international journal of pure and applied mathematicstheorem 1. (duggal, [4]) let (m ,g) be an...

International Journal of Pure and Applied Mathematics————————————————————————–Volume 1 No. 4 2002, 389-413

NULL CURVES AND 2-SURFACES OF

GLOBALLY NULL MANIFOLDS

K.L. Duggal

Dept. of MathematicsUniversity of Windsor

Windsor, Ontario, CANADA N9B3P4

e-mail: [email protected]

Abstract: In this paper we develop a technique which provides a way toreduce problems of null geometry to problems of Riemannian geometry for null2-surfaces and 3-dimensional globally null manifold (see Definition 1) with anintegrable screen distribution.

AMS Subject Classification: 53C20, 53C50, 83C40Key Words: null manifold, null geodesics, 2-surfaces

1. Introduction

During last quarter of the twentieth century the research on massless objectshas produced considerable insight information on our physical universe. Unfor-tunately, very limited information is available (other than some papers dealingwith specific problems) on the general geometric theory of null curves (which,in particular, represent one dimensional massless particles), null 2-surfaces andhigher dimensional null (lightlike) manifolds, needed as a mathematical foun-dation and its use in physics. Also, compared to extensive research on globalRiemannian and Lorentzian geometries (see Berger [2] and Beem et al [1]), the

Received: March 4, 2002 c© 2002, Academic Publications Ltd.

390 K.L. Duggal

study on global geometry of null manifolds is quite rare (see, however, Duggal[4], [5]). It is reasonabie to believe thai any global study of null manifolds,must be based on extensive knowledge of null curves and 2-surfaces, which isthe objective of this paper. Traditional way of studying of geometric objectsusing submanifold theory and physical objects in 4-dimensional spacetimes hasrecently changed. For example, for physical study, the dimension of landingspace depends on the type of problem. Now one needs eleven dimensions tounite all the forces of the universe with a single equation, which is still at pre-liminary stages. Consequently, the study of geometric and or physical objects(independent of their landing manifold) has recently drawn considerable atten-tion. In line with this latest trend, we assume, in this paper, that although thedegenerate metric of a null manifold M must come from some non-degeneratemetric of a higher dimensional manifold, say M̄ , but M need not be landedin M̄ . This seems to be a reasonable assumption for a global study of nullcurves and 2-surfaces since they do naturally arise as geometric and physicalobjects. For specific examples and computing metric components, we use asuitable landing manifold with a nondegenerate metric.

An outline of the paper is as follows. In Section 2, we define globally nullmanifolds (M, g) which admit a global null vector field, a complete Rieman-nian hypersurface and a metric (Levi-Civita) connection with respect to thedegenerate metric g of M . For proof of Theorem 1 and details we refer Duggal[4]. Section 3 is devoted to a study of null curves, denoted by C, in a globallynull manifold M . We construct a global Frenet frame (see Proposition 1) andrecall that there exists a special parameter p with respect to which C is a nullgeodesic curve on M . It is known that M is a global product manifold of a1-dimensional null manifold and a complete Riemannian manifold, if and onlyif its screen bundle space S(TM) is integrable (see Theorem 2). The materialpresented in this section can be effectively used in reducing the problems ofnull geometry to problems of Riemannian geometry, which further opens a wayto study on global null geometry by using several key results of Riemanniangeometry (see, for example, Berger [2]). Some work in this direction has beendone by the present author [5]. In Section 4 we study null 2-surfaces N ofa 3-dimensional globally null manifold M . We show that the geometry of Nis essentially the same as that of the family of its spacelike curves (see Theo-rem 3). In particular, we study ruled null surfaces in 4-dimensional Minkowskior curved spacetimes M̄ . Initially, the notion of ruled null surfaces was in-troduced by Schild [10] as a null geodesic string on the null cone (with onedimension suppressed) of M̄ . We show that Schild’s null geodesic 2-surface canbe obtained by null geodesics of a sequence of a class of 4-dimensional globally

NULL CURVES AND 2-SURFACES OF... 391

null manifolds, tangent to the null cone of M̄ . Finally, in Section 5 we studyspacelike 2-surfaces of a globally null manifold. We show that the geometry ofa 3-dimensional globally null manifold M is the same as the Riemannian ge-ometry of its spacelike 2-surface generated by its integrable screen distributionS(TM). This is a step further than a problem discussed by Duggal in [5], onconstant curvature of globally null manifolds. The results, in this paper, are notrestricted to a constant scalar curvature of M . The paper contains examples ofits main results.

