn1

V1⟨(V1 V2 V3 V4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minuse1 minuse2 e3 e41199091 1199092 1199093 1199094

1199091015840

11199091015840

21199091015840

31199091015840

4

11990910158401015840

111990910158401015840

211990910158401015840

311990910158401015840

4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟩

= plusmn1

V1⟨k 120574 (119904) and 1205741015840 (119904) and 12057410158401015840 (119904)⟩

(44)

Since k isin 119863119867 is a singular point then

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (45)

6 Journal of Function Spaces

and we have

120574 (119904) and 1205741015840(119904) and 120574

10158401015840(119904) = 120574 (119904) and 120574

1015840(119904)

and (120574 (119904) + 120581119892 (119904)n (119904))

= 120581119892 (119904) (120574 (119904) and t (119904) and n (119904))

= 120581119892 (119904) e (119904)

(46)

Therefore

det119860 = plusmn1

V1⟨sinh 120579120574 (119904)

+sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) 120581119892 (119904) e (119904)⟩ = plusmnsinh 120579V1

= 0

(47)

In summary119867 is a versal unfolding of ℎV this completes theproof

5 Main Result

The main result in this paper is in this section We nowconsider the following conditions

(A1) The number of points 119901 of 120574120579(119904) where LCV0(120579) at 119901

have four-point contact with the curve 120574120579 is finite

(A2) There is no point 119901 of 120574120579(119904) where LCV0(120579) at 119901 have

five-point contact or greater with the curve 120574120579

Our main result is as follows

Theorem 5 Let 120574 119868 rarr 1198673

1be a unit regular spacelike curve

k0 = 119871plusmn

120579(1199040 1205830) and

119871119862V0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (48)

Then one has the following(1) 119871119862V0(120579) and 120574120579 have at least 2-point contact at 1199040(2) 119871119862V0(120579) and 120574120579 have 3-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (49)

and 120590(1199040) = 1205811015840

119892(1199040) ∓ 120581119892(1199040)120591119892(1199040) = 0 Under this condition

the germ of image 119871plusmn120579at (1199040 1205830) is diffeomorphic to the cuspidal

edge 119862 timesR (Figure 1)(3) 119871119862V0(120579) and 120574120579 have 4-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (50)

120590(1199040) = 0 and 1205901015840(1199040) = 0 Under this condition the germ of

image 119871plusmn

120579at (1199040 1205830) is diffeomorphic to the swallowtail 119878119882

(Figure 2)

Here 119862 = (1199091 1199092) | 1199092

1= 1199093

2 and 119878119882 = (1199091 1199092 1199093) |

1199091 = 31199064+ 1199062V 1199092 = 4119906

3+ 2119906V 1199093 = V

Figure 1 Cuspidal edge

Figure 2 Swallowtail

Proof Let 120574 119868 rarr 1198673

1be a spacelike regular curve and

⟨t1015840(119904) t1015840(119904)⟩ = minus1 As k0 = 119871plusmn

1205790(1199040 1205830) we give a function

119867 1198673

1(minussinh21205790) 997888rarr R (51)

by119867(119906) = ⟨119906 V0⟩ + sinh21205790 then we assume that

ℎV0 1205790 (119904) = 119867 (1205741205790(119904)) (52)

Because 119867minus1(0) = LCV0(1205790) and 0 is a regular value of 119867

1205741205790

and LCV0(1205790) have (119896 + 1)-point contact at 1199040 if andonly if ℎV0 1205790(119904) has the 119860119896-singularity at 1199040 By Proposition 1Theorem 3 and Proposition 4 we get the results

6 Example

In this section we construct the one-parameter null hyper-surfaces associated with a spacelike curve and two sections of

Journal of Function Spaces 7

the one-parameter null hypersurfaces The two sections arenull surfaces and they are also the wavefronts of spacelikecurves By calculatingwe get the singularities of null surfacesIt is useful to understand the one-parameter null hypersur-faces

Let 120574(119904) = (radic2 cosh(2119904) radic3 cosh(radic7119904) + 2 sinh(radic7119904)radic2 sinh(2119904)radic3 sinh(radic7119904)+2 cosh(radic7119904)) be a spacelike curvein1198673

1 where 119904 is the arc-length parameter Then

t (119904) = (2radic2 sinh (2119904) radic21 sinh (radic7119904) + 2radic7

sdot cosh (radic7119904) 2radic2 cosh (2119904) radic21 cosh (radic7119904) + 2radic7

sdot sinh (radic7119904))

n (119904) = (cosh (2119904) radic6 cosh (radic7119904) + 2radic2

sdot sinh (radic7119904) sinh (2119904) radic6 sinh (radic7119904) + 2radic2

sdot cosh (radic7119904))

(53)

We have

e (119904) = 120574 (119904) and t (119904) and n (119904)

= (radic7 sinh (2119904) 2radic6 sinh (radic7119904)

+ 4radic2 cosh (radic7119904) radic7 cosh (2119904) 2radic6 cosh (radic7119904)

+ 4radic2 sinh (radic7119904))

(54)

119871plusmn

120579(119904 120583) = sinh 120579120574(119904)+120583(n(119904)plusmne(119904)) be the one-parameter

null hypersurfaces of 120574(119904) At the moment we can calculatethat 120581119892(119904) = 3radic2 and 120591119892(119904) = 2radic7 two sections of 119871plusmn

120579(119904 120583)

with 1205791 = arcsinh(12) 1205792 = arcsinh(radic22) are as follows

119871+

1205791(119904 120583) =

1

2120574 (119904) + 120583 (n (119904) + e (119904)) = ((

radic2

2+ 120583)

sdot cosh (2119904) + radic7120583 sinh (2119904) (radic3

2+ radic6120583 + 4radic2120583)

sdot cosh (radic7119904) + (1 + 2radic2120583 + 2radic6120583)

sdot sinh (radic7119904) radic7120583 cosh (2119904) + (radic2

2+ 120583)

sdot sinh (2119904) (1 + 2radic2120583 + 2radic6120583)

sdot cosh (radic7119904) + (radic3

2+ radic6120583 + 4radic2120583)

sdot sinh (radic7119904))

Figure 3 Projection of 119871+1205791(119904 120583) on 119909211990931199094-space

Figure 4 Projection of 119871minus1205792(119904 120583) on 119909111990921199093-space

119871minus

1205792(119904 120583) =

radic2

2120574 (119904) + 120583 (n (119904) minus e (119904)) = ((1 + 120583)

sdot cosh (2119904) minus radic7120583 sinh (2119904) (radic6

2+ radic6120583 minus 4radic2120583)

sdot cosh (radic7119904) + (radic2 + 2radic2120583 minus 2radic6120583) sinh (radic7119904)

minus radic7120583 cosh (2119904) + (1 + 120583)

sdot sinh (2119904) (radic2 + 2radic2120583 minus 2radic6120583) cosh (radic7119904)

+ (radic6

2+ radic6120583 minus 4radic2120583) sinh (radic7119904))

(55)

The pictures of 1205791-null hypersurface 119871+

1205791(119904 120583) and its

singularities 119871+1205791(119904 radic212) can be seen in Figure 3 And the

pictures of 1205792-null hypersurface 119871minus

1205792(119904 120583) and its singularities

119871minus

1205792(119904 16) can be seen in Figure 4

Competing Interests

The authors declare that they have no competing interests

8 Journal of Function Spaces

Acknowledgments

This work was partially supported by NSF of China (nos11271063 and 11501051) and NCET of China (no 05-0319)The first author was partially supported by the Project of Sci-ence and Technology of Heilongjiang Provincial EducationDepartment of China (no UNPYSCT-2015103)

References

[1] AMahmut E Soley and TMurat ldquoBeltrami-Meusnier formu-las of generalized semi ruled surfaces in semi Euclidean spacerdquoKuwait Journal of Science vol 41 no 2 pp 65ndash83 2014

[2] E Tulay andGM Ali ldquoSome characterizations of quaternionicrectifying curves in the semi-Euclidean space E2

4rdquo HonamMathematical Journal vol 36 no 1 pp 67ndash83 2014

[3] H Liu ldquoCurves in affine and semi-Euclidean spacesrdquo Results inMathematics vol 65 no 1-2 pp 235ndash249 2014

[4] E Soley and T Murat ldquoTimelike Bertrand curves in semi-Euclidean spacerdquo International Journal of Mathematics andStatistics vol 14 no 2 pp 78ndash89 2013

[5] M Sakaki ldquoBi-null Cartan curves in semi-Euclidean spaces ofindex 2rdquo Beitrage zur Algebra und Geometrie Contributions toAlgebra and Geometry vol 53 no 2 pp 421ndash436 2012

[6] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[7] G Clement ldquoBlack holes with a null Killing vector in three-dimensional massive gravityrdquo Classical and Quantum Gravityvol 26 no 16 Article ID 165002 11 pages 2009

[8] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[9] S W Hawking ldquoBlack holes in general relativityrdquo Communica-tions in Mathematical Physics vol 25 pp 152ndash166 1972

[10] M Korzynski J Lewandowski and T Pawlowski ldquoMechanicsof multidimensional isolated horizonsrdquo Classical and QuantumGravity vol 22 no 11 pp 2001ndash2016 2005

[11] V Moncrief and J Isenberg ldquoSymmetries of cosmologicalCauchy horizonsrdquo Communications in Mathematical Physicsvol 89 no 3 pp 387ndash413 1983

[12] L Kong and D Pei ldquoOn spacelike curves in hyperbolic spacetimes sphererdquo International Journal of Geometric Methods inModern Physics vol 11 no 3 2014

[13] I R Porteous ldquoThe normal singularities of a submanifoldrdquoJournal of Differential Geometry vol 5 pp 543ndash564 1971

[14] L Chen S Izumiya and D Pei ldquoTimelike hypersurfaces inthe anti-de Sitter space from a contact viewpointrdquo Journal ofMathematical Sciences vol 199 no 6 pp 629ndash645 2014

[15] J Sun and D Pei ldquoNull surfaces of null curves on 3-null conerdquoPhysics Letters A vol 378 no 14-15 pp 1010ndash1016 2014

[16] J Sun and D Pei ldquoNull Cartan Bertrand curves of AW(k)-typein Minkowski 4-spacerdquo Physics Letters A vol 376 no 33 pp2230ndash2233 2012

[17] T Fusho and S Izumiya ldquoLightlike surfaces of spacelike curvesin de Sitter 3-spacerdquo Journal of Geometry vol 88 no 1-2 pp19ndash29 2008

[18] L Chen Q Han D Pei and W Sun ldquoThe singularities ofnull surfaces in anti de Sitter 3-spacerdquo Journal of MathematicalAnalysis and Applications vol 366 no 1 pp 256ndash265 2010

[19] Y Wang and D Pei ldquoSingularities for normal hypersurfacesof de Sitter timelike curves in Minkowski 4-spacerdquo Journal ofSingularities vol 12 pp 207ndash214 2015

[20] J W Bruce and P J Giblin Curves and Singularities CambridgeUniversity Press 1992

Submit your manuscripts athttpwwwhindawicom

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

studied in preceding literature As we all know the horizon ofthe black hole is a null hypersurface or a part However to thebest of the authorsrsquo knowledge no literature exists regardingthe singularities of one-parameter null hypersurfaces as theyrelate to spacelike curves in anti-de Sitter 3-space Thusthe current study hopes to serve such a need Therefore inthis paper we stick to the one-parameter null hypersurfaceswhich are generated by spacelike curves in anti-de Sitter 3-space When the parameter is fixed the sections of one-parameter null hypersurfaces are null surfaces Moreoverthe null ruled surfaces in [18] are the sections of one-parameter null hypersurfaces And the one-parameter nullhypersurfaces can be taken as the most elementary case forthe study of the lowest codimensional submanifolds in semi-Euclidean space with index 2

A singularity is a point at which a function blows up Itis a point at which a function is at a maximumminimumor a submanifold is no longer smooth and regular In [19]we have discussed the singularities of normal hypersurfaceassociated with a timelike curve In this paper we first con-sider spacelike curves in anti-de Sitter 3-space and then definethe one-parameter null hypersurfaces which are bundlesalong spacelike curves whose fibres are null lines or timelikecurves We also define the one-parameter height functionson spacelike curves and apply the versal unfolding theoryof functions to discuss them the functions can be used toinvestigate the geometric properties of one-parameter nullhypersurfaces In fact one-parameter null hypersurfaces arethe discriminant sets of these functions (the discriminantsets of one-parameter height functions are precisely thewavefronts of spacelike curves) the singularities of nullhypersurfaces are119860119896-singularities (119896 ge 2) of these functionsThe main result in this paper is Theorem 5 This theoremcharacterizes the contact of spacelike curves with null conesin semi-Euclidean space with index 2

A brief description of the organization of this paper is asfollows In Section 2 we review the concepts of submanifoldsin semi-Euclidean space with index 2 In Section 3 wegive one-parameter height functions of a spacelike curveby which we can obtain several geometric invariants ofthe spacelike curve We also get the singularities of one-parameter null hypersurfaces and the geometric meaning ofTheorem 5 is described in this section The preparations forthe proof of Theorem 5 are in Section 4 We give the proof ofTheorem 5 in Section 5 In Section 6 we give an example toillustrate the results of Theorem 5

We will assume throughout the whole paper that allmanifolds and maps are 119862infin unless the contrary is stated

2 Preliminaries

Let R4 be a 4-dimensional vector space For any two vectorsx = (1199091 1199092 1199093 1199094) and y = (1199101 1199102 1199103 1199104) in R4 theirpseudoscalar product is defined by

⟨x y⟩ = minus11990911199101 minus 11990921199102 + 11990931199103 + 11990941199104 (1)

The space (R4 ⟨ ⟩) is called semi-Euclidean 4-space withindex 2 and denoted by R4

2

For three vectors x = (1199091 1199092 1199093 1199094) y = (1199101 1199102 1199103 1199104)and z = (1199111 1199112 1199113 1199114) isin R4

2 we define a vector x and y and z by

x and y and z =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minuse1 minuse2 e3 e41199091 1199092 1199093 1199094

1199101 1199102 1199103 1199104

1199111 1199112 1199113 1199114

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(2)

where e1 e2 e3 e4 is the canonical basis of R42 We have

⟨x0 xandyandz⟩ = det(x0 x y z) so xandyandz is pseudoorthogonalto x y and z A nonzero vector x isin R4

2is called spacelike null

or timelike if ⟨x x⟩ gt 0 ⟨x x⟩ = 0 or ⟨x x⟩ lt 0 respectivelyThe norm of x isin R4

2is defined by x = (sign(x)⟨x x⟩)12

where sign(x) denotes the signature of x which is given bysign(x) = 1 0 or minus1 if x is a spacelike null or timelike vectorrespectively

Let 120574 119868 rarr R42be a regular curve in R4

2(ie (119905) =

119889120574119889119905 = 0) where 119868 is an open interval For any 119905 isin 119868 thecurve 120574 is called spacelike null or timelike if ⟨(119905) (119905)⟩ gt 0⟨(119905) (119905)⟩ = 0 or ⟨(119905) (119905)⟩ lt 0 respectively We call 120574a nonnull curve if 120574 is a spacelike or timelike curve The arc-length of a nonnull curve 120574 measured from 120574(1199050)(1199050 isin 119868) is119904(119905) = int

119905

1199050(119905)119889119905

The parameter 119904 is determined such that 1205741015840(119904) = 1 forthe nonnull curve where 1205741015840(119904) = 119889120574119889119904 is the unit tangentvector of 120574 Some submanifolds in R4

2are as follows

The anti-de Sitter space is defined by

1198673

1= x isin R

4

2| ⟨x x⟩ = minus1 (3)

and the lightlike cone by

LC119901 = x isin R4

2| ⟨x minus p x minus p⟩ = 0 (4)