2. Globally Null Manifolds

Let (M, g ) be a real n - dimensional smooth and paracompact manifold, whereg is a symmetric tensor field of type (0 , 2). Denote by RadTM a radical (null)distribution of the tangent bundle space TM of M , which is defined by

RadTM = {ξ ∈ Γ(TM) ; g ( ξ, X ) = 0 for all X ∈ Γ(TM) }, (1)

where Γ(TM) denotes the set of all tangent vector fields on M . The dimension,say r, of RadTM is called nullity degree of g. Clearly, g is degenerate ornon-degenerate on M iff r > 0 or r = 0, respectively. We say that (M ,g )is a lightlike manifold if 0 < r ≤ n. For a lightlike M , a complementarydistribution S (TM) to RadTM in TM is called (for details, see Duggal [4]) ascreen distribution on M , whose existence is secured for paracompact M . It iseasy to see that S (TM) is semi-Riemannian, and we have

TM = RadTM ⊕ S(TM). (2)

Example 1. Let S31 be the unit pseudo sphere of Minkowski space R41,

given by −t2 + x2 + y2 + z2 = 1. Cut S31 by the hypersurface t − x = 0and obtain a lightlike surface M of S31 with RadTM spanned by a null vectorξ = ∂t + ∂x. Take a screen distribution S(TM) spanned by a spacelike vectorW = z ∂y − y ∂z. Thus, M is lightlike with r = 1 and S(TM) Riemannian.

Theorem 1. (Duggal, [4]) Let (M ,g ) be an n - dimensional lightlike man-ifold, with RadTM of rank r = 1. Then, RadTM is a Killing distribution,and there exists a metric (Levi-Civita) connection ∇ on M with respect to thedegenerate tensor field g.

In this paper we study a special class of lightlike manifolds, introduced bythe present author in [4] as follows:

392 K.L. Duggal

Definition 1. (Duggal [4]) A lightlike (M, g ) is called a globally nullmanifold, if it admits a global null vector field and a complete Riemannianhypersurface.

Since the 1-dimensional RadTM is obviously integrable, using Theorem 1we construct an (n + 1)-dimensional manifold, denoted by (M̄, ḡ), with localcoordinates (x, xa, y), where (x, xa) are coordinates on a globally null manifold(M, g), induced by the foliation determined by RadTM and (y) is a coordinateon 1-dimensional fiber of its vector bundle structure. In this way g can be takenas a degenerate metric on a family of globally null hypersurfaces M , inducedby the metric ḡ of M̄ . The embedding condition (see O’Neill [9]) implies that ḡmust be a Lorentz metric. Since null spaces do arise quite naturally, which maynot be embedded in any higher dimensional manifold, in this paper we have thefollowing two approaches:

(a) Consider null and spacelike spaces in a globally null manifold which re-quire no mention of an ambient manifold.

(b) For the purpose of computing the coefficients of the degenerate metric gof M , we assume that g is an induced metric of the Lorentz metric ḡ ofthe manifold M̄ , constructed using 1-dimensional RadTM , as explainedabove.

The approach (b) will also be followed for any example of a null curve, a 2-surface and or a globally null manifold, as explained below:

Example 2. Let (M̄, ḡ) be an (n + 1) - dimensional globally hyperbolicspacetime [1], with the line element of the metric ḡ given by

ds2 = − dt2 + dx1 + ḡab dxa dxb, ( a, b = 2, . . . , n ) (3)

with respect to a coordinate system ( t , x1 , . . . , xn ) on M̄ . Choose the range0 < x1 < ∞ so that the metric (3) is non - singular. Take two null coordinatesu and v such that u = t + x1 and v = t − x1. Thus, (3) transforms into anon - singular metric:

ds2 = − du dv + ḡab dxa xb.

The absence of du2 and dv2 in this transformed metric implies that {v = con-stant} and {u = constant} are lightlike hypersurfaces of N . Let (M, g, r =1, v = constant) be one of this lightlike pair and let D be the 1 - dimensionaldistribution generated by the null vector {∂v}, in M̄ . Denote by L the 1 -dimensional integral manifold of D. A leaf M ′ of the (n − 1) - dimensional


screen distribution of M is Riemannian with metric dΩ2 = ḡab xa xb and is

the intersection of the two lightlike hypersurfaces. In particular, there will bemany global timelike vector fields in globally hyperbolic spacetimes M̄ . If oneis given a fixed global time function, then its gradient is a global timelike vectorfield in a given M̄ . With this choice of a global timelike vector field in M̄ , weconclude that both its lightlike hypersurfaces admit a global null vector field.Now, the celebrated Hopf -Rinow theorem allows to assume that M ′ is a com-plete Riemannian hypersurface of M . Thus, there exists a pair of globally nullhypersurfaces of a globally hyperbolic spacetime. In particular, a Minkowskispace and a De - Sitter spacetime M̄ have a pair of globally null hypersurfaces.Proceeding similar to above example for 4 - dimensional M̄ , one can show thatRobertson - Walker, Reissner -Nordström and Kerr spacetimes, all have pairs ofglobally null hypersurfaces (see Hawking - Ellis [6]).