We also define the one-parameter anti-de Sitter space by

1198673

1(minussinh2120579) = x isin R

4

2| ⟨x x⟩ = minussinh2120579 (5)

Let 120574 119868 rarr 1198673

1be a spacelike regular curve that is 120574

satisfies ⟨(119905) (119905)⟩ gt 0 119905 isin 119868 Since the curve 120574 is spacelikewe can reparametrize it by the arc-length 119904 Then we havethe tangent vector t(119904) = 120574

1015840(119904) obviously t(119904) = 1 When

⟨t1015840(119904) t1015840(119904)⟩ = minus1 we define a unit vector

n (119904) = t1015840 (119904) minus 120574 (119904)1003817100381710038171003817t1015840 (119904) minus 120574 (119904)

1003817100381710038171003817

(6)

Let e(119904) = 120574(119904)andt(119904)andn(119904) thenwehave a pseudoorthonormalframe 120574(119904) t(119904)n(119904) e(119904) of R4

2along 120574 By direct calculat-

ing the following Frenet-Serret type is displayed under theassumption that ⟨t1015840(119904) t1015840(119904)⟩ = minus1

1205741015840(119904) = t (119904)

t1015840 (119904) = 120574 (119904) + 120581119892 (119904)n (119904)

n1015840 (119904) = minus120575120581119892 (119904) t (119904) + 120575120591119892 (119904) e (119904)

e1015840 (119904) = 120575120591119892 (119904)n (119904)

(7)

Journal of Function Spaces 3

Here 120581119892(119904) = t1015840(119904) minus 120574(119904) is the geodesic curvature

120591119892 (119904) = minus120581minus2

119892(119904) det (120574 (119904) 1205741015840 (119904) 12057410158401015840 (119904) 120574101584010158401015840 (119904)) (8)

is the geodesic torsion and 120575 = sign(119899(119904)) If ⟨t1015840(119904) t1015840(119904)⟩ = minus1we can obtain 120581119892(119904) = 0 it means that 120574(119904) is a geodesic curvein 1198673

1 We consider ⟨t1015840(119904) t1015840(119904)⟩ = minus1 (ie 120581119892(119904) = 0) in the

following sectionsLet 120574 119868 rarr 119867

3

1be a unit speed spacelike curve we write

120574120579= sinh 120579120574 and define 119871plusmn

120579 119868 timesR rarr 119867

3

1(minussinh2120579) by

119871plusmn

120579(119904 120583) = sinh 120579120574 (119904) + 120583 (n (119904) plusmn e (119904))

= 120574120579+ 120583 (n (119904) plusmn e (119904)) 120579 isin (0 +infin)

(9)

We call 119871plusmn120579(119904 120583) the one-parameter null hypersurfaces associ-

ated with 120574(119904) We also define the following model surfaceFor any k0 isin 119867

3

1(minussinh2120579)

LCV0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (10)

On the other hand let 119865 1198673

1rarr R be a submersion and

let 120574 119868 rarr 1198673

1be a spacelike curve We say that 120574 and 119865minus1(0)

have 119896-point contact at 119905 = 1199050 if the function 119892(119905) = 119865 ∘ 120574(119905)

satisfies 119892(1199050) = 1198921015840(1199050) = sdot sdot sdot = 119892

(119896minus1)(1199050) = 0 119892(119896)(1199050) = 0

We also have that 120574 and 119865minus1(0) have at least 119896-point contact

at 119905 = 1199050 if the function119892(119905) = 119865∘120574(119905) satisfies119892(1199050) = 1198921015840(1199050) =

sdot sdot sdot = 119892(119896minus1)

(1199050) = 0

3 One-Parameter HeightFunctions and the Singularities ofOne-Parameter Null Surfaces

In this section we discuss a kind of Lorentzian invariantfunctions on a spacelike curve in119867

3

1 It is useful to study the

null hypersurfaces of the spacelike curve Let 120574 119868 rarr 1198673

1be

a unit spacelike curve We now define a function

119867 119868 times 1198673

1(minussinh2120579) times (0 +infin) 997888rarr R (11)

by 119867(119904 k 120579) = ⟨120574120579(119904) k⟩ + sinh2120579 we call 119867 one-parameter

height function on the spacelike curve 120574 We denote thatℎV120579(119904) = 119867(119904 k 120579) (k 120579) isin 119867

3

1(minussinh2120579) times (0 +infin) Then

we have the following proposition

Proposition 1 Let 120574 119868 rarr 1198673

1be a unit spacelike curve and

(k 120579) isin 1198673

1(minussinh2120579) times (0 +infin) Then one has the following

(1) ℎV120579(119904) = ℎ1015840

V120579(119904) = 0 if and only if there exists 120583 isin R

such that k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904))

(2) ℎV120579(119904) = ℎ1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = 0 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (12)

(3) ℎV120579(119904) = ℎ1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = ℎ101584010158401015840

V120579(119904) = 0 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (13)

and 120590 = 0(4) ℎV120579(119904) = ℎ

1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = ℎ101584010158401015840

V120579(119904) = ℎ(4)

V120579(119904) = 0 if andonly if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (14)

and 120590 = 1205901015840= 0

Proof (1) Since k isin 1198673

1(minussinh2120579) we can find that 120578 120572 120583 120573 isin

R with minus1205782 + 1205722+ 1205751205832minus 1205751205732= minussinh2120579 such that k = 120578120574(119904) +

120572t(119904) + 120583n(119904) + 120573e(119904) Because

ℎV120579 (119904) = ⟨sinh 120579120574 (119904) k⟩ + sinh2120579 = 0 (15)

we can get 120578 = sinh 120579 when ℎ1015840

V120579(119904) = 0 it means that 120572 = 0

and 120573 = plusmn120583 Therefore k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904)) theconverse direction also holds

(2) By (1) an easy computation shows that

sinh 120579 ⟨t1015840 (119904) k⟩ = sinh 120579 ⟨120574 (119904) + 120581119892 (119904)n (119904) k⟩

= sinh 120579 ⟨120574 (119904) + 120581119892 (119904)n (119904) sinh 120579120574 (119904)

+ 120583 (n (119904) plusmn e (119904))⟩ = 0

(16)

we get 120583 = sinh 120579120575120581119892(119904) therefore

k = sinh 120579120574 (119904) + (sinh 120579

(120575120581119892 (119904))

) (n (119904) plusmn e (119904)) (17)

(3) Under the assumption that

ℎV120579 (119904) = ℎ1015840

V120579 (119904) = ℎ10158401015840

V120579 (119904) = 0

ℎ101584010158401015840

V120579 (119904)

sinh 120579= ⟨(1 minus 120575120581

2

119892(119904)) t (119904) + 120581

1015840

119892(119904)n (119904)

+ 120575120581119892 (119904) 120591119892 (119904) k⟩

(18)

we can get 120590(119904) = 1205811015840

119892(119904)∓120581119892(119904)120591119892(119904) = 0 assertion (3) follows

(4) Based on the assumption that

ℎV120579 (119904) = ℎ1015840

V120579 (119904) = ℎ10158401015840

V120579 (119904) = ℎ101584010158401015840

V120579 (119904) = 0 (19)

the relation

ℎ(4)

V120579 (119904)

sinh 120579= ⟨(1 minus 120575120581

2

119892(119904)) t (119904) + 120581

1015840

119892(119904)n (119904)

minus 120575120581119892 (119904) 120591119892 (119904) e (119904) k⟩1015840

(20)

follows the fact that ℎ(4)V120579(119904) = 0 is equivalent to

12058110158401015840

119892(119904) ∓ 120581

1015840

119892(119904) 120591119892 (119904) ∓ 120581119892 (119904) 120591

1015840

119892(s) = 0 (21)

so 1205901015840 = 0 This proves assertion (4)

4 Journal of Function Spaces

Now we do research on some properties of one-parameter null hypersurfaces of the spacelike curve in 119867

3

1

As we can know the functions 120581119892(119904) 120591119892(119904) and 120590(119904) haveparticular meanings Here we consider the case when theone-parameter null hypersurfaces have the most degeneratesingularities We have the following proposition

Proposition 2 Let 120574 119868 rarr 1198673

1be a unit spacelike curveThen

one has the following

(1) The set 119871plusmn120579(119904 120583) | 120583 = sinh 120579120575120581119892(119904) is the singulari-

ties of one-parameter null hypersurfaces 119871plusmn120579(119904 120583)

(2) If k0 = 119871plusmn

120579(119904 sinh 120579120575120581119892(119904)) is a constant vector one

has 120574120579(119904) isin 119871119862V0(120579) for any 119904 isin 119868 at the same time

120590(119904) = 0

Proof By calculations we have

120597119871plusmn

120579(119904 120583)

120597120579= minus cosh 120579120574 (119904) (22)

120597119871plusmn

120579(119904 120583)

120597120583= e (119904) plusmn n (119904) (23)

120597119871plusmn

120579(119904 120583)

120597119904

= sinh 120579t (119904)

+ 120583 (minus120575120581119892 (119904) t (119904) + 120575120591119892 (119904) e (119904) plusmn 120575120591119892 (119904)n (119904))

= (sinh 120579 minus 120575120583120581119892 (119904)) t (119904)

+ 120575120583120591119892 (119904) (e (119904) plusmn n (119904))

(24)

(1) If the above three vectors are linearly dependent wecan get the singularities of 119871plusmn

120579(119904 120583) if and only if sinh 120579 minus

120575120583120581119892(119904) = 0 Therefore assertion (1) holds(2) For any fixed 120579 isin (0 +infin) if

119891 (119904) = sinh 120579120574 (119904) + 120583 (119904) (n (119904) plusmn e (119904)) (25)

is a constant then

119889119891

119889119904= (sinh 120579 minus 120575120583 (119904) 120581119892 (119904)) t (119904)

+ (1205831015840(119904) minus 120575120583 (119904) 120591119892 (119904)) (n (119904) plusmn e (119904)) = 0

(26)

Since

120583 (119904) =sinh 120579120575120581119892 (119904)

1205831015840(119904) minus 120575120583 (119904) 120591119892 (119904) = 0 (27)

then

120590 = 1205811015840

119892(119904) ∓ 120581119892 (119904) 120591119892 (119904) = 0 (28)

We have

⟨120574120579(119904) k0⟩

= ⟨120574120579(119904) sinh 120579120574 (119904) + sinh 120579

120575120581119892 (119904)(n (119904) plusmn e (119904))⟩

= minussinh2120579

(29)

This completes the proof

4 Unfoldings of One-ParameterHeight Functions

In this section we classify singularities of the one-parameternull hypersurfaces along 120574 as an application of the unfoldingtheory of functions

Let 119865 (R times R119903 (1199040 x0)) rarr R be a function germ119891(119904) = 1198651199090

(119904 x0) We call 119865 an 119903-parameter unfolding of 119891If 119891(119901)(1199040) = 0 for all 1 le 119901 le 119896 and 119891

(119896+1)(1199040) = 0 we say

that 119891 has 119860119896-singularity at 1199040 We also say that 119891 has 119860ge119896-singularity at 1199040 if 119891

(119901)(1199040) = 0 for all 1 le 119901 le 119896 Let 119865 be an

119903-parameter unfolding of 119891 and 119891 has 119860119896-singularity (119896 ge 1)

at 1199040 we define the (119896 minus 1)-jet of the partial derivative 120597119865120597119909119894at 1199040 as

119895(119896minus1) 120597119865

120597119909119894

(119904 x0) (1199040) =119896+1

sum

119895=1

120572119895119894 (119904 minus 1199040)119895

(119894 = 1 119903)

(30)

If the rank of 119896 times 119903 matrix (1205720119894 120572119895119894) is 119896 (119896 le 119903) then 119865 iscalled a versal unfolding of 119891 where 1205720119894 = (120597119865120597119909119894)(1199040 x0)The discriminant set of 119865 is defined by

119863119865 = x isin R119903| exist119904 isin R 119865 (119904 x) = 120597119865

120597119904(119904 x) = 0 (31)

There has been the following famous result [20]

Theorem 3 Let 119865 (R timesR119903 (1199040 x0)) rarr R be an 119903-parameterunfolding of 119891(119904) which has 119860119896-singularity at 1199040 suppose that119865 is a V119890119903119904119886119897 119906119899119891119900119897119889119894119899119892 of 119891 Then one has the following

(a) If 119896 = 1 then119863119865 is locally diffeomorphic to 0 timesR119903minus1(b) If 119896 = 2 then119863119865 is locally diffeomorphic to 119862 timesR119903minus2(c) If 119896 = 3 then119863119865 is locally diffeomorphic to 119878119882timesR119903minus3

By Proposition 1 the discriminant set of the timelikeheight function119867(119904 k 120579) is given by

119863119867 = sinh 120579120574 (119904) + 120583 (n (119904) plusmn e (119904)) | 119904 120583 isin 119868 120579

isin (0 +infin)

(32)

Proposition 4 Let119867(119904 k 120579) be a one-parameter height func-tion on the spacelike curve 120574 k isin 119863119867 If ℎV has 119860119896-singularityat s (119896 = 1 2 3 4) then119867 is a versal unfolding of ℎV

Journal of Function Spaces 5

Proof Let 120574(119904) = (1199091(119904) 1199092(119904) 1199093(119904) 1199094(119904)) isin 1198673

1and k =

(V1 V2 V3 V4) isin 1198673

1(minussinh2120579)

Then119867(119904 k 120579)

= sinh 120579 (minus1199091V1 minus 1199092V2 + 1199093V3 + 1199094V4 + sinh 120579)

120579 isin (0 +infin)

(33)

Let V1 = plusmnradicminusV22+ V23+ V24+ sinh2120579 so

120597119867

120597V2(119904 k) = sinh 120579 (minus1199092 minus

V2V11199091)

1205972119867

120597119904120597V2(119904 k) = sinh 120579 (minus1199091015840

2minusV2V11199091015840

1)

1205973119867

1205972119904120597V2(119904 k) = sinh 120579 (minus11990910158401015840

2minusV2V111990910158401015840

1)

120597119867

120597V119894(119904 k) = sinh 120579 (119909119894 +

V119894V11199091)

1205972119867

120597119904120597V119894(119904 k) = sinh 120579 (1199091015840

119894+

V119894V11199091015840

1)

1205973119867

1205972119904120597V119894(119904 k) = sinh 120579 (11990910158401015840

119894+

V119894V111990910158401015840

1) (119894 = 3 4)

120597119867

120597120579(119904 k) = 120593120579 (119904)

1205972119867

120597119904120597120579(119904 k) = 120593

1015840

120579(119904)

1205973119867

1205972119904120597120579(119904 k) = 120593

10158401015840

120579(119904)

(34)

The 2-119895119890119905 of (120597119867120597V2)(119904 k) at 1199040 is given by

sinh 120579 (minus11990910158402119904 minus

1

211990910158401015840

21199042) +

V2V1

(minus1199091015840

1119904 minus

1

211990910158401015840

11199042) (35)

the 2-119895119890119905 of (120597119867120597V119894)(119904 k) at 1199040 is given by

sinh 120579 (1199091015840119894119904 +

1

211990910158401015840

1198941199042) +

V119894V1

(1199091015840

1119904 +

1

211990910158401015840

11199042)

(119894 = 3 4)

(36)

(1) By Proposition 1 ℎV has the 1198601-singularity at 1199040 if andonly if k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904)) Since the curve 120574(119904) isregular the rank of

(minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579) (37)

is 1(2) We can get that ℎ has the 1198602-singularity at 1199040 if and

only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (38)

and 120590 = 0 When ℎ has the 119860ge2-singularity at 1199040 we requirethe 2 times 4matrix