3. Null Curves

Let C be a smooth null curve in a globally null n - dimensional manifold (M, g)given by

xi = xi(t), t ∈ I ⊂ R, i ∈ {1, . . . , n},

for a coordinate neighborhood U on C. Then, the tangent vector field

d

dt= (

dx1

dt, . . . ,

dxn

dt)

on U satisfies

g (d

dt,

d

dt) = 0, i.e., gij

dxi

dt

dxj

dt= 0,

where gij = g (∂i, ∂j) and i, j ∈ {1, . . . , n}. Denote by T C the tangent bundleof C, which is a vector sub bundle of TM , and is of rank 1. Then

T C⊥ = {V ∈ TM ; g (V, ξ ) = 0} = TM ,

where ξ is null vector field tangent over C. We consider a class of null curves,such that RadTM = T C, and both are generated by ξ. Then we have

TM = RadTM ⊕ S(TM) = T C ⊕ S(TM). (4)

394 K.L. Duggal

Proposition 1. Let (M, g ) be an n-dimensional globally null manifold.Then, there exists a quasi - orthonormal frame

F = {ξ, W1, . . . ,Wn−1 } , g(Wa , Wa) = δab , g(ξ, Wa) = 0, (5)

for all a ∈ {1, . . . , n−1 }, along a null curve C, generated by a null vector fieldξ on M , where Γ(S(TM)) is spanned by an orthonormal frame {W1, . . . ,Wn−1 }.

Proof. Since g is a metric tensor on M (see Theorem 1), we have

(LX g )(Y, Z) = X ( g (Y, Z)) − g ( [X, Y ], Z ) − g (Y, [X, Z] ) = 0

for any X,Y,Z ∈ Γ(TM). Using this, ddt ≡ ξ null and (4), we obtain thefollowing differential equations:

∇ξ ξ = h ξ,

∇ξ W1 = − k1 ξ + k3 W2 + k4 W3,

∇ξ W2 = − k2 ξ − k3 W1 + k5 W3 + k6 W4,

∇ξ W3 = − k4 W1 − k5 W2 + k7 W4 + k8 W5, (6)

......................

......................

∇ξ Wn−2 = −kn−1 Wn−4 − kn Wn−3 + kn+2 Wn−1 + kn+3 Wn,

∇ξ Wn−1 = −k2n−4 Wn−3 − k2n−3 Wn−2,

provided n ≥ 5, where h and {k1, . . . , k2n−3} are smooth functions on U and{W1, . . . ,Wn−1} is an orthonormal basis of Γ(S(TM)U ). For n < 5 aboveequations reduce to the following cases:

Case 1 (n = 2).

∇ξ ξ = h ξ,

∇ξ W1 = − k1 ξ.

Case 2 (n = 3).

∇ξ ξ = h ξ,

∇ξ W1 = − k1 ξ + k3 W2,

∇ξ W2 = − k2 ξ − k3 W1.


Case 3 (n = 4).

∇ξ ξ = h ξ,

∇ξ W1 = − k1 ξ + k3 W2 + k4 W3,

∇ξ W2 = − k2 ξ − k3 W1 + k5 W3,

∇ξ W3 = − k4 W1 − k5 W2.

In general, for any n > 1 we call F (given by (5)) a Frenet frame onM along C with respect to the screen distribution S (TM). The functions{ k1, . . . , k2n−3 ) and the differential equations (6) (along with three cases forn < 5) are called curvature functions of C and Frenet equations for F , re-spectively. Thus, F , given by (5), is a quasi - orthonormal Frenet frame whichcompletes the proof. �

Example 3. Let R41 be a 4-dimensional Minkowski spacetime with aLorentz metric of signature (− + + + ) and local coordinates (x, x1, x2, y ).Following the approach (b), as stated in Section 2, let (M, g ) be a globally nullhypersurface of R41 such that (x, x

1, x2, y = constant ) are coordinates on M ,induced by RadTM . Consider a curve C in M defined by

x = f(t) , x1 = − f(t) , x2 = a1

, y = a2

, p ∈ I ⊂ R,

where a1

and a2

are suitable constants. Then,

d

dt= ( f ′(t) , − f ′(t) , 0 , 0 ) and g(

d

dt,

d

dt) = 0.

Thus, C is a null curve of M , generated by a null vector field, say ξ ≡ ddt .Choose a quasi-orthonormal set { ξ, W1, W2 } on M along C, where

W1 = ( b f(t) , − b f(t) , 1 , 0 ) ; W2 = ( c f(t) , − c f(t) , 0 , 1 )

are orthonormal spacelike vectors which generate a screen distribution of M , band c are suitable constants. Following are three Frenet equations:

∇ξξ = h ξ , h ≡

f ′′(t)

f(t),

∇ξW1 = b ξ + 0W2 , ∇ξ W2 = c ξ + 0W1

such that, according to the case 2 (n = 3) , k1 = − b , k2 = − c , k3 = 0.

396 K.L. Duggal

Consider, with respect to a given screen distribution S(TM), two Frenetframes F and F ⋆ along two neighborhoods U and U⋆, respectively, with non-nullintersection. Then we have

ξ⋆ =dt

dt⋆ξ , (7)

W ⋆a = Aba Wb, a, b ∈ {1, . . . , n − 1 }, (8)

where Aba are smooth functions on U ∩ U⋆ and the matrix [Aba (x) ] is an element

of the orthogonal group O(n) for any x ∈ U ∩ U⋆.

Proposition 2. Let C be a null curve of an n-dimensional globally nullmanifold M, and F , F ∗ be two Frenet frames on U and U∗ with curvaturefunctions {k1, . . . , k2n−3} and {k

∗1 , . . . , k

∗2n−3}, respectively, induced by the same

screen vector bundle S(TC⊥). Suppose U∩U∗ 6= φ and Π2n−2α=1

kα 6= 0 on U∩U∗.

Then at any point of U ∩ U∗ we have

k⋆1 = k1 A1, k⋆2 = k2 A2, (9)

k∗a = Aa kadt

dt∗, a ∈ {3, . . . , 2n − 3 }, where Aa = ± 1. (10)

Proof. Since[

Aba]

is an orthogonal matrix we infer Aaa = Aa = ±1 andAba = 0 for all a, b ∈ { 1, . . . , n − 1 }. Then from the second and the thirdequations in (6) with respect to both F and F ∗ and taking into account thatka 6= 0 implies k

∗a 6= 0 for a = 4, 5 we obtain the relations (9) and (10) for

a = 1, . . . , 5. Similarly, we obtain all the relations of (10). �

Corollary 1. Let C be a null curve as given in Proposition 2. Then,k1 and k2 are invariant functions up to a sign, with respect to any parametertransformations on C.

Next, let F ={

ddt , N, W1, . . . ,Wn−1

}

and F̄ ={

ddt̄ , N̄ , W̄1, . . . , W̄n−1

}

betwo Frenet frames with respect to

(

t, S(TC⊥),U)

and(

t̄, S̄(TC⊥), Ū)

, respectively. Then the general transformations that relate el-ements of F and F̄ on U ∩ Ū 6= φ, are given by

d

dt̄=

dt

dt̄

d

dt, (11)

W̄a = Bba

(

Wb −dt

dt̄c

b

d

dt

)

, (12)


where cb

and Bba are smooth functions on U ∩Ū and the (n−1)× (n−1) matrix[

Bba(x)]

is an element of O(m) for each x ∈ U ∩ Ū . Thus, by using (11) and thefirst equation in (6) for both F and F̄ , we obtain

h̄ =d2t

dt̄2dt̄

dt+ h

dt

dt̄. (13)

Proposition 3. (Duggal, [4]) Let C be a null curve of a globally nullmanifold M , with Frenet equations (6). Then, it is possible to find a specialparameter, say p, on C such that the function h vanishes. Moreover, C is a nullgeodesic with respect to such a special parameter p.

Consider two Frenet frames F and F̄ for two screen distributions S(TM)and S̄(TM), respectively. Using the transformation equations (11) and (12)we conclude that the Proposition 2 will also hold for any screen distribution.Also, let p and p̄ be two special parameters induced by t and t̄, with respect tothe same screen distribution. Then for both p and p̄, one can obtain a specialparameter p̄ = a p + b, a 6= 0. Thus, we have the following result:

Corollary 2. Let M be a globally null manifold. The existence of a nullgeodesic curve C of M , is independent of both the parameter transformationson C and the screen distribution transformations.