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091 1199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

) (39)

to have rank 2 which follows from the proof of the next case(3) It also follows from Proposition 1 that ℎ has the 119860ge3-

singularity at 1199040 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (40)

and 120590 = 0 1205901015840 = 0We require the 3 times 4matrix

(

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

12059310158401015840

120579

sinh 120579

)

)

(41)

to have rank 3Let 3 times 3matrix

119860 = (

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

) (42)

We denote

119860 (119894 119895 119896) = det(

119909119894 119909119895 119909119896

1199091015840

1198941199091015840

1198951199091015840

119896

11990910158401015840

11989411990910158401015840

11989511990910158401015840

119896

) (43)

thendet119860

= minus119860 (2 3 4) minusV2V1119860 (1 3 4) minus

V3V1119860 (2 1 4)

minusV4V1119860 (2 3 1)

= plusmn1

V1⟨(V1 V2 V3 V4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minuse1 minuse2 e3 e41199091 1199092 1199093 1199094

1199091015840

11199091015840

21199091015840

31199091015840

4

11990910158401015840

111990910158401015840

211990910158401015840

311990910158401015840

4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟩

= plusmn1

V1⟨k 120574 (119904) and 1205741015840 (119904) and 12057410158401015840 (119904)⟩

(44)

Since k isin 119863119867 is a singular point then

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (45)

6 Journal of Function Spaces

and we have

120574 (119904) and 1205741015840(119904) and 120574

10158401015840(119904) = 120574 (119904) and 120574

1015840(119904)

and (120574 (119904) + 120581119892 (119904)n (119904))

= 120581119892 (119904) (120574 (119904) and t (119904) and n (119904))

= 120581119892 (119904) e (119904)

(46)

Therefore

det119860 = plusmn1

V1⟨sinh 120579120574 (119904)

+sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) 120581119892 (119904) e (119904)⟩ = plusmnsinh 120579V1

= 0

(47)

In summary119867 is a versal unfolding of ℎV this completes theproof

5 Main Result

The main result in this paper is in this section We nowconsider the following conditions

(A1) The number of points 119901 of 120574120579(119904) where LCV0(120579) at 119901

have four-point contact with the curve 120574120579 is finite

(A2) There is no point 119901 of 120574120579(119904) where LCV0(120579) at 119901 have

five-point contact or greater with the curve 120574120579

Our main result is as follows

Theorem 5 Let 120574 119868 rarr 1198673

1be a unit regular spacelike curve

k0 = 119871plusmn

120579(1199040 1205830) and

119871119862V0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (48)

Then one has the following(1) 119871119862V0(120579) and 120574120579 have at least 2-point contact at 1199040(2) 119871119862V0(120579) and 120574120579 have 3-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (49)

and 120590(1199040) = 1205811015840

119892(1199040) ∓ 120581119892(1199040)120591119892(1199040) = 0 Under this condition

the germ of image 119871plusmn120579at (1199040 1205830) is diffeomorphic to the cuspidal

edge 119862 timesR (Figure 1)(3) 119871119862V0(120579) and 120574120579 have 4-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (50)

120590(1199040) = 0 and 1205901015840(1199040) = 0 Under this condition the germ of

image 119871plusmn

120579at (1199040 1205830) is diffeomorphic to the swallowtail 119878119882

(Figure 2)

Here 119862 = (1199091 1199092) | 1199092

1= 1199093

2 and 119878119882 = (1199091 1199092 1199093) |

1199091 = 31199064+ 1199062V 1199092 = 4119906

3+ 2119906V 1199093 = V

Figure 1 Cuspidal edge

Figure 2 Swallowtail

Proof Let 120574 119868 rarr 1198673

1be a spacelike regular curve and

⟨t1015840(119904) t1015840(119904)⟩ = minus1 As k0 = 119871plusmn

1205790(1199040 1205830) we give a function

119867 1198673

1(minussinh21205790) 997888rarr R (51)

by119867(119906) = ⟨119906 V0⟩ + sinh21205790 then we assume that

ℎV0 1205790 (119904) = 119867 (1205741205790(119904)) (52)

Because 119867minus1(0) = LCV0(1205790) and 0 is a regular value of 119867

1205741205790

and LCV0(1205790) have (119896 + 1)-point contact at 1199040 if andonly if ℎV0 1205790(119904) has the 119860119896-singularity at 1199040 By Proposition 1Theorem 3 and Proposition 4 we get the results

6 Example

In this section we construct the one-parameter null hyper-surfaces associated with a spacelike curve and two sections of

Journal of Function Spaces 7

the one-parameter null hypersurfaces The two sections arenull surfaces and they are also the wavefronts of spacelikecurves By calculatingwe get the singularities of null surfacesIt is useful to understand the one-parameter null hypersur-faces

Let 120574(119904) = (radic2 cosh(2119904) radic3 cosh(radic7119904) + 2 sinh(radic7119904)radic2 sinh(2119904)radic3 sinh(radic7119904)+2 cosh(radic7119904)) be a spacelike curvein1198673

1 where 119904 is the arc-length parameter Then

t (119904) = (2radic2 sinh (2119904) radic21 sinh (radic7119904) + 2radic7

sdot cosh (radic7119904) 2radic2 cosh (2119904) radic21 cosh (radic7119904) + 2radic7

sdot sinh (radic7119904))

n (119904) = (cosh (2119904) radic6 cosh (radic7119904) + 2radic2

sdot sinh (radic7119904) sinh (2119904) radic6 sinh (radic7119904) + 2radic2

sdot cosh (radic7119904))

(53)

We have

e (119904) = 120574 (119904) and t (119904) and n (119904)

= (radic7 sinh (2119904) 2radic6 sinh (radic7119904)

+ 4radic2 cosh (radic7119904) radic7 cosh (2119904) 2radic6 cosh (radic7119904)

+ 4radic2 sinh (radic7119904))

(54)

119871plusmn

120579(119904 120583) = sinh 120579120574(119904)+120583(n(119904)plusmne(119904)) be the one-parameter

null hypersurfaces of 120574(119904) At the moment we can calculatethat 120581119892(119904) = 3radic2 and 120591119892(119904) = 2radic7 two sections of 119871plusmn

120579(119904 120583)

with 1205791 = arcsinh(12) 1205792 = arcsinh(radic22) are as follows

119871+

1205791(119904 120583) =

1

2120574 (119904) + 120583 (n (119904) + e (119904)) = ((

radic2

2+ 120583)

sdot cosh (2119904) + radic7120583 sinh (2119904) (radic3

2+ radic6120583 + 4radic2120583)

sdot cosh (radic7119904) + (1 + 2radic2120583 + 2radic6120583)

sdot sinh (radic7119904) radic7120583 cosh (2119904) + (radic2

2+ 120583)

sdot sinh (2119904) (1 + 2radic2120583 + 2radic6120583)

sdot cosh (radic7119904) + (radic3

2+ radic6120583 + 4radic2120583)

sdot sinh (radic7119904))

Figure 3 Projection of 119871+1205791(119904 120583) on 119909211990931199094-space

Figure 4 Projection of 119871minus1205792(119904 120583) on 119909111990921199093-space

119871minus

1205792(119904 120583) =

radic2

2120574 (119904) + 120583 (n (119904) minus e (119904)) = ((1 + 120583)

sdot cosh (2119904) minus radic7120583 sinh (2119904) (radic6

2+ radic6120583 minus 4radic2120583)

sdot cosh (radic7119904) + (radic2 + 2radic2120583 minus 2radic6120583) sinh (radic7119904)

minus radic7120583 cosh (2119904) + (1 + 120583)

sdot sinh (2119904) (radic2 + 2radic2120583 minus 2radic6120583) cosh (radic7119904)

+ (radic6

2+ radic6120583 minus 4radic2120583) sinh (radic7119904))

(55)

The pictures of 1205791-null hypersurface 119871+

1205791(119904 120583) and its

singularities 119871+1205791(119904 radic212) can be seen in Figure 3 And the

pictures of 1205792-null hypersurface 119871minus

1205792(119904 120583) and its singularities

119871minus

1205792(119904 16) can be seen in Figure 4

Competing Interests

The authors declare that they have no competing interests

8 Journal of Function Spaces

Acknowledgments

This work was partially supported by NSF of China (nos11271063 and 11501051) and NCET of China (no 05-0319)The first author was partially supported by the Project of Sci-ence and Technology of Heilongjiang Provincial EducationDepartment of China (no UNPYSCT-2015103)

References

[1] AMahmut E Soley and TMurat ldquoBeltrami-Meusnier formu-las of generalized semi ruled surfaces in semi Euclidean spacerdquoKuwait Journal of Science vol 41 no 2 pp 65ndash83 2014

[2] E Tulay andGM Ali ldquoSome characterizations of quaternionicrectifying curves in the semi-Euclidean space E2

4rdquo HonamMathematical Journal vol 36 no 1 pp 67ndash83 2014

[3] H Liu ldquoCurves in affine and semi-Euclidean spacesrdquo Results inMathematics vol 65 no 1-2 pp 235ndash249 2014

[4] E Soley and T Murat ldquoTimelike Bertrand curves in semi-Euclidean spacerdquo International Journal of Mathematics andStatistics vol 14 no 2 pp 78ndash89 2013

[5] M Sakaki ldquoBi-null Cartan curves in semi-Euclidean spaces ofindex 2rdquo Beitrage zur Algebra und Geometrie Contributions toAlgebra and Geometry vol 53 no 2 pp 421ndash436 2012

[6] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[7] G Clement ldquoBlack holes with a null Killing vector in three-dimensional massive gravityrdquo Classical and Quantum Gravityvol 26 no 16 Article ID 165002 11 pages 2009

[8] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[9] S W Hawking ldquoBlack holes in general relativityrdquo Communica-tions in Mathematical Physics vol 25 pp 152ndash166 1972

[10] M Korzynski J Lewandowski and T Pawlowski ldquoMechanicsof multidimensional isolated horizonsrdquo Classical and QuantumGravity vol 22 no 11 pp 2001ndash2016 2005

[11] V Moncrief and J Isenberg ldquoSymmetries of cosmologicalCauchy horizonsrdquo Communications in Mathematical Physicsvol 89 no 3 pp 387ndash413 1983

[12] L Kong and D Pei ldquoOn spacelike curves in hyperbolic spacetimes sphererdquo International Journal of Geometric Methods inModern Physics vol 11 no 3 2014

[13] I R Porteous ldquoThe normal singularities of a submanifoldrdquoJournal of Differential Geometry vol 5 pp 543ndash564 1971

[14] L Chen S Izumiya and D Pei ldquoTimelike hypersurfaces inthe anti-de Sitter space from a contact viewpointrdquo Journal ofMathematical Sciences vol 199 no 6 pp 629ndash645 2014

[15] J Sun and D Pei ldquoNull surfaces of null curves on 3-null conerdquoPhysics Letters A vol 378 no 14-15 pp 1010ndash1016 2014

[16] J Sun and D Pei ldquoNull Cartan Bertrand curves of AW(k)-typein Minkowski 4-spacerdquo Physics Letters A vol 376 no 33 pp2230ndash2233 2012

[17] T Fusho and S Izumiya ldquoLightlike surfaces of spacelike curvesin de Sitter 3-spacerdquo Journal of Geometry vol 88 no 1-2 pp19ndash29 2008

[18] L Chen Q Han D Pei and W Sun ldquoThe singularities ofnull surfaces in anti de Sitter 3-spacerdquo Journal of MathematicalAnalysis and Applications vol 366 no 1 pp 256ndash265 2010

[19] Y Wang and D Pei ldquoSingularities for normal hypersurfacesof de Sitter timelike curves in Minkowski 4-spacerdquo Journal ofSingularities vol 12 pp 207ndash214 2015

[20] J W Bruce and P J Giblin Curves and Singularities CambridgeUniversity Press 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

Here 120581119892(119904) = t1015840(119904) minus 120574(119904) is the geodesic curvature

120591119892 (119904) = minus120581minus2

119892(119904) det (120574 (119904) 1205741015840 (119904) 12057410158401015840 (119904) 120574101584010158401015840 (119904)) (8)

is the geodesic torsion and 120575 = sign(119899(119904)) If ⟨t1015840(119904) t1015840(119904)⟩ = minus1we can obtain 120581119892(119904) = 0 it means that 120574(119904) is a geodesic curvein 1198673

1 We consider ⟨t1015840(119904) t1015840(119904)⟩ = minus1 (ie 120581119892(119904) = 0) in the

following sectionsLet 120574 119868 rarr 119867

3

1be a unit speed spacelike curve we write

120574120579= sinh 120579120574 and define 119871plusmn

120579 119868 timesR rarr 119867

3

1(minussinh2120579) by

119871plusmn

120579(119904 120583) = sinh 120579120574 (119904) + 120583 (n (119904) plusmn e (119904))

= 120574120579+ 120583 (n (119904) plusmn e (119904)) 120579 isin (0 +infin)

(9)

We call 119871plusmn120579(119904 120583) the one-parameter null hypersurfaces associ-

ated with 120574(119904) We also define the following model surfaceFor any k0 isin 119867

3

1(minussinh2120579)

LCV0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (10)

On the other hand let 119865 1198673

1rarr R be a submersion and

let 120574 119868 rarr 1198673

1be a spacelike curve We say that 120574 and 119865minus1(0)

have 119896-point contact at 119905 = 1199050 if the function 119892(119905) = 119865 ∘ 120574(119905)

satisfies 119892(1199050) = 1198921015840(1199050) = sdot sdot sdot = 119892

(119896minus1)(1199050) = 0 119892(119896)(1199050) = 0

We also have that 120574 and 119865minus1(0) have at least 119896-point contact

at 119905 = 1199050 if the function119892(119905) = 119865∘120574(119905) satisfies119892(1199050) = 1198921015840(1199050) =

sdot sdot sdot = 119892(119896minus1)

(1199050) = 0

3 One-Parameter HeightFunctions and the Singularities ofOne-Parameter Null Surfaces

In this section we discuss a kind of Lorentzian invariantfunctions on a spacelike curve in119867

3

1 It is useful to study the

null hypersurfaces of the spacelike curve Let 120574 119868 rarr 1198673

1be

a unit spacelike curve We now define a function

119867 119868 times 1198673

1(minussinh2120579) times (0 +infin) 997888rarr R (11)

by 119867(119904 k 120579) = ⟨120574120579(119904) k⟩ + sinh2120579 we call 119867 one-parameter

height function on the spacelike curve 120574 We denote thatℎV120579(119904) = 119867(119904 k 120579) (k 120579) isin 119867

3

1(minussinh2120579) times (0 +infin) Then

we have the following proposition

Proposition 1 Let 120574 119868 rarr 1198673

1be a unit spacelike curve and

(k 120579) isin 1198673

1(minussinh2120579) times (0 +infin) Then one has the following

(1) ℎV120579(119904) = ℎ1015840

V120579(119904) = 0 if and only if there exists 120583 isin R

such that k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904))

(2) ℎV120579(119904) = ℎ1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = 0 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (12)

(3) ℎV120579(119904) = ℎ1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = ℎ101584010158401015840

V120579(119904) = 0 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (13)

and 120590 = 0(4) ℎV120579(119904) = ℎ

1015840

V120579(119904) = ℎ10158401015840

V120579(119904) = ℎ101584010158401015840

V120579(119904) = ℎ(4)

V120579(119904) = 0 if andonly if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (14)

and 120590 = 1205901015840= 0

Proof (1) Since k isin 1198673

1(minussinh2120579) we can find that 120578 120572 120583 120573 isin

R with minus1205782 + 1205722+ 1205751205832minus 1205751205732= minussinh2120579 such that k = 120578120574(119904) +