Example 4. Let C be the null curve as given in Example 3. It follows

that C is a null geodesic, if h ≡ f′′(t)f(t) = 0. This implies that f

′′(t) = 0. Thus

f(t) = c1t + c

2is a linear function. For this case we may take t = p, a special

parameter. Other two Frenet equations (from Example 3) will be same.

Theorem 2. (Duggal, [4]) Let (M, g) be a globally null manifold. Then,the following assertions are equivalent:

(a) The screen distribution S(TM) is integrable.

(b) M = L × M ′ is a global product manifold, where M ′ is a leaf of S(TM)and L is a 1-dimensional integral manifold of a null curve C in M .

(c) S(TM) is parallel with respect to the metric connection ∇ on M .

A Mathematical Model. Theorem 1 tells us that any lightlike manifoldwith dim(RadTM) = 1, has a null Killing vector field, say ξ. In particular,

398 K.L. Duggal

for a globally null manifold M , ξ is globally defined. Using this information,we assume that M has a smooth 1 - parameter group G of isometries, whoseorbits are global null curves in M , such that ξ is the infinitesimal generator ofG. Let M ′ be the orbit space of the action G ≈ L, where we denote L by a1 - dimensional null line in M . Then M ′ is a smooth Riemannian hypersurfaceof M and the projection

π : M → M ′

is a principle L - bundle, with null fiber G. The global existence of null vectorfield implies that M ′ is Hausdorff and paracompact. The metric g restricted tothe screen bundle space S(TM), induces a Riemannian metric g′ on M ′. Sincenull vector field ξ is non - vanishing on M , we can take ξ = ∂∂t a global nullcoordinate vector field for some global function t on M . Thus the null functiont induces a diffeomorphism on M such that (M, g) is a global product manifoldof the form

M = L × M ′, g = π⋆ g′.

There exists a connection 1 - form η for the L - bundle π, determined byg and the null coordinate function t, such that η(ξ) = 0. Thus we have amathematical model of globally null product manifolds (M, g) with an orbitdata (M ′, g′, t, η ).

Proposition 4. Under the hypothesis of Theorem 1, if the screen distribu-tion S(TM) of M , is integrable, then the first two curvature functions k1 andk2 vanish in the Frenet equations (6).

Proof. S(TM) integrable implies from (a) of Theorem 2 that M is a productmanifold. Also it follows from (c) of Theorem 2, that there exists a metricconnection ∇ on M , which preserves RadTM and S(TM). Thus k1 and k2 (inthe second and third equations of (6)) must vanish. �

Remark 1. Using Theorem 2, one can take (M ′, g′) a complete spacelikehypersurface of the globally null manifold M with induced Riemannian metricg′ expressed by

g′ = w1 ⊗ w1 + . . . + wn−1 ⊗ wn−1,

where {w1, . . . , wn−1} are duals of the orthonormal basis {W1, . . . ,Wn−1} ofΓ S(TM). Clearly, g′ being Riemannian metric its inverse exists and is alsoRiemannian. In this way, any tensor (including degenerate metric g) on M canbe projected onto its screen distribution, and all the analysis on M can be doneon its integral spacelike hypersurface M ′. In particular, for above mathematical


model one can use the orbit data (M ′, g′, t, η ) to study null geometry of M .Consequently, Theorem 2 provides a way to reduce, as far as possible, problemsof null geometry to problems of Riemannian geometry. The present author hasrecently used this idea and the technique of warped product to study someproperties of 4-dimensional globally null manifolds. For details see Duggal [5],where we solved the following two problems:

(a) Let M = (L × B ×f F, g ) be a 4 - dimensional globally null warpedproduct manifold, B = (a, b) an open connected subset of real line with positivedefinite metric dr2 and −∞ ≤ a < b ≤ +∞, and the 2 - dimensional fiber spaceF be of constant scalar curvature. Then the metric g admits a warping functionf(r) for which M has a constant scalar curvature.

(b) Let M = (L × B ×f F, g) be a 4 - dimensional globally null warpedproduct manifold, (B, g

B) a Riemannian surface with scalar curvature SB and

F = (a, b) an open connected subset of real line with positive definite metricdx2 and −∞ ≤ a < b ≤ +∞. Then the metric g admits infinitely manywarped functions for which M has constant scalar curvature.