120572t(119904) + 120583n(119904) + 120573e(119904) Because

ℎV120579 (119904) = ⟨sinh 120579120574 (119904) k⟩ + sinh2120579 = 0 (15)

we can get 120578 = sinh 120579 when ℎ1015840

V120579(119904) = 0 it means that 120572 = 0

and 120573 = plusmn120583 Therefore k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904)) theconverse direction also holds

(2) By (1) an easy computation shows that

sinh 120579 ⟨t1015840 (119904) k⟩ = sinh 120579 ⟨120574 (119904) + 120581119892 (119904)n (119904) k⟩

= sinh 120579 ⟨120574 (119904) + 120581119892 (119904)n (119904) sinh 120579120574 (119904)

+ 120583 (n (119904) plusmn e (119904))⟩ = 0

(16)

we get 120583 = sinh 120579120575120581119892(119904) therefore

k = sinh 120579120574 (119904) + (sinh 120579

(120575120581119892 (119904))

) (n (119904) plusmn e (119904)) (17)

(3) Under the assumption that

ℎV120579 (119904) = ℎ1015840

V120579 (119904) = ℎ10158401015840

V120579 (119904) = 0

ℎ101584010158401015840

V120579 (119904)

sinh 120579= ⟨(1 minus 120575120581

2

119892(119904)) t (119904) + 120581

1015840

119892(119904)n (119904)

+ 120575120581119892 (119904) 120591119892 (119904) k⟩

(18)

we can get 120590(119904) = 1205811015840

119892(119904)∓120581119892(119904)120591119892(119904) = 0 assertion (3) follows

(4) Based on the assumption that

ℎV120579 (119904) = ℎ1015840

V120579 (119904) = ℎ10158401015840

V120579 (119904) = ℎ101584010158401015840

V120579 (119904) = 0 (19)

the relation

ℎ(4)

V120579 (119904)

sinh 120579= ⟨(1 minus 120575120581

2

119892(119904)) t (119904) + 120581

1015840

119892(119904)n (119904)

minus 120575120581119892 (119904) 120591119892 (119904) e (119904) k⟩1015840

(20)

follows the fact that ℎ(4)V120579(119904) = 0 is equivalent to

12058110158401015840

119892(119904) ∓ 120581

1015840

119892(119904) 120591119892 (119904) ∓ 120581119892 (119904) 120591

1015840

119892(s) = 0 (21)

so 1205901015840 = 0 This proves assertion (4)

4 Journal of Function Spaces

Now we do research on some properties of one-parameter null hypersurfaces of the spacelike curve in 119867

3

1

As we can know the functions 120581119892(119904) 120591119892(119904) and 120590(119904) haveparticular meanings Here we consider the case when theone-parameter null hypersurfaces have the most degeneratesingularities We have the following proposition

Proposition 2 Let 120574 119868 rarr 1198673

1be a unit spacelike curveThen

one has the following

(1) The set 119871plusmn120579(119904 120583) | 120583 = sinh 120579120575120581119892(119904) is the singulari-

ties of one-parameter null hypersurfaces 119871plusmn120579(119904 120583)

(2) If k0 = 119871plusmn

120579(119904 sinh 120579120575120581119892(119904)) is a constant vector one

has 120574120579(119904) isin 119871119862V0(120579) for any 119904 isin 119868 at the same time

120590(119904) = 0

Proof By calculations we have

120597119871plusmn

120579(119904 120583)

120597120579= minus cosh 120579120574 (119904) (22)

120597119871plusmn

120579(119904 120583)

120597120583= e (119904) plusmn n (119904) (23)

120597119871plusmn

120579(119904 120583)

120597119904

= sinh 120579t (119904)

+ 120583 (minus120575120581119892 (119904) t (119904) + 120575120591119892 (119904) e (119904) plusmn 120575120591119892 (119904)n (119904))

= (sinh 120579 minus 120575120583120581119892 (119904)) t (119904)

+ 120575120583120591119892 (119904) (e (119904) plusmn n (119904))

(24)

(1) If the above three vectors are linearly dependent wecan get the singularities of 119871plusmn

120579(119904 120583) if and only if sinh 120579 minus

120575120583120581119892(119904) = 0 Therefore assertion (1) holds(2) For any fixed 120579 isin (0 +infin) if

119891 (119904) = sinh 120579120574 (119904) + 120583 (119904) (n (119904) plusmn e (119904)) (25)

is a constant then

119889119891

119889119904= (sinh 120579 minus 120575120583 (119904) 120581119892 (119904)) t (119904)

+ (1205831015840(119904) minus 120575120583 (119904) 120591119892 (119904)) (n (119904) plusmn e (119904)) = 0

(26)

Since

120583 (119904) =sinh 120579120575120581119892 (119904)

1205831015840(119904) minus 120575120583 (119904) 120591119892 (119904) = 0 (27)

then

120590 = 1205811015840

119892(119904) ∓ 120581119892 (119904) 120591119892 (119904) = 0 (28)

We have

⟨120574120579(119904) k0⟩

= ⟨120574120579(119904) sinh 120579120574 (119904) + sinh 120579

120575120581119892 (119904)(n (119904) plusmn e (119904))⟩

= minussinh2120579

(29)

This completes the proof

4 Unfoldings of One-ParameterHeight Functions

In this section we classify singularities of the one-parameternull hypersurfaces along 120574 as an application of the unfoldingtheory of functions

Let 119865 (R times R119903 (1199040 x0)) rarr R be a function germ119891(119904) = 1198651199090

(119904 x0) We call 119865 an 119903-parameter unfolding of 119891If 119891(119901)(1199040) = 0 for all 1 le 119901 le 119896 and 119891

(119896+1)(1199040) = 0 we say

that 119891 has 119860119896-singularity at 1199040 We also say that 119891 has 119860ge119896-singularity at 1199040 if 119891

(119901)(1199040) = 0 for all 1 le 119901 le 119896 Let 119865 be an

119903-parameter unfolding of 119891 and 119891 has 119860119896-singularity (119896 ge 1)

at 1199040 we define the (119896 minus 1)-jet of the partial derivative 120597119865120597119909119894at 1199040 as

119895(119896minus1) 120597119865

120597119909119894

(119904 x0) (1199040) =119896+1

sum

119895=1

120572119895119894 (119904 minus 1199040)119895

(119894 = 1 119903)

(30)

If the rank of 119896 times 119903 matrix (1205720119894 120572119895119894) is 119896 (119896 le 119903) then 119865 iscalled a versal unfolding of 119891 where 1205720119894 = (120597119865120597119909119894)(1199040 x0)The discriminant set of 119865 is defined by

119863119865 = x isin R119903| exist119904 isin R 119865 (119904 x) = 120597119865

120597119904(119904 x) = 0 (31)

There has been the following famous result [20]

Theorem 3 Let 119865 (R timesR119903 (1199040 x0)) rarr R be an 119903-parameterunfolding of 119891(119904) which has 119860119896-singularity at 1199040 suppose that119865 is a V119890119903119904119886119897 119906119899119891119900119897119889119894119899119892 of 119891 Then one has the following

(a) If 119896 = 1 then119863119865 is locally diffeomorphic to 0 timesR119903minus1(b) If 119896 = 2 then119863119865 is locally diffeomorphic to 119862 timesR119903minus2(c) If 119896 = 3 then119863119865 is locally diffeomorphic to 119878119882timesR119903minus3

By Proposition 1 the discriminant set of the timelikeheight function119867(119904 k 120579) is given by

119863119867 = sinh 120579120574 (119904) + 120583 (n (119904) plusmn e (119904)) | 119904 120583 isin 119868 120579

isin (0 +infin)

(32)

Proposition 4 Let119867(119904 k 120579) be a one-parameter height func-tion on the spacelike curve 120574 k isin 119863119867 If ℎV has 119860119896-singularityat s (119896 = 1 2 3 4) then119867 is a versal unfolding of ℎV

Journal of Function Spaces 5

Proof Let 120574(119904) = (1199091(119904) 1199092(119904) 1199093(119904) 1199094(119904)) isin 1198673

1and k =

(V1 V2 V3 V4) isin 1198673

1(minussinh2120579)

Then119867(119904 k 120579)

= sinh 120579 (minus1199091V1 minus 1199092V2 + 1199093V3 + 1199094V4 + sinh 120579)

120579 isin (0 +infin)

(33)

Let V1 = plusmnradicminusV22+ V23+ V24+ sinh2120579 so

120597119867

120597V2(119904 k) = sinh 120579 (minus1199092 minus

V2V11199091)

1205972119867

120597119904120597V2(119904 k) = sinh 120579 (minus1199091015840

2minusV2V11199091015840

1)

1205973119867

1205972119904120597V2(119904 k) = sinh 120579 (minus11990910158401015840

2minusV2V111990910158401015840

1)

120597119867

120597V119894(119904 k) = sinh 120579 (119909119894 +

V119894V11199091)

1205972119867

120597119904120597V119894(119904 k) = sinh 120579 (1199091015840

119894+

V119894V11199091015840

1)

1205973119867

1205972119904120597V119894(119904 k) = sinh 120579 (11990910158401015840

119894+

V119894V111990910158401015840

1) (119894 = 3 4)

120597119867

120597120579(119904 k) = 120593120579 (119904)

1205972119867

120597119904120597120579(119904 k) = 120593

1015840

120579(119904)

1205973119867

1205972119904120597120579(119904 k) = 120593

10158401015840

120579(119904)

(34)

The 2-119895119890119905 of (120597119867120597V2)(119904 k) at 1199040 is given by

sinh 120579 (minus11990910158402119904 minus

1

211990910158401015840

21199042) +

V2V1

(minus1199091015840

1119904 minus

1

211990910158401015840

11199042) (35)

the 2-119895119890119905 of (120597119867120597V119894)(119904 k) at 1199040 is given by

sinh 120579 (1199091015840119894119904 +

1

211990910158401015840

1198941199042) +

V119894V1

(1199091015840

1119904 +

1

211990910158401015840

11199042)

(119894 = 3 4)

(36)

(1) By Proposition 1 ℎV has the 1198601-singularity at 1199040 if andonly if k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904)) Since the curve 120574(119904) isregular the rank of

(minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579) (37)

is 1(2) We can get that ℎ has the 1198602-singularity at 1199040 if and

only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (38)

and 120590 = 0 When ℎ has the 119860ge2-singularity at 1199040 we requirethe 2 times 4matrix

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091 1199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

) (39)

to have rank 2 which follows from the proof of the next case(3) It also follows from Proposition 1 that ℎ has the 119860ge3-

singularity at 1199040 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (40)

and 120590 = 0 1205901015840 = 0We require the 3 times 4matrix

(

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

12059310158401015840

120579

sinh 120579

)

)

(41)

to have rank 3Let 3 times 3matrix

119860 = (

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

) (42)

We denote

119860 (119894 119895 119896) = det(

119909119894 119909119895 119909119896

1199091015840

1198941199091015840

1198951199091015840

119896

11990910158401015840

11989411990910158401015840

11989511990910158401015840

119896

) (43)

thendet119860

= minus119860 (2 3 4) minusV2V1119860 (1 3 4) minus

V3V1119860 (2 1 4)

minusV4V1119860 (2 3 1)

= plusmn1

V1⟨(V1 V2 V3 V4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minuse1 minuse2 e3 e41199091 1199092 1199093 1199094

1199091015840

11199091015840

21199091015840

31199091015840

4

11990910158401015840

111990910158401015840

211990910158401015840

311990910158401015840

4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟩

= plusmn1

V1⟨k 120574 (119904) and 1205741015840 (119904) and 12057410158401015840 (119904)⟩

(44)

Since k isin 119863119867 is a singular point then

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (45)

6 Journal of Function Spaces

and we have

120574 (119904) and 1205741015840(119904) and 120574

10158401015840(119904) = 120574 (119904) and 120574

1015840(119904)

and (120574 (119904) + 120581119892 (119904)n (119904))

= 120581119892 (119904) (120574 (119904) and t (119904) and n (119904))

= 120581119892 (119904) e (119904)

(46)

Therefore

det119860 = plusmn1

V1⟨sinh 120579120574 (119904)

+sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) 120581119892 (119904) e (119904)⟩ = plusmnsinh 120579V1

= 0

(47)

In summary119867 is a versal unfolding of ℎV this completes theproof

5 Main Result

The main result in this paper is in this section We nowconsider the following conditions

(A1) The number of points 119901 of 120574120579(119904) where LCV0(120579) at 119901

have four-point contact with the curve 120574120579 is finite

(A2) There is no point 119901 of 120574120579(119904) where LCV0(120579) at 119901 have

five-point contact or greater with the curve 120574120579

Our main result is as follows

Theorem 5 Let 120574 119868 rarr 1198673

1be a unit regular spacelike curve

k0 = 119871plusmn

120579(1199040 1205830) and

119871119862V0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (48)

Then one has the following(1) 119871119862V0(120579) and 120574120579 have at least 2-point contact at 1199040(2) 119871119862V0(120579) and 120574120579 have 3-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (49)

and 120590(1199040) = 1205811015840

119892(1199040) ∓ 120581119892(1199040)120591119892(1199040) = 0 Under this condition

the germ of image 119871plusmn120579at (1199040 1205830) is diffeomorphic to the cuspidal

edge 119862 timesR (Figure 1)(3) 119871119862V0(120579) and 120574120579 have 4-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (50)

120590(1199040) = 0 and 1205901015840(1199040) = 0 Under this condition the germ of

image 119871plusmn

120579at (1199040 1205830) is diffeomorphic to the swallowtail 119878119882

(Figure 2)

Here 119862 = (1199091 1199092) | 1199092

1= 1199093

2 and 119878119882 = (1199091 1199092 1199093) |

1199091 = 31199064+ 1199062V 1199092 = 4119906

3+ 2119906V 1199093 = V

Figure 1 Cuspidal edge

Figure 2 Swallowtail

Proof Let 120574 119868 rarr 1198673

1be a spacelike regular curve and

⟨t1015840(119904) t1015840(119904)⟩ = minus1 As k0 = 119871plusmn

1205790(1199040 1205830) we give a function

119867 1198673

1(minussinh21205790) 997888rarr R (51)

by119867(119906) = ⟨119906 V0⟩ + sinh21205790 then we assume that

ℎV0 1205790 (119904) = 119867 (1205741205790(119904)) (52)

Because 119867minus1(0) = LCV0(1205790) and 0 is a regular value of 119867

1205741205790

and LCV0(1205790) have (119896 + 1)-point contact at 1199040 if andonly if ℎV0 1205790(119904) has the 119860119896-singularity at 1199040 By Proposition 1Theorem 3 and Proposition 4 we get the results

6 Example

In this section we construct the one-parameter null hyper-surfaces associated with a spacelike curve and two sections of

Journal of Function Spaces 7

the one-parameter null hypersurfaces The two sections arenull surfaces and they are also the wavefronts of spacelikecurves By calculatingwe get the singularities of null surfacesIt is useful to understand the one-parameter null hypersur-faces

Let 120574(119904) = (radic2 cosh(2119904) radic3 cosh(radic7119904) + 2 sinh(radic7119904)radic2 sinh(2119904)radic3 sinh(radic7119904)+2 cosh(radic7119904)) be a spacelike curvein1198673

1 where 119904 is the arc-length parameter Then

t (119904) = (2radic2 sinh (2119904) radic21 sinh (radic7119904) + 2radic7

sdot cosh (radic7119904) 2radic2 cosh (2119904) radic21 cosh (radic7119904) + 2radic7

sdot sinh (radic7119904))

n (119904) = (cosh (2119904) radic6 cosh (radic7119904) + 2radic2

sdot sinh (radic7119904) sinh (2119904) radic6 sinh (radic7119904) + 2radic2

sdot cosh (radic7119904))

(53)

We have

e (119904) = 120574 (119904) and t (119904) and n (119904)

= (radic7 sinh (2119904) 2radic6 sinh (radic7119904)

+ 4radic2 cosh (radic7119904) radic7 cosh (2119904) 2radic6 cosh (radic7119904)

+ 4radic2 sinh (radic7119904))

(54)

119871plusmn

120579(119904 120583) = sinh 120579120574(119904)+120583(n(119904)plusmne(119904)) be the one-parameter

null hypersurfaces of 120574(119904) At the moment we can calculatethat 120581119892(119904) = 3radic2 and 120591119892(119904) = 2radic7 two sections of 119871plusmn

120579(119904 120583)

with 1205791 = arcsinh(12) 1205792 = arcsinh(radic22) are as follows

119871+

1205791(119904 120583) =

1

2120574 (119904) + 120583 (n (119904) + e (119904)) = ((

radic2

2+ 120583)

sdot cosh (2119904) + radic7120583 sinh (2119904) (radic3

2+ radic6120583 + 4radic2120583)

sdot cosh (radic7119904) + (1 + 2radic2120583 + 2radic6120583)

sdot sinh (radic7119904) radic7120583 cosh (2119904) + (radic2

2+ 120583)

sdot sinh (2119904) (1 + 2radic2120583 + 2radic6120583)

sdot cosh (radic7119904) + (radic3

2+ radic6120583 + 4radic2120583)

sdot sinh (radic7119904))

Figure 3 Projection of 119871+1205791(119904 120583) on 119909211990931199094-space

Figure 4 Projection of 119871minus1205792(119904 120583) on 119909111990921199093-space

119871minus

1205792(119904 120583) =

radic2

2120574 (119904) + 120583 (n (119904) minus e (119904)) = ((1 + 120583)

sdot cosh (2119904) minus radic7120583 sinh (2119904) (radic6

2+ radic6120583 minus 4radic2120583)

sdot cosh (radic7119904) + (radic2 + 2radic2120583 minus 2radic6120583) sinh (radic7119904)

minus radic7120583 cosh (2119904) + (1 + 120583)

sdot sinh (2119904) (radic2 + 2radic2120583 minus 2radic6120583) cosh (radic7119904)

+ (radic6

2+ radic6120583 minus 4radic2120583) sinh (radic7119904))

(55)

The pictures of 1205791-null hypersurface 119871+

1205791(119904 120583) and its

singularities 119871+1205791(119904 radic212) can be seen in Figure 3 And the

pictures of 1205792-null hypersurface 119871minus

1205792(119904 120583) and its singularities

119871minus

1205792(119904 16) can be seen in Figure 4

Competing Interests

The authors declare that they have no competing interests

8 Journal of Function Spaces

Acknowledgments

This work was partially supported by NSF of China (nos11271063 and 11501051) and NCET of China (no 05-0319)The first author was partially supported by the Project of Sci-ence and Technology of Heilongjiang Provincial EducationDepartment of China (no UNPYSCT-2015103)

References

[1] AMahmut E Soley and TMurat ldquoBeltrami-Meusnier formu-las of generalized semi ruled surfaces in semi Euclidean spacerdquoKuwait Journal of Science vol 41 no 2 pp 65ndash83 2014

[2] E Tulay andGM Ali ldquoSome characterizations of quaternionicrectifying curves in the semi-Euclidean space E2

4rdquo HonamMathematical Journal vol 36 no 1 pp 67ndash83 2014

[3] H Liu ldquoCurves in affine and semi-Euclidean spacesrdquo Results inMathematics vol 65 no 1-2 pp 235ndash249 2014

[4] E Soley and T Murat ldquoTimelike Bertrand curves in semi-Euclidean spacerdquo International Journal of Mathematics andStatistics vol 14 no 2 pp 78ndash89 2013

[5] M Sakaki ldquoBi-null Cartan curves in semi-Euclidean spaces ofindex 2rdquo Beitrage zur Algebra und Geometrie Contributions toAlgebra and Geometry vol 53 no 2 pp 421ndash436 2012

[6] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[7] G Clement ldquoBlack holes with a null Killing vector in three-dimensional massive gravityrdquo Classical and Quantum Gravityvol 26 no 16 Article ID 165002 11 pages 2009

[8] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[9] S W Hawking ldquoBlack holes in general relativityrdquo Communica-tions in Mathematical Physics vol 25 pp 152ndash166 1972

[10] M Korzynski J Lewandowski and T Pawlowski ldquoMechanicsof multidimensional isolated horizonsrdquo Classical and QuantumGravity vol 22 no 11 pp 2001ndash2016 2005

[11] V Moncrief and J Isenberg ldquoSymmetries of cosmologicalCauchy horizonsrdquo Communications in Mathematical Physicsvol 89 no 3 pp 387ndash413 1983

[12] L Kong and D Pei ldquoOn spacelike curves in hyperbolic spacetimes sphererdquo International Journal of Geometric Methods inModern Physics vol 11 no 3 2014

[13] I R Porteous ldquoThe normal singularities of a submanifoldrdquoJournal of Differential Geometry vol 5 pp 543ndash564 1971

[14] L Chen S Izumiya and D Pei ldquoTimelike hypersurfaces inthe anti-de Sitter space from a contact viewpointrdquo Journal ofMathematical Sciences vol 199 no 6 pp 629ndash645 2014

[15] J Sun and D Pei ldquoNull surfaces of null curves on 3-null conerdquoPhysics Letters A vol 378 no 14-15 pp 1010ndash1016 2014

[16] J Sun and D Pei ldquoNull Cartan Bertrand curves of AW(k)-typein Minkowski 4-spacerdquo Physics Letters A vol 376 no 33 pp2230ndash2233 2012

[17] T Fusho and S Izumiya ldquoLightlike surfaces of spacelike curvesin de Sitter 3-spacerdquo Journal of Geometry vol 88 no 1-2 pp19ndash29 2008

[18] L Chen Q Han D Pei and W Sun ldquoThe singularities ofnull surfaces in anti de Sitter 3-spacerdquo Journal of MathematicalAnalysis and Applications vol 366 no 1 pp 256ndash265 2010

[19] Y Wang and D Pei ldquoSingularities for normal hypersurfacesof de Sitter timelike curves in Minkowski 4-spacerdquo Journal ofSingularities vol 12 pp 207ndash214 2015

[20] J W Bruce and P J Giblin Curves and Singularities CambridgeUniversity Press 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

Now we do research on some properties of one-parameter null hypersurfaces of the spacelike curve in 119867

3

1

As we can know the functions 120581119892(119904) 120591119892(119904) and 120590(119904) haveparticular meanings Here we consider the case when theone-parameter null hypersurfaces have the most degeneratesingularities We have the following proposition

Proposition 2 Let 120574 119868 rarr 1198673

1be a unit spacelike curveThen

one has the following

(1) The set 119871plusmn120579(119904 120583) | 120583 = sinh 120579120575120581119892(119904) is the singulari-

ties of one-parameter null hypersurfaces 119871plusmn120579(119904 120583)

(2) If k0 = 119871plusmn

120579(119904 sinh 120579120575120581119892(119904)) is a constant vector one

has 120574120579(119904) isin 119871119862V0(120579) for any 119904 isin 119868 at the same time

120590(119904) = 0

Proof By calculations we have

120597119871plusmn

120579(119904 120583)

120597120579= minus cosh 120579120574 (119904) (22)

120597119871plusmn

120579(119904 120583)

120597120583= e (119904) plusmn n (119904) (23)

120597119871plusmn

120579(119904 120583)

120597119904

= sinh 120579t (119904)

+ 120583 (minus120575120581119892 (119904) t (119904) + 120575120591119892 (119904) e (119904) plusmn 120575120591119892 (119904)n (119904))

= (sinh 120579 minus 120575120583120581119892 (119904)) t (119904)

+ 120575120583120591119892 (119904) (e (119904) plusmn n (119904))

(24)

(1) If the above three vectors are linearly dependent wecan get the singularities of 119871plusmn

120579(119904 120583) if and only if sinh 120579 minus

120575120583120581119892(119904) = 0 Therefore assertion (1) holds(2) For any fixed 120579 isin (0 +infin) if

119891 (119904) = sinh 120579120574 (119904) + 120583 (119904) (n (119904) plusmn e (119904)) (25)

is a constant then

119889119891

119889119904= (sinh 120579 minus 120575120583 (119904) 120581119892 (119904)) t (119904)

+ (1205831015840(119904) minus 120575120583 (119904) 120591119892 (119904)) (n (119904) plusmn e (119904)) = 0

(26)

Since

120583 (119904) =sinh 120579120575120581119892 (119904)

1205831015840(119904) minus 120575120583 (119904) 120591119892 (119904) = 0 (27)

then

120590 = 1205811015840

119892(119904) ∓ 120581119892 (119904) 120591119892 (119904) = 0 (28)

We have

⟨120574120579(119904) k0⟩

= ⟨120574120579(119904) sinh 120579120574 (119904) + sinh 120579

120575120581119892 (119904)(n (119904) plusmn e (119904))⟩

= minussinh2120579

(29)

This completes the proof

4 Unfoldings of One-ParameterHeight Functions

In this section we classify singularities of the one-parameternull hypersurfaces along 120574 as an application of the unfoldingtheory of functions

Let 119865 (R times R119903 (1199040 x0)) rarr R be a function germ119891(119904) = 1198651199090

(119904 x0) We call 119865 an 119903-parameter unfolding of 119891If 119891(119901)(1199040) = 0 for all 1 le 119901 le 119896 and 119891

(119896+1)(1199040) = 0 we say

that 119891 has 119860119896-singularity at 1199040 We also say that 119891 has 119860ge119896-singularity at 1199040 if 119891

(119901)(1199040) = 0 for all 1 le 119901 le 119896 Let 119865 be an

119903-parameter unfolding of 119891 and 119891 has 119860119896-singularity (119896 ge 1)

at 1199040 we define the (119896 minus 1)-jet of the partial derivative 120597119865120597119909119894at 1199040 as

119895(119896minus1) 120597119865

120597119909119894

(119904 x0) (1199040) =119896+1

sum

119895=1

120572119895119894 (119904 minus 1199040)119895

(119894 = 1 119903)

(30)

If the rank of 119896 times 119903 matrix (1205720119894 120572119895119894) is 119896 (119896 le 119903) then 119865 iscalled a versal unfolding of 119891 where 1205720119894 = (120597119865120597119909119894)(1199040 x0)The discriminant set of 119865 is defined by

119863119865 = x isin R119903| exist119904 isin R 119865 (119904 x) = 120597119865

120597119904(119904 x) = 0 (31)

There has been the following famous result [20]

Theorem 3 Let 119865 (R timesR119903 (1199040 x0)) rarr R be an 119903-parameterunfolding of 119891(119904) which has 119860119896-singularity at 1199040 suppose that119865 is a V119890119903119904119886119897 119906119899119891119900119897119889119894119899119892 of 119891 Then one has the following

(a) If 119896 = 1 then119863119865 is locally diffeomorphic to 0 timesR119903minus1(b) If 119896 = 2 then119863119865 is locally diffeomorphic to 119862 timesR119903minus2(c) If 119896 = 3 then119863119865 is locally diffeomorphic to 119878119882timesR119903minus3

By Proposition 1 the discriminant set of the timelikeheight function119867(119904 k 120579) is given by

119863119867 = sinh 120579120574 (119904) + 120583 (n (119904) plusmn e (119904)) | 119904 120583 isin 119868 120579

isin (0 +infin)

(32)

Proposition 4 Let119867(119904 k 120579) be a one-parameter height func-tion on the spacelike curve 120574 k isin 119863119867 If ℎV has 119860119896-singularityat s (119896 = 1 2 3 4) then119867 is a versal unfolding of ℎV

Journal of Function Spaces 5

Proof Let 120574(119904) = (1199091(119904) 1199092(119904) 1199093(119904) 1199094(119904)) isin 1198673

1and k =

(V1 V2 V3 V4) isin 1198673

1(minussinh2120579)

Then119867(119904 k 120579)

= sinh 120579 (minus1199091V1 minus 1199092V2 + 1199093V3 + 1199094V4 + sinh 120579)

120579 isin (0 +infin)

(33)

Let V1 = plusmnradicminusV22+ V23+ V24+ sinh2120579 so

120597119867

120597V2(119904 k) = sinh 120579 (minus1199092 minus

V2V11199091)

1205972119867

120597119904120597V2(119904 k) = sinh 120579 (minus1199091015840

2minusV2V11199091015840

1)

1205973119867

1205972119904120597V2(119904 k) = sinh 120579 (minus11990910158401015840

2minusV2V111990910158401015840

1)

120597119867

120597V119894(119904 k) = sinh 120579 (119909119894 +

V119894V11199091)

1205972119867

120597119904120597V119894(119904 k) = sinh 120579 (1199091015840

119894+

V119894V11199091015840

1)

1205973119867

1205972119904120597V119894(119904 k) = sinh 120579 (11990910158401015840

119894+

V119894V111990910158401015840

1) (119894 = 3 4)

120597119867

120597120579(119904 k) = 120593120579 (119904)

1205972119867

120597119904120597120579(119904 k) = 120593

1015840

120579(119904)

1205973119867

1205972119904120597120579(119904 k) = 120593

10158401015840

120579(119904)

(34)

The 2-119895119890119905 of (120597119867120597V2)(119904 k) at 1199040 is given by

sinh 120579 (minus11990910158402119904 minus

1

211990910158401015840

21199042) +

V2V1

(minus1199091015840

1119904 minus

1

211990910158401015840

11199042) (35)

the 2-119895119890119905 of (120597119867120597V119894)(119904 k) at 1199040 is given by

sinh 120579 (1199091015840119894119904 +

1

211990910158401015840

1198941199042) +

V119894V1

(1199091015840

1119904 +

1

211990910158401015840

11199042)

(119894 = 3 4)

(36)

(1) By Proposition 1 ℎV has the 1198601-singularity at 1199040 if andonly if k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904)) Since the curve 120574(119904) isregular the rank of

(minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579) (37)

is 1(2) We can get that ℎ has the 1198602-singularity at 1199040 if and

only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (38)

and 120590 = 0 When ℎ has the 119860ge2-singularity at 1199040 we requirethe 2 times 4matrix

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091 1199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

) (39)

to have rank 2 which follows from the proof of the next case(3) It also follows from Proposition 1 that ℎ has the 119860ge3-

singularity at 1199040 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (40)

and 120590 = 0 1205901015840 = 0We require the 3 times 4matrix

(

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

12059310158401015840

120579

sinh 120579

)

)

(41)

to have rank 3Let 3 times 3matrix

119860 = (

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

) (42)

We denote

119860 (119894 119895 119896) = det(

119909119894 119909119895 119909119896

1199091015840

1198941199091015840

1198951199091015840

119896

11990910158401015840

11989411990910158401015840

11989511990910158401015840

119896

) (43)

thendet119860

= minus119860 (2 3 4) minusV2V1119860 (1 3 4) minus

V3V1119860 (2 1 4)

minusV4V1119860 (2 3 1)

= plusmn1

V1⟨(V1 V2 V3 V4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minuse1 minuse2 e3 e41199091 1199092 1199093 1199094

1199091015840

11199091015840

21199091015840

31199091015840

4

11990910158401015840

111990910158401015840

211990910158401015840

311990910158401015840

4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟩

= plusmn1

V1⟨k 120574 (119904) and 1205741015840 (119904) and 12057410158401015840 (119904)⟩

(44)

Since k isin 119863119867 is a singular point then

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (45)