Remark 2. It follows from the Remark 1 that a specific technique on theglobal study of null curves, presented in this paper and its use in [5], has beeneffective in reducing the problems of null geometry to problems of Riemanniangeometry. We certainly hope that this opens a way to further study on globalnull geometry by using several key results of Riemannian geometry (see, forexample, Berger [2]). In next section we show how this specific technique canbe used to study curvature properties of null 2-surfaces.

4. Null 2-surfaces

Let (N, h) be a 2 - surface of an (n + 2) - dimensional globally null (M, g) man-ifold, n > 0, where h is the induced tensor field on N of g, i.e.

h(X, Y ) = g(X, Y ), for all X,Y ∈ Γ(TN).

It follows from Section 2 that N is a null 2-surface of M , if

RadTM = RadTN ,

which we assume in this section. Since S(TM) is Riemannian, it is alwayspossible to decompose it such that

S(TM) = TH ⊕ TV, (14)

400 K.L. Duggal

where TH and TV = TH⊥ are horizontal and vertical distributions of S(TM),respectively. In this section, we assume that dim(TH) = 1 and, therefore,dim(TV ) = n. Thus using (2) and (14), we have

TM|N = (RadTN ⊕ TH) ⊕ TV = TN ⊕ TV. (15)

Since both the distributions RadTN and TH are of rank 1 on N , they areintegrable. Therefore, there exists an atlas of local charts

{U ; u0 , u1, u2, . . . , un+1 } ,

such that { ∂∂u0

, ∂∂u1

} ∈ Γ(TN|U). Thus the matrix of the degenerate metric g

on M with respect to the natural frames field { ∂∂uo , . . . ,∂

∂un+1}, is as follows

[g] =

[

0 00 g

ij(uo, . . . , un+1)

]

, (16)

where

gij = g

(

∂

∂ui,

∂

∂uj

)

, i, j ∈ {1, . . . , n + 1}, det[gij ] 6= 0

and ga1

= g1a = 0

for all a ∈ {2, . . . , n + 1 }. According to the general transformations on afoliated manifold, we have

ūo = ūo(uo, u1, . . . , un+1),

ūi = ūi(u1, . . . , un+1) .

Using above transformations, a well-known procedure is available to obtaina local field of frames on M adapted to the decomposition (15). However,with respect to the tangent bundle space TM , we show that there exists apseudo-orthonormal Frenet frame F = { ξ, W1, W2, . . . , Wn+1 } on M alongN , adapted to the decomposition (15), such that TN is spanned by { ξ, W1 }and TV is spanned by {W2, . . . , Wn+1 }. At this point, for simplicity andfor physical reasons, we restrict to n = 1 so that M is a 3-dimensional glob-ally null manifold (higher dimensional case is too lengthy but straightforward).Following the approach (b), as stated in Section 2, we assume that the degen-erate metric g comes from a Lorentzian metric ḡ of a 4-dimensional Minkowskispacetime R41. Suppose N is given by

xA = xA(u, v), A ∈ {0, 1, 2} . (17)


Then the tangent bundle of N is spanned by

{

∂

∂u=

∂xA

∂u

∂

∂xA;

∂

∂v=

∂xA

∂v

∂

∂xA

}

.

By considering a vector field

ξ = α∂

∂u+ β

∂

∂v, (18)

we find that N is null, if and only if the homogeneous linear system with (α, β )as variables

α

(

2∑

a = 1

(

∂xa

∂u

)2

−

(

∂x0

∂u

)2)

+ β

(

2∑

a=1

∂xa

∂u

∂xa

∂v−

∂x0

∂u

∂x0

∂v

)

= 0,

α

(

2∑

a=1

∂xa

∂u

∂xa

∂v−

∂x0

∂u

∂x0

∂v

)

+ β

(

2∑

a=1

(

∂xa

∂v

)2

−

(

∂x0

∂v

)2)

= 0 ,

has non-trivial solutions. Denote

DAB =

∣

∣

∣

∣

∣

∣

∣

∂xA

∂u∂xB

∂u

∂xA

∂v∂xB

∂v

∣

∣

∣

∣

∣

∣

∣

. (19)

Proposition 5. A 2-surface N of a 3-dimensional globally null manifoldM is null, if and only if on each coordinate neighborhood U ⊂ N we have

2∑

a =1

(D0 a )2 =∑

1≤a

402 K.L. Duggal

where {ǫB} is the signature of the basis{

∂∂xB

}

with respect to the Minkowskimetric ḡ. The corresponding 1-dimensional screen distribution S(TN) = TH(see equation (15)) is spanned by a spacelike vector field

U =∂x0

∂v

∂

∂u−

∂x0

∂u

∂

∂v= D10

∂

∂x1+ D20

∂

∂x2. (23)

Finally, by using the decomposition (15) we obtain the spacelike vector bundleTH⊥ = TV spanned by

V = D20∂

∂x1− D10

∂

∂x2. (24)

Summing up, we have a quasi-orthonormal Frenet frame field, adapted to thecomposition (15), as follows:

F =

{

ξ, W1 =1

(∆)1/2U, W2 =

1

(∆)1/2V

}

, ∆ =

2∑

a = 1

(D0a )2. (25)

The adapted Frenet frame (25) on M will relate the material of Section 3with the following discussion on the geometry of null 2-surfaces.

Using the terminology of differential geometry, we say that x = x(u, v)is a class Cm (m ≥ 1) regular parametric representation of N , defined on acoordinates neighborhood U , if:

(1) x is of class Cm ∈ U .

(2) At least one 2 × 2 determinant DAB of (19) is non-zero.A curve x(u, v = v

0) is called a u-parameter curve on N . Similarly, a curve

x(u = u0, v) is called a v-parameter curve on N . To cover any possible singular

points, we take necessary overlapping allowable coordinate patches and useelementary topology, so that N is at least smooth. If this is not possible, thenwe restrict the parameters to regular points. Thus subject to above topologicalconstraints, this parametric representation can cover a regular N with a nullfamily and a spacelike family of curves. Then xu(u0 , v0) and xv (u0 , v0) arevectors tangent to the u-parameter and the v-parameter curves, respectively,at their intersecting point P (u

0, v

0) of N . In this paper we assume that u-

parameter curve is null.

Null Tangent Plane. Let P ∈ M and ℓ be a null vector of TP

M . A planeT

P(N) of T

PM is called a null plane directed by ℓ, if it contains ℓ, g

P(ℓ, W ) = 0

for any W ∈ TP(N), and there exists W0 ∈ TP (N) such that g(W0, W0) 6= 0. In


particular, given a regular parametric representation x(u, v) of a null surfaceN and any point P ∈ N , a null plane T

P(N) through P parallel to xu and

xv at P is called the null tangent plane to N at P . One can verify, that itis independent on the patch containing P and that a nonzero vector of M istangent to N at P if and only if it is parallel to T

P(N). Thus T

x(N) at any x

on N is given by

y = x + Axu + B xv , −∞ < A, B > ∞ (26)

Example 5. Construct a 4-dimensional Minkowski spacetime(R41, ḡ), with local coordinates (x

0, x1, x2, y = a), where (x0, x1, x2) are coor-dinates on a 3-dimensional globally null hypersurface (M, g) of R41 and a is aconstant. Let N be a surface of M given by

x = x(u, v) =(

x0 = f(u, v), x1 = −f(u, v), x2 = − v2)

, (27)

where f is an arbitrary smooth function of two variables u and v and of classCm (m ≥ 1). This parametrisation is admissible since from (21) α = 0 butβ = −4 v2 6= 0. Then from (19) we obtain

D10 = 0 , D20 = v fu 6= 0 , D12 = −v fu 6= 0, fu ≡

∂f

∂u.

Using above values in (20) we verify that N is a null surface of M . Moreover,since not all DAB’s vanish, N is a regular null surface of class Cm (m ≥ 1).From (19) and (22)-(24) we obtain

ξ = − v2 fu

(

∂

∂x0+

∂

∂x1

)

, U = fu∂

∂x2, V = fu

∂

∂x1.

Thus we have the following quasi-orthonormal Frenet frames field on Malong the null surface N :

F ={

ξ, W1 =∂

∂x2, W2 =

∂

∂x1

}

.

Alsoxu = ( fu, − fu, 0); xv = ( fv, − fv, − 2v)

are null and spacelike vectors, respectively. Thus u-parameter and v-parametercurves are null and spacelike, respectively. N is regular, of class Cm (m ≥1), since not all DAB’s vanish. {xu , xv } is a linearly independent set. Inparticular, let f(u, v) = u2 − v2 and P (1, 1) a point on N . Then

x(1, 1) = (0, 0, − 1); xu(1, 1) = (2, − 2, 0) ;

xv (2, 1) = (−2, 2, − 2).