6 Journal of Function Spaces

and we have

120574 (119904) and 1205741015840(119904) and 120574

10158401015840(119904) = 120574 (119904) and 120574

1015840(119904)

and (120574 (119904) + 120581119892 (119904)n (119904))

= 120581119892 (119904) (120574 (119904) and t (119904) and n (119904))

= 120581119892 (119904) e (119904)

(46)

Therefore

det119860 = plusmn1

V1⟨sinh 120579120574 (119904)

+sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) 120581119892 (119904) e (119904)⟩ = plusmnsinh 120579V1

= 0

(47)

In summary119867 is a versal unfolding of ℎV this completes theproof

5 Main Result

The main result in this paper is in this section We nowconsider the following conditions

(A1) The number of points 119901 of 120574120579(119904) where LCV0(120579) at 119901

have four-point contact with the curve 120574120579 is finite

(A2) There is no point 119901 of 120574120579(119904) where LCV0(120579) at 119901 have

five-point contact or greater with the curve 120574120579

Our main result is as follows

Theorem 5 Let 120574 119868 rarr 1198673

1be a unit regular spacelike curve

k0 = 119871plusmn

120579(1199040 1205830) and

119871119862V0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (48)

Then one has the following(1) 119871119862V0(120579) and 120574120579 have at least 2-point contact at 1199040(2) 119871119862V0(120579) and 120574120579 have 3-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (49)

and 120590(1199040) = 1205811015840

119892(1199040) ∓ 120581119892(1199040)120591119892(1199040) = 0 Under this condition

the germ of image 119871plusmn120579at (1199040 1205830) is diffeomorphic to the cuspidal

edge 119862 timesR (Figure 1)(3) 119871119862V0(120579) and 120574120579 have 4-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (50)

120590(1199040) = 0 and 1205901015840(1199040) = 0 Under this condition the germ of

image 119871plusmn

120579at (1199040 1205830) is diffeomorphic to the swallowtail 119878119882

(Figure 2)

Here 119862 = (1199091 1199092) | 1199092

1= 1199093

2 and 119878119882 = (1199091 1199092 1199093) |

1199091 = 31199064+ 1199062V 1199092 = 4119906

3+ 2119906V 1199093 = V

Figure 1 Cuspidal edge

Figure 2 Swallowtail

Proof Let 120574 119868 rarr 1198673

1be a spacelike regular curve and

⟨t1015840(119904) t1015840(119904)⟩ = minus1 As k0 = 119871plusmn

1205790(1199040 1205830) we give a function

119867 1198673

1(minussinh21205790) 997888rarr R (51)

by119867(119906) = ⟨119906 V0⟩ + sinh21205790 then we assume that

ℎV0 1205790 (119904) = 119867 (1205741205790(119904)) (52)

Because 119867minus1(0) = LCV0(1205790) and 0 is a regular value of 119867

1205741205790

and LCV0(1205790) have (119896 + 1)-point contact at 1199040 if andonly if ℎV0 1205790(119904) has the 119860119896-singularity at 1199040 By Proposition 1Theorem 3 and Proposition 4 we get the results

6 Example

In this section we construct the one-parameter null hyper-surfaces associated with a spacelike curve and two sections of

Journal of Function Spaces 7

the one-parameter null hypersurfaces The two sections arenull surfaces and they are also the wavefronts of spacelikecurves By calculatingwe get the singularities of null surfacesIt is useful to understand the one-parameter null hypersur-faces

Let 120574(119904) = (radic2 cosh(2119904) radic3 cosh(radic7119904) + 2 sinh(radic7119904)radic2 sinh(2119904)radic3 sinh(radic7119904)+2 cosh(radic7119904)) be a spacelike curvein1198673

1 where 119904 is the arc-length parameter Then

t (119904) = (2radic2 sinh (2119904) radic21 sinh (radic7119904) + 2radic7

sdot cosh (radic7119904) 2radic2 cosh (2119904) radic21 cosh (radic7119904) + 2radic7

sdot sinh (radic7119904))

n (119904) = (cosh (2119904) radic6 cosh (radic7119904) + 2radic2

sdot sinh (radic7119904) sinh (2119904) radic6 sinh (radic7119904) + 2radic2

sdot cosh (radic7119904))

(53)

We have

e (119904) = 120574 (119904) and t (119904) and n (119904)

= (radic7 sinh (2119904) 2radic6 sinh (radic7119904)

+ 4radic2 cosh (radic7119904) radic7 cosh (2119904) 2radic6 cosh (radic7119904)

+ 4radic2 sinh (radic7119904))

(54)

119871plusmn

120579(119904 120583) = sinh 120579120574(119904)+120583(n(119904)plusmne(119904)) be the one-parameter

null hypersurfaces of 120574(119904) At the moment we can calculatethat 120581119892(119904) = 3radic2 and 120591119892(119904) = 2radic7 two sections of 119871plusmn

120579(119904 120583)

with 1205791 = arcsinh(12) 1205792 = arcsinh(radic22) are as follows

119871+

1205791(119904 120583) =

1

2120574 (119904) + 120583 (n (119904) + e (119904)) = ((

radic2

2+ 120583)

sdot cosh (2119904) + radic7120583 sinh (2119904) (radic3

2+ radic6120583 + 4radic2120583)

sdot cosh (radic7119904) + (1 + 2radic2120583 + 2radic6120583)

sdot sinh (radic7119904) radic7120583 cosh (2119904) + (radic2

2+ 120583)

sdot sinh (2119904) (1 + 2radic2120583 + 2radic6120583)

sdot cosh (radic7119904) + (radic3

2+ radic6120583 + 4radic2120583)

sdot sinh (radic7119904))

Figure 3 Projection of 119871+1205791(119904 120583) on 119909211990931199094-space

Figure 4 Projection of 119871minus1205792(119904 120583) on 119909111990921199093-space

119871minus

1205792(119904 120583) =

radic2

2120574 (119904) + 120583 (n (119904) minus e (119904)) = ((1 + 120583)

sdot cosh (2119904) minus radic7120583 sinh (2119904) (radic6

2+ radic6120583 minus 4radic2120583)

sdot cosh (radic7119904) + (radic2 + 2radic2120583 minus 2radic6120583) sinh (radic7119904)

minus radic7120583 cosh (2119904) + (1 + 120583)

sdot sinh (2119904) (radic2 + 2radic2120583 minus 2radic6120583) cosh (radic7119904)

+ (radic6

2+ radic6120583 minus 4radic2120583) sinh (radic7119904))

(55)

The pictures of 1205791-null hypersurface 119871+

1205791(119904 120583) and its

singularities 119871+1205791(119904 radic212) can be seen in Figure 3 And the

pictures of 1205792-null hypersurface 119871minus

1205792(119904 120583) and its singularities

119871minus

1205792(119904 16) can be seen in Figure 4

Competing Interests

The authors declare that they have no competing interests

8 Journal of Function Spaces

Acknowledgments

This work was partially supported by NSF of China (nos11271063 and 11501051) and NCET of China (no 05-0319)The first author was partially supported by the Project of Sci-ence and Technology of Heilongjiang Provincial EducationDepartment of China (no UNPYSCT-2015103)

References

[1] AMahmut E Soley and TMurat ldquoBeltrami-Meusnier formu-las of generalized semi ruled surfaces in semi Euclidean spacerdquoKuwait Journal of Science vol 41 no 2 pp 65ndash83 2014

[2] E Tulay andGM Ali ldquoSome characterizations of quaternionicrectifying curves in the semi-Euclidean space E2

4rdquo HonamMathematical Journal vol 36 no 1 pp 67ndash83 2014

[3] H Liu ldquoCurves in affine and semi-Euclidean spacesrdquo Results inMathematics vol 65 no 1-2 pp 235ndash249 2014

[4] E Soley and T Murat ldquoTimelike Bertrand curves in semi-Euclidean spacerdquo International Journal of Mathematics andStatistics vol 14 no 2 pp 78ndash89 2013

[5] M Sakaki ldquoBi-null Cartan curves in semi-Euclidean spaces ofindex 2rdquo Beitrage zur Algebra und Geometrie Contributions toAlgebra and Geometry vol 53 no 2 pp 421ndash436 2012

[6] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[7] G Clement ldquoBlack holes with a null Killing vector in three-dimensional massive gravityrdquo Classical and Quantum Gravityvol 26 no 16 Article ID 165002 11 pages 2009

[8] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[9] S W Hawking ldquoBlack holes in general relativityrdquo Communica-tions in Mathematical Physics vol 25 pp 152ndash166 1972

[10] M Korzynski J Lewandowski and T Pawlowski ldquoMechanicsof multidimensional isolated horizonsrdquo Classical and QuantumGravity vol 22 no 11 pp 2001ndash2016 2005

[11] V Moncrief and J Isenberg ldquoSymmetries of cosmologicalCauchy horizonsrdquo Communications in Mathematical Physicsvol 89 no 3 pp 387ndash413 1983

[12] L Kong and D Pei ldquoOn spacelike curves in hyperbolic spacetimes sphererdquo International Journal of Geometric Methods inModern Physics vol 11 no 3 2014

[13] I R Porteous ldquoThe normal singularities of a submanifoldrdquoJournal of Differential Geometry vol 5 pp 543ndash564 1971

[14] L Chen S Izumiya and D Pei ldquoTimelike hypersurfaces inthe anti-de Sitter space from a contact viewpointrdquo Journal ofMathematical Sciences vol 199 no 6 pp 629ndash645 2014

[15] J Sun and D Pei ldquoNull surfaces of null curves on 3-null conerdquoPhysics Letters A vol 378 no 14-15 pp 1010ndash1016 2014

[16] J Sun and D Pei ldquoNull Cartan Bertrand curves of AW(k)-typein Minkowski 4-spacerdquo Physics Letters A vol 376 no 33 pp2230ndash2233 2012

[17] T Fusho and S Izumiya ldquoLightlike surfaces of spacelike curvesin de Sitter 3-spacerdquo Journal of Geometry vol 88 no 1-2 pp19ndash29 2008

[18] L Chen Q Han D Pei and W Sun ldquoThe singularities ofnull surfaces in anti de Sitter 3-spacerdquo Journal of MathematicalAnalysis and Applications vol 366 no 1 pp 256ndash265 2010

[19] Y Wang and D Pei ldquoSingularities for normal hypersurfacesof de Sitter timelike curves in Minkowski 4-spacerdquo Journal ofSingularities vol 12 pp 207ndash214 2015

[20] J W Bruce and P J Giblin Curves and Singularities CambridgeUniversity Press 1992

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

Proof Let 120574(119904) = (1199091(119904) 1199092(119904) 1199093(119904) 1199094(119904)) isin 1198673

1and k =

(V1 V2 V3 V4) isin 1198673

1(minussinh2120579)

Then119867(119904 k 120579)

= sinh 120579 (minus1199091V1 minus 1199092V2 + 1199093V3 + 1199094V4 + sinh 120579)

120579 isin (0 +infin)

(33)

Let V1 = plusmnradicminusV22+ V23+ V24+ sinh2120579 so

120597119867

120597V2(119904 k) = sinh 120579 (minus1199092 minus

V2V11199091)

1205972119867

120597119904120597V2(119904 k) = sinh 120579 (minus1199091015840

2minusV2V11199091015840

1)

1205973119867

1205972119904120597V2(119904 k) = sinh 120579 (minus11990910158401015840

2minusV2V111990910158401015840

1)

120597119867

120597V119894(119904 k) = sinh 120579 (119909119894 +

V119894V11199091)

1205972119867

120597119904120597V119894(119904 k) = sinh 120579 (1199091015840

119894+

V119894V11199091015840

1)

1205973119867

1205972119904120597V119894(119904 k) = sinh 120579 (11990910158401015840

119894+

V119894V111990910158401015840

1) (119894 = 3 4)

120597119867

120597120579(119904 k) = 120593120579 (119904)

1205972119867

120597119904120597120579(119904 k) = 120593

1015840

120579(119904)

1205973119867

1205972119904120597120579(119904 k) = 120593

10158401015840

120579(119904)

(34)

The 2-119895119890119905 of (120597119867120597V2)(119904 k) at 1199040 is given by

sinh 120579 (minus11990910158402119904 minus

1

211990910158401015840

21199042) +

V2V1

(minus1199091015840

1119904 minus

1

211990910158401015840

11199042) (35)

the 2-119895119890119905 of (120597119867120597V119894)(119904 k) at 1199040 is given by

sinh 120579 (1199091015840119894119904 +

1

211990910158401015840

1198941199042) +

V119894V1

(1199091015840

1119904 +

1

211990910158401015840

11199042)

(119894 = 3 4)

(36)

(1) By Proposition 1 ℎV has the 1198601-singularity at 1199040 if andonly if k = sinh 120579120574(119904) + 120583(n(119904) plusmn e(119904)) Since the curve 120574(119904) isregular the rank of

(minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579) (37)

is 1(2) We can get that ℎ has the 1198602-singularity at 1199040 if and

only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (38)

and 120590 = 0 When ℎ has the 119860ge2-singularity at 1199040 we requirethe 2 times 4matrix

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091 1199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

) (39)

to have rank 2 which follows from the proof of the next case(3) It also follows from Proposition 1 that ℎ has the 119860ge3-

singularity at 1199040 if and only if

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (40)

and 120590 = 0 1205901015840 = 0We require the 3 times 4matrix

(

(

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

120593120579

sinh 120579

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

1205931015840

120579

sinh 120579

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

12059310158401015840

120579

sinh 120579

)

)

(41)

to have rank 3Let 3 times 3matrix

119860 = (

minus1199092 minusV2V11199091 1199093 +

V3V11199091 1199094 +

V4V11199091

minus1199091015840

2minusV2V11199091015840

11199091015840

3+V3V11199091015840

11199091015840

4+V4V11199091015840

1

minus11990910158401015840

2minusV2V111990910158401015840

111990910158401015840

3+V3V111990910158401015840

111990910158401015840

4+V4V111990910158401015840

1

) (42)

We denote

119860 (119894 119895 119896) = det(

119909119894 119909119895 119909119896

1199091015840

1198941199091015840

1198951199091015840

119896

11990910158401015840

11989411990910158401015840

11989511990910158401015840

119896

) (43)

thendet119860

= minus119860 (2 3 4) minusV2V1119860 (1 3 4) minus

V3V1119860 (2 1 4)

minusV4V1119860 (2 3 1)

= plusmn1

V1⟨(V1 V2 V3 V4)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minuse1 minuse2 e3 e41199091 1199092 1199093 1199094

1199091015840

11199091015840

21199091015840

31199091015840

4

11990910158401015840

111990910158401015840

211990910158401015840

311990910158401015840

4

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟩

= plusmn1

V1⟨k 120574 (119904) and 1205741015840 (119904) and 12057410158401015840 (119904)⟩

(44)

Since k isin 119863119867 is a singular point then

k = sinh 120579120574 (119904) + sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) (45)

6 Journal of Function Spaces

and we have

120574 (119904) and 1205741015840(119904) and 120574

10158401015840(119904) = 120574 (119904) and 120574

1015840(119904)

and (120574 (119904) + 120581119892 (119904)n (119904))

= 120581119892 (119904) (120574 (119904) and t (119904) and n (119904))

= 120581119892 (119904) e (119904)

(46)

Therefore

det119860 = plusmn1

V1⟨sinh 120579120574 (119904)

+sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) 120581119892 (119904) e (119904)⟩ = plusmnsinh 120579V1

= 0

(47)

In summary119867 is a versal unfolding of ℎV this completes theproof

5 Main Result

The main result in this paper is in this section We nowconsider the following conditions

(A1) The number of points 119901 of 120574120579(119904) where LCV0(120579) at 119901

have four-point contact with the curve 120574120579 is finite

(A2) There is no point 119901 of 120574120579(119904) where LCV0(120579) at 119901 have

five-point contact or greater with the curve 120574120579

Our main result is as follows

Theorem 5 Let 120574 119868 rarr 1198673

1be a unit regular spacelike curve

k0 = 119871plusmn

120579(1199040 1205830) and

119871119862V0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (48)

Then one has the following(1) 119871119862V0(120579) and 120574120579 have at least 2-point contact at 1199040(2) 119871119862V0(120579) and 120574120579 have 3-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (49)

and 120590(1199040) = 1205811015840

119892(1199040) ∓ 120581119892(1199040)120591119892(1199040) = 0 Under this condition

the germ of image 119871plusmn120579at (1199040 1205830) is diffeomorphic to the cuspidal

edge 119862 timesR (Figure 1)(3) 119871119862V0(120579) and 120574120579 have 4-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (50)

120590(1199040) = 0 and 1205901015840(1199040) = 0 Under this condition the germ of

image 119871plusmn

120579at (1199040 1205830) is diffeomorphic to the swallowtail 119878119882

(Figure 2)

Here 119862 = (1199091 1199092) | 1199092

1= 1199093

2 and 119878119882 = (1199091 1199092 1199093) |

1199091 = 31199064+ 1199062V 1199092 = 4119906

3+ 2119906V 1199093 = V

Figure 1 Cuspidal edge

Figure 2 Swallowtail

Proof Let 120574 119868 rarr 1198673

1be a spacelike regular curve and

⟨t1015840(119904) t1015840(119904)⟩ = minus1 As k0 = 119871plusmn

1205790(1199040 1205830) we give a function

119867 1198673

1(minussinh21205790) 997888rarr R (51)

by119867(119906) = ⟨119906 V0⟩ + sinh21205790 then we assume that

ℎV0 1205790 (119904) = 119867 (1205741205790(119904)) (52)

Because 119867minus1(0) = LCV0(1205790) and 0 is a regular value of 119867

1205741205790

and LCV0(1205790) have (119896 + 1)-point contact at 1199040 if andonly if ℎV0 1205790(119904) has the 119860119896-singularity at 1199040 By Proposition 1Theorem 3 and Proposition 4 we get the results

6 Example

In this section we construct the one-parameter null hyper-surfaces associated with a spacelike curve and two sections of

Journal of Function Spaces 7

the one-parameter null hypersurfaces The two sections arenull surfaces and they are also the wavefronts of spacelikecurves By calculatingwe get the singularities of null surfacesIt is useful to understand the one-parameter null hypersur-faces

Let 120574(119904) = (radic2 cosh(2119904) radic3 cosh(radic7119904) + 2 sinh(radic7119904)radic2 sinh(2119904)radic3 sinh(radic7119904)+2 cosh(radic7119904)) be a spacelike curvein1198673

1 where 119904 is the arc-length parameter Then

t (119904) = (2radic2 sinh (2119904) radic21 sinh (radic7119904) + 2radic7

sdot cosh (radic7119904) 2radic2 cosh (2119904) radic21 cosh (radic7119904) + 2radic7

sdot sinh (radic7119904))

n (119904) = (cosh (2119904) radic6 cosh (radic7119904) + 2radic2

sdot sinh (radic7119904) sinh (2119904) radic6 sinh (radic7119904) + 2radic2

sdot cosh (radic7119904))

(53)

We have

e (119904) = 120574 (119904) and t (119904) and n (119904)

= (radic7 sinh (2119904) 2radic6 sinh (radic7119904)

+ 4radic2 cosh (radic7119904) radic7 cosh (2119904) 2radic6 cosh (radic7119904)

+ 4radic2 sinh (radic7119904))

(54)

119871plusmn

120579(119904 120583) = sinh 120579120574(119904)+120583(n(119904)plusmne(119904)) be the one-parameter

null hypersurfaces of 120574(119904) At the moment we can calculatethat 120581119892(119904) = 3radic2 and 120591119892(119904) = 2radic7 two sections of 119871plusmn

120579(119904 120583)

with 1205791 = arcsinh(12) 1205792 = arcsinh(radic22) are as follows

119871+

1205791(119904 120583) =

1

2120574 (119904) + 120583 (n (119904) + e (119904)) = ((

radic2

2+ 120583)

sdot cosh (2119904) + radic7120583 sinh (2119904) (radic3

2+ radic6120583 + 4radic2120583)

sdot cosh (radic7119904) + (1 + 2radic2120583 + 2radic6120583)

sdot sinh (radic7119904) radic7120583 cosh (2119904) + (radic2

2+ 120583)

sdot sinh (2119904) (1 + 2radic2120583 + 2radic6120583)

sdot cosh (radic7119904) + (radic3

2+ radic6120583 + 4radic2120583)

sdot sinh (radic7119904))

Figure 3 Projection of 119871+1205791(119904 120583) on 119909211990931199094-space

Figure 4 Projection of 119871minus1205792(119904 120583) on 119909111990921199093-space

119871minus

1205792(119904 120583) =

radic2

2120574 (119904) + 120583 (n (119904) minus e (119904)) = ((1 + 120583)

sdot cosh (2119904) minus radic7120583 sinh (2119904) (radic6

2+ radic6120583 minus 4radic2120583)

sdot cosh (radic7119904) + (radic2 + 2radic2120583 minus 2radic6120583) sinh (radic7119904)

minus radic7120583 cosh (2119904) + (1 + 120583)

sdot sinh (2119904) (radic2 + 2radic2120583 minus 2radic6120583) cosh (radic7119904)

+ (radic6

2+ radic6120583 minus 4radic2120583) sinh (radic7119904))

(55)

The pictures of 1205791-null hypersurface 119871+

1205791(119904 120583) and its

singularities 119871+1205791(119904 radic212) can be seen in Figure 3 And the

pictures of 1205792-null hypersurface 119871minus

1205792(119904 120583) and its singularities

119871minus

1205792(119904 16) can be seen in Figure 4

Competing Interests

The authors declare that they have no competing interests

8 Journal of Function Spaces

Acknowledgments

This work was partially supported by NSF of China (nos11271063 and 11501051) and NCET of China (no 05-0319)The first author was partially supported by the Project of Sci-ence and Technology of Heilongjiang Provincial EducationDepartment of China (no UNPYSCT-2015103)

References

[1] AMahmut E Soley and TMurat ldquoBeltrami-Meusnier formu-las of generalized semi ruled surfaces in semi Euclidean spacerdquoKuwait Journal of Science vol 41 no 2 pp 65ndash83 2014

[2] E Tulay andGM Ali ldquoSome characterizations of quaternionicrectifying curves in the semi-Euclidean space E2

4rdquo HonamMathematical Journal vol 36 no 1 pp 67ndash83 2014

[3] H Liu ldquoCurves in affine and semi-Euclidean spacesrdquo Results inMathematics vol 65 no 1-2 pp 235ndash249 2014

[4] E Soley and T Murat ldquoTimelike Bertrand curves in semi-Euclidean spacerdquo International Journal of Mathematics andStatistics vol 14 no 2 pp 78ndash89 2013

[5] M Sakaki ldquoBi-null Cartan curves in semi-Euclidean spaces ofindex 2rdquo Beitrage zur Algebra und Geometrie Contributions toAlgebra and Geometry vol 53 no 2 pp 421ndash436 2012

[6] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[7] G Clement ldquoBlack holes with a null Killing vector in three-dimensional massive gravityrdquo Classical and Quantum Gravityvol 26 no 16 Article ID 165002 11 pages 2009

[8] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[9] S W Hawking ldquoBlack holes in general relativityrdquo Communica-tions in Mathematical Physics vol 25 pp 152ndash166 1972

[10] M Korzynski J Lewandowski and T Pawlowski ldquoMechanicsof multidimensional isolated horizonsrdquo Classical and QuantumGravity vol 22 no 11 pp 2001ndash2016 2005

[11] V Moncrief and J Isenberg ldquoSymmetries of cosmologicalCauchy horizonsrdquo Communications in Mathematical Physicsvol 89 no 3 pp 387ndash413 1983

[12] L Kong and D Pei ldquoOn spacelike curves in hyperbolic spacetimes sphererdquo International Journal of Geometric Methods inModern Physics vol 11 no 3 2014

[13] I R Porteous ldquoThe normal singularities of a submanifoldrdquoJournal of Differential Geometry vol 5 pp 543ndash564 1971

[14] L Chen S Izumiya and D Pei ldquoTimelike hypersurfaces inthe anti-de Sitter space from a contact viewpointrdquo Journal ofMathematical Sciences vol 199 no 6 pp 629ndash645 2014

[15] J Sun and D Pei ldquoNull surfaces of null curves on 3-null conerdquoPhysics Letters A vol 378 no 14-15 pp 1010ndash1016 2014

[16] J Sun and D Pei ldquoNull Cartan Bertrand curves of AW(k)-typein Minkowski 4-spacerdquo Physics Letters A vol 376 no 33 pp2230ndash2233 2012

[17] T Fusho and S Izumiya ldquoLightlike surfaces of spacelike curvesin de Sitter 3-spacerdquo Journal of Geometry vol 88 no 1-2 pp19ndash29 2008

[18] L Chen Q Han D Pei and W Sun ldquoThe singularities ofnull surfaces in anti de Sitter 3-spacerdquo Journal of MathematicalAnalysis and Applications vol 366 no 1 pp 256ndash265 2010

[19] Y Wang and D Pei ldquoSingularities for normal hypersurfacesof de Sitter timelike curves in Minkowski 4-spacerdquo Journal ofSingularities vol 12 pp 207ndash214 2015

[20] J W Bruce and P J Giblin Curves and Singularities CambridgeUniversity Press 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

and we have

120574 (119904) and 1205741015840(119904) and 120574

10158401015840(119904) = 120574 (119904) and 120574

1015840(119904)

and (120574 (119904) + 120581119892 (119904)n (119904))

= 120581119892 (119904) (120574 (119904) and t (119904) and n (119904))

= 120581119892 (119904) e (119904)

(46)

Therefore

det119860 = plusmn1

V1⟨sinh 120579120574 (119904)

+sinh 120579120575120581119892 (119904)

(n (119904) plusmn e (119904)) 120581119892 (119904) e (119904)⟩ = plusmnsinh 120579V1

= 0

(47)

In summary119867 is a versal unfolding of ℎV this completes theproof

5 Main Result

The main result in this paper is in this section We nowconsider the following conditions

(A1) The number of points 119901 of 120574120579(119904) where LCV0(120579) at 119901

have four-point contact with the curve 120574120579 is finite

(A2) There is no point 119901 of 120574120579(119904) where LCV0(120579) at 119901 have

five-point contact or greater with the curve 120574120579

Our main result is as follows

Theorem 5 Let 120574 119868 rarr 1198673

1be a unit regular spacelike curve

k0 = 119871plusmn

120579(1199040 1205830) and

119871119862V0 (120579) = u isin 1198673

1(minussinh2120579) | ⟨u k0⟩ = minussinh2120579 (48)

Then one has the following(1) 119871119862V0(120579) and 120574120579 have at least 2-point contact at 1199040(2) 119871119862V0(120579) and 120574120579 have 3-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (49)

and 120590(1199040) = 1205811015840

119892(1199040) ∓ 120581119892(1199040)120591119892(1199040) = 0 Under this condition

the germ of image 119871plusmn120579at (1199040 1205830) is diffeomorphic to the cuspidal

edge 119862 timesR (Figure 1)(3) 119871119862V0(120579) and 120574120579 have 4-point contact at 1199040 if and only if

k0 = sinh 120579120574 (1199040) +sinh 120579120575120581119892 (1199040)

(n (1199040) plusmn e (1199040)) (50)

120590(1199040) = 0 and 1205901015840(1199040) = 0 Under this condition the germ of

image 119871plusmn

120579at (1199040 1205830) is diffeomorphic to the swallowtail 119878119882

(Figure 2)

Here 119862 = (1199091 1199092) | 1199092

1= 1199093

2 and 119878119882 = (1199091 1199092 1199093) |

1199091 = 31199064+ 1199062V 1199092 = 4119906

3+ 2119906V 1199093 = V

Figure 1 Cuspidal edge

Figure 2 Swallowtail

Proof Let 120574 119868 rarr 1198673

1be a spacelike regular curve and

⟨t1015840(119904) t1015840(119904)⟩ = minus1 As k0 = 119871plusmn

1205790(1199040 1205830) we give a function

119867 1198673

1(minussinh21205790) 997888rarr R (51)

by119867(119906) = ⟨119906 V0⟩ + sinh21205790 then we assume that

ℎV0 1205790 (119904) = 119867 (1205741205790(119904)) (52)

Because 119867minus1(0) = LCV0(1205790) and 0 is a regular value of 119867

1205741205790

and LCV0(1205790) have (119896 + 1)-point contact at 1199040 if andonly if ℎV0 1205790(119904) has the 119860119896-singularity at 1199040 By Proposition 1Theorem 3 and Proposition 4 we get the results

6 Example

In this section we construct the one-parameter null hyper-surfaces associated with a spacelike curve and two sections of

Journal of Function Spaces 7

the one-parameter null hypersurfaces The two sections arenull surfaces and they are also the wavefronts of spacelikecurves By calculatingwe get the singularities of null surfacesIt is useful to understand the one-parameter null hypersur-faces

Let 120574(119904) = (radic2 cosh(2119904) radic3 cosh(radic7119904) + 2 sinh(radic7119904)radic2 sinh(2119904)radic3 sinh(radic7119904)+2 cosh(radic7119904)) be a spacelike curvein1198673

1 where 119904 is the arc-length parameter Then

t (119904) = (2radic2 sinh (2119904) radic21 sinh (radic7119904) + 2radic7

sdot cosh (radic7119904) 2radic2 cosh (2119904) radic21 cosh (radic7119904) + 2radic7

sdot sinh (radic7119904))

n (119904) = (cosh (2119904) radic6 cosh (radic7119904) + 2radic2

sdot sinh (radic7119904) sinh (2119904) radic6 sinh (radic7119904) + 2radic2

sdot cosh (radic7119904))

(53)

We have

e (119904) = 120574 (119904) and t (119904) and n (119904)

= (radic7 sinh (2119904) 2radic6 sinh (radic7119904)

+ 4radic2 cosh (radic7119904) radic7 cosh (2119904) 2radic6 cosh (radic7119904)

+ 4radic2 sinh (radic7119904))

(54)

119871plusmn

120579(119904 120583) = sinh 120579120574(119904)+120583(n(119904)plusmne(119904)) be the one-parameter

null hypersurfaces of 120574(119904) At the moment we can calculatethat 120581119892(119904) = 3radic2 and 120591119892(119904) = 2radic7 two sections of 119871plusmn

120579(119904 120583)

with 1205791 = arcsinh(12) 1205792 = arcsinh(radic22) are as follows

119871+

1205791(119904 120583) =

1

2120574 (119904) + 120583 (n (119904) + e (119904)) = ((

radic2

2+ 120583)

sdot cosh (2119904) + radic7120583 sinh (2119904) (radic3

2+ radic6120583 + 4radic2120583)

sdot cosh (radic7119904) + (1 + 2radic2120583 + 2radic6120583)

sdot sinh (radic7119904) radic7120583 cosh (2119904) + (radic2

2+ 120583)

sdot sinh (2119904) (1 + 2radic2120583 + 2radic6120583)

sdot cosh (radic7119904) + (radic3

2+ radic6120583 + 4radic2120583)

sdot sinh (radic7119904))

Figure 3 Projection of 119871+1205791(119904 120583) on 119909211990931199094-space

Figure 4 Projection of 119871minus1205792(119904 120583) on 119909111990921199093-space