404 K.L. Duggal

Thus

y = x(1, 1) + Axu(1, 1) + B xv (1, 1)

=(

2A − 2B , 2B − 2A , − (1 + B))

is the equation of tangent plane at P (1, 1) ∈ N .Contrary to the Riemannian or semi-Riemannian case, unfortunately, the

normal vector field xu × xv falls back in TP (N). Thus one fails to use, in theusual way, the theory of non-degenerate surfaces to study the geometry of N .To overcome this difficulty, we proceed as follows:

Let x(u, v) be a coordinate patch on N of class ≥ 1. Then, the differentialdx = xu du + xv dv is parallel to Tx(N) at x(u, v) and the quantity

I = dx • dx = (xu du + xv dv) • (xu du + xv dv)

= h11

d v2,

since xu • xu = xv • xv = 0; (xv • xv ) ≡ h11 6= 0.As in the Riemannian case we call the function I = h

11d v2 the first fun-

damental form of the null surface N , whose only one surviving component isalways in the direction of the family, say φ(u, v) = c (constant), of spacelikecurves on N . It is known that I is independent of any coordinates transforma-tion, however the coefficient h

11varies from point to point on N . Since any

2-dimensional manifold is Einstein, the Ricci tensor of N will also have onesurviving component in the direction of φ(u, v) = c. Thus we state

Theorem 3. Let (N, h) be a null 2-surface of a 3-dimensional globally nullmanifold (M, g). Then the geometry of N is essentially the geometry of its1-dimensional spacelike integral manifold, say N ′, generated by the family ofits spacelike curves φ(u, v) = c.

Using above theorem and a well-known procedure to compute curvaturequantities of a family of spacelike curves, one can find curvature properties ofa null 2-surface N .

Ruled Null Surfaces. Let α(u) be a null curve in a 4-dimensional Lorentzmanifold (M̄ , ḡ), where u ∈ I ⊂ R. Consider a null vector field ℓ(u) of α(u).Then, by definition

x(u, v) = α(u) + v ℓ(u) ; v ∈ I ⊂ R,

is called a ruled null 2-surface, say N , of M̄ , which is generated by ℓ(u) andα is called the base curve of N . Its u-parameter and v-parameter curves both


are families of null curves. Therefore, it is also called a totally null surfacewith RadTN = TN . In particular, N is ruled by null geodesics if α is ageodesic curve. The notion of a null geodesic ruled surface was first introducedby Schild [10] in the form of a geodesic null string of a null hypersurface ofa 4-dimensional Minkowski or curved spacetime. By null strings we mean 2-dimensional ruled null surfaces on the null cone (with one dimension suppressed)of M̄ . Since Schild’s paper, there has been considerable work done on geodesicand non-geodesic null strings (see, for example, IIyenko [7] and many otherscited therein).

Example 6. Consider a Minkowski spacetime (R41, ḡ) with the distanceelement given by

ds2 = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3)2.

Here we set x0 = t, the time coordinate and x1, x2 and x3 are the three spacecoordinates. It is well-known, that for this metric, the spacelike hypersurfaces(t = constant) are a family of Cauchy hypersurfaces which cover the whole ofR41. Thus, R

41 is a product space

(R41 = R × B, ḡ = −dt2 ⊕ G),

where (B, G) is a 3-dimensional Euclidean space. It is important to mentionthat not every spacelike hypersurface of R41 is a Cauchy hypersurface (for de-tails seen Hawking-Ellis [6, page 119]). Choose a spherical coordinate system(t, r, θ, φ) with x1 = r sin θ sin φ, x2 = r sin θ cos φ, x3 = r cos θ. Then, abovemetric transforms into

ds2 = −dt2 + dr2 + r2(dθ2 + sin2 θdφ2),

which is singular at r = 0 and sin θ = 0. We, therefore, choose the ranges0 < r < ∞ 0 < θ < π and 0 < φ < 2π for which it is a regular metric.Actually two such charts are needed to cover the full R41. Now we take twonull coordinates u and v, with respect to a pseudo-orthonormal basis, such thatu = t + r and w = t − r (u ≥ w). Thus above metric transforms as

ds2 = −dudw +1

4(u − w)2(dθ2 + sin2 θdφ2),

where −∞ < u,w < ∞. The absence of du2 and dw2 in above transformedmetric imply that the hypersurfaces {u = constant} and {w = constant } arenull hypersurfaces since v;av;bη

ab = 0 = u;au;bηab. Thus, there exists a pair of

406 K.L. Duggal

null hypersurfaces of R41. Relating this example with the discussion on globallynull manifolds, we say that a leaf of the 2-dimensional screen distribution Sis topologically a 2-sphere