119871minus

1205792(119904 120583) =

radic2

2120574 (119904) + 120583 (n (119904) minus e (119904)) = ((1 + 120583)

sdot cosh (2119904) minus radic7120583 sinh (2119904) (radic6

2+ radic6120583 minus 4radic2120583)

sdot cosh (radic7119904) + (radic2 + 2radic2120583 minus 2radic6120583) sinh (radic7119904)

minus radic7120583 cosh (2119904) + (1 + 120583)

sdot sinh (2119904) (radic2 + 2radic2120583 minus 2radic6120583) cosh (radic7119904)

+ (radic6

2+ radic6120583 minus 4radic2120583) sinh (radic7119904))

(55)

The pictures of 1205791-null hypersurface 119871+

1205791(119904 120583) and its

singularities 119871+1205791(119904 radic212) can be seen in Figure 3 And the

pictures of 1205792-null hypersurface 119871minus

1205792(119904 120583) and its singularities

119871minus

1205792(119904 16) can be seen in Figure 4

Competing Interests

The authors declare that they have no competing interests

8 Journal of Function Spaces

Acknowledgments

This work was partially supported by NSF of China (nos11271063 and 11501051) and NCET of China (no 05-0319)The first author was partially supported by the Project of Sci-ence and Technology of Heilongjiang Provincial EducationDepartment of China (no UNPYSCT-2015103)

References

[1] AMahmut E Soley and TMurat ldquoBeltrami-Meusnier formu-las of generalized semi ruled surfaces in semi Euclidean spacerdquoKuwait Journal of Science vol 41 no 2 pp 65ndash83 2014

[2] E Tulay andGM Ali ldquoSome characterizations of quaternionicrectifying curves in the semi-Euclidean space E2

4rdquo HonamMathematical Journal vol 36 no 1 pp 67ndash83 2014

[3] H Liu ldquoCurves in affine and semi-Euclidean spacesrdquo Results inMathematics vol 65 no 1-2 pp 235ndash249 2014

[4] E Soley and T Murat ldquoTimelike Bertrand curves in semi-Euclidean spacerdquo International Journal of Mathematics andStatistics vol 14 no 2 pp 78ndash89 2013

[5] M Sakaki ldquoBi-null Cartan curves in semi-Euclidean spaces ofindex 2rdquo Beitrage zur Algebra und Geometrie Contributions toAlgebra and Geometry vol 53 no 2 pp 421ndash436 2012

[6] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[7] G Clement ldquoBlack holes with a null Killing vector in three-dimensional massive gravityrdquo Classical and Quantum Gravityvol 26 no 16 Article ID 165002 11 pages 2009

[8] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[9] S W Hawking ldquoBlack holes in general relativityrdquo Communica-tions in Mathematical Physics vol 25 pp 152ndash166 1972

[10] M Korzynski J Lewandowski and T Pawlowski ldquoMechanicsof multidimensional isolated horizonsrdquo Classical and QuantumGravity vol 22 no 11 pp 2001ndash2016 2005

[11] V Moncrief and J Isenberg ldquoSymmetries of cosmologicalCauchy horizonsrdquo Communications in Mathematical Physicsvol 89 no 3 pp 387ndash413 1983

[12] L Kong and D Pei ldquoOn spacelike curves in hyperbolic spacetimes sphererdquo International Journal of Geometric Methods inModern Physics vol 11 no 3 2014

[13] I R Porteous ldquoThe normal singularities of a submanifoldrdquoJournal of Differential Geometry vol 5 pp 543ndash564 1971

[14] L Chen S Izumiya and D Pei ldquoTimelike hypersurfaces inthe anti-de Sitter space from a contact viewpointrdquo Journal ofMathematical Sciences vol 199 no 6 pp 629ndash645 2014

[15] J Sun and D Pei ldquoNull surfaces of null curves on 3-null conerdquoPhysics Letters A vol 378 no 14-15 pp 1010ndash1016 2014

[16] J Sun and D Pei ldquoNull Cartan Bertrand curves of AW(k)-typein Minkowski 4-spacerdquo Physics Letters A vol 376 no 33 pp2230ndash2233 2012

[17] T Fusho and S Izumiya ldquoLightlike surfaces of spacelike curvesin de Sitter 3-spacerdquo Journal of Geometry vol 88 no 1-2 pp19ndash29 2008

[18] L Chen Q Han D Pei and W Sun ldquoThe singularities ofnull surfaces in anti de Sitter 3-spacerdquo Journal of MathematicalAnalysis and Applications vol 366 no 1 pp 256ndash265 2010

[19] Y Wang and D Pei ldquoSingularities for normal hypersurfacesof de Sitter timelike curves in Minkowski 4-spacerdquo Journal ofSingularities vol 12 pp 207ndash214 2015

[20] J W Bruce and P J Giblin Curves and Singularities CambridgeUniversity Press 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 7

the one-parameter null hypersurfaces The two sections arenull surfaces and they are also the wavefronts of spacelikecurves By calculatingwe get the singularities of null surfacesIt is useful to understand the one-parameter null hypersur-faces

Let 120574(119904) = (radic2 cosh(2119904) radic3 cosh(radic7119904) + 2 sinh(radic7119904)radic2 sinh(2119904)radic3 sinh(radic7119904)+2 cosh(radic7119904)) be a spacelike curvein1198673

1 where 119904 is the arc-length parameter Then

t (119904) = (2radic2 sinh (2119904) radic21 sinh (radic7119904) + 2radic7

sdot cosh (radic7119904) 2radic2 cosh (2119904) radic21 cosh (radic7119904) + 2radic7

sdot sinh (radic7119904))

n (119904) = (cosh (2119904) radic6 cosh (radic7119904) + 2radic2

sdot sinh (radic7119904) sinh (2119904) radic6 sinh (radic7119904) + 2radic2

sdot cosh (radic7119904))

(53)

We have

e (119904) = 120574 (119904) and t (119904) and n (119904)

= (radic7 sinh (2119904) 2radic6 sinh (radic7119904)

+ 4radic2 cosh (radic7119904) radic7 cosh (2119904) 2radic6 cosh (radic7119904)

+ 4radic2 sinh (radic7119904))

(54)

119871plusmn

120579(119904 120583) = sinh 120579120574(119904)+120583(n(119904)plusmne(119904)) be the one-parameter

null hypersurfaces of 120574(119904) At the moment we can calculatethat 120581119892(119904) = 3radic2 and 120591119892(119904) = 2radic7 two sections of 119871plusmn

120579(119904 120583)

with 1205791 = arcsinh(12) 1205792 = arcsinh(radic22) are as follows

119871+

1205791(119904 120583) =

1

2120574 (119904) + 120583 (n (119904) + e (119904)) = ((

radic2

2+ 120583)

sdot cosh (2119904) + radic7120583 sinh (2119904) (radic3

2+ radic6120583 + 4radic2120583)

sdot cosh (radic7119904) + (1 + 2radic2120583 + 2radic6120583)

sdot sinh (radic7119904) radic7120583 cosh (2119904) + (radic2

2+ 120583)

sdot sinh (2119904) (1 + 2radic2120583 + 2radic6120583)

sdot cosh (radic7119904) + (radic3

2+ radic6120583 + 4radic2120583)

sdot sinh (radic7119904))

Figure 3 Projection of 119871+1205791(119904 120583) on 119909211990931199094-space

Figure 4 Projection of 119871minus1205792(119904 120583) on 119909111990921199093-space

119871minus

1205792(119904 120583) =

radic2

2120574 (119904) + 120583 (n (119904) minus e (119904)) = ((1 + 120583)

sdot cosh (2119904) minus radic7120583 sinh (2119904) (radic6

2+ radic6120583 minus 4radic2120583)

sdot cosh (radic7119904) + (radic2 + 2radic2120583 minus 2radic6120583) sinh (radic7119904)

minus radic7120583 cosh (2119904) + (1 + 120583)

sdot sinh (2119904) (radic2 + 2radic2120583 minus 2radic6120583) cosh (radic7119904)

+ (radic6

2+ radic6120583 minus 4radic2120583) sinh (radic7119904))

(55)

The pictures of 1205791-null hypersurface 119871+

1205791(119904 120583) and its

singularities 119871+1205791(119904 radic212) can be seen in Figure 3 And the

pictures of 1205792-null hypersurface 119871minus

1205792(119904 120583) and its singularities

119871minus

1205792(119904 16) can be seen in Figure 4

Competing Interests

The authors declare that they have no competing interests

8 Journal of Function Spaces

Acknowledgments

This work was partially supported by NSF of China (nos11271063 and 11501051) and NCET of China (no 05-0319)The first author was partially supported by the Project of Sci-ence and Technology of Heilongjiang Provincial EducationDepartment of China (no UNPYSCT-2015103)

References

[1] AMahmut E Soley and TMurat ldquoBeltrami-Meusnier formu-las of generalized semi ruled surfaces in semi Euclidean spacerdquoKuwait Journal of Science vol 41 no 2 pp 65ndash83 2014

[2] E Tulay andGM Ali ldquoSome characterizations of quaternionicrectifying curves in the semi-Euclidean space E2

4rdquo HonamMathematical Journal vol 36 no 1 pp 67ndash83 2014

[3] H Liu ldquoCurves in affine and semi-Euclidean spacesrdquo Results inMathematics vol 65 no 1-2 pp 235ndash249 2014

[4] E Soley and T Murat ldquoTimelike Bertrand curves in semi-Euclidean spacerdquo International Journal of Mathematics andStatistics vol 14 no 2 pp 78ndash89 2013

[5] M Sakaki ldquoBi-null Cartan curves in semi-Euclidean spaces ofindex 2rdquo Beitrage zur Algebra und Geometrie Contributions toAlgebra and Geometry vol 53 no 2 pp 421ndash436 2012

[6] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[7] G Clement ldquoBlack holes with a null Killing vector in three-dimensional massive gravityrdquo Classical and Quantum Gravityvol 26 no 16 Article ID 165002 11 pages 2009

[8] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[9] S W Hawking ldquoBlack holes in general relativityrdquo Communica-tions in Mathematical Physics vol 25 pp 152ndash166 1972

[10] M Korzynski J Lewandowski and T Pawlowski ldquoMechanicsof multidimensional isolated horizonsrdquo Classical and QuantumGravity vol 22 no 11 pp 2001ndash2016 2005

[11] V Moncrief and J Isenberg ldquoSymmetries of cosmologicalCauchy horizonsrdquo Communications in Mathematical Physicsvol 89 no 3 pp 387ndash413 1983

[12] L Kong and D Pei ldquoOn spacelike curves in hyperbolic spacetimes sphererdquo International Journal of Geometric Methods inModern Physics vol 11 no 3 2014

[13] I R Porteous ldquoThe normal singularities of a submanifoldrdquoJournal of Differential Geometry vol 5 pp 543ndash564 1971

[14] L Chen S Izumiya and D Pei ldquoTimelike hypersurfaces inthe anti-de Sitter space from a contact viewpointrdquo Journal ofMathematical Sciences vol 199 no 6 pp 629ndash645 2014

[15] J Sun and D Pei ldquoNull surfaces of null curves on 3-null conerdquoPhysics Letters A vol 378 no 14-15 pp 1010ndash1016 2014

[16] J Sun and D Pei ldquoNull Cartan Bertrand curves of AW(k)-typein Minkowski 4-spacerdquo Physics Letters A vol 376 no 33 pp2230ndash2233 2012

[17] T Fusho and S Izumiya ldquoLightlike surfaces of spacelike curvesin de Sitter 3-spacerdquo Journal of Geometry vol 88 no 1-2 pp19ndash29 2008

[18] L Chen Q Han D Pei and W Sun ldquoThe singularities ofnull surfaces in anti de Sitter 3-spacerdquo Journal of MathematicalAnalysis and Applications vol 366 no 1 pp 256ndash265 2010

[19] Y Wang and D Pei ldquoSingularities for normal hypersurfacesof de Sitter timelike curves in Minkowski 4-spacerdquo Journal ofSingularities vol 12 pp 207ndash214 2015

[20] J W Bruce and P J Giblin Curves and Singularities CambridgeUniversity Press 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces

Acknowledgments

This work was partially supported by NSF of China (nos11271063 and 11501051) and NCET of China (no 05-0319)The first author was partially supported by the Project of Sci-ence and Technology of Heilongjiang Provincial EducationDepartment of China (no UNPYSCT-2015103)

References

[1] AMahmut E Soley and TMurat ldquoBeltrami-Meusnier formu-las of generalized semi ruled surfaces in semi Euclidean spacerdquoKuwait Journal of Science vol 41 no 2 pp 65ndash83 2014

[2] E Tulay andGM Ali ldquoSome characterizations of quaternionicrectifying curves in the semi-Euclidean space E2

4rdquo HonamMathematical Journal vol 36 no 1 pp 67ndash83 2014

[3] H Liu ldquoCurves in affine and semi-Euclidean spacesrdquo Results inMathematics vol 65 no 1-2 pp 235ndash249 2014

[4] E Soley and T Murat ldquoTimelike Bertrand curves in semi-Euclidean spacerdquo International Journal of Mathematics andStatistics vol 14 no 2 pp 78ndash89 2013

[5] M Sakaki ldquoBi-null Cartan curves in semi-Euclidean spaces ofindex 2rdquo Beitrage zur Algebra und Geometrie Contributions toAlgebra and Geometry vol 53 no 2 pp 421ndash436 2012

[6] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[7] G Clement ldquoBlack holes with a null Killing vector in three-dimensional massive gravityrdquo Classical and Quantum Gravityvol 26 no 16 Article ID 165002 11 pages 2009

[8] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[9] S W Hawking ldquoBlack holes in general relativityrdquo Communica-tions in Mathematical Physics vol 25 pp 152ndash166 1972

[10] M Korzynski J Lewandowski and T Pawlowski ldquoMechanicsof multidimensional isolated horizonsrdquo Classical and QuantumGravity vol 22 no 11 pp 2001ndash2016 2005

[11] V Moncrief and J Isenberg ldquoSymmetries of cosmologicalCauchy horizonsrdquo Communications in Mathematical Physicsvol 89 no 3 pp 387ndash413 1983

[12] L Kong and D Pei ldquoOn spacelike curves in hyperbolic spacetimes sphererdquo International Journal of Geometric Methods inModern Physics vol 11 no 3 2014

[13] I R Porteous ldquoThe normal singularities of a submanifoldrdquoJournal of Differential Geometry vol 5 pp 543ndash564 1971

[14] L Chen S Izumiya and D Pei ldquoTimelike hypersurfaces inthe anti-de Sitter space from a contact viewpointrdquo Journal ofMathematical Sciences vol 199 no 6 pp 629ndash645 2014

[15] J Sun and D Pei ldquoNull surfaces of null curves on 3-null conerdquoPhysics Letters A vol 378 no 14-15 pp 1010ndash1016 2014

[16] J Sun and D Pei ldquoNull Cartan Bertrand curves of AW(k)-typein Minkowski 4-spacerdquo Physics Letters A vol 376 no 33 pp2230ndash2233 2012

[17] T Fusho and S Izumiya ldquoLightlike surfaces of spacelike curvesin de Sitter 3-spacerdquo Journal of Geometry vol 88 no 1-2 pp19ndash29 2008

[18] L Chen Q Han D Pei and W Sun ldquoThe singularities ofnull surfaces in anti de Sitter 3-spacerdquo Journal of MathematicalAnalysis and Applications vol 366 no 1 pp 256ndash265 2010

[19] Y Wang and D Pei ldquoSingularities for normal hypersurfacesof de Sitter timelike curves in Minkowski 4-spacerdquo Journal ofSingularities vol 12 pp 207ndash214 2015

[20] J W Bruce and P J Giblin Curves and Singularities CambridgeUniversity Press 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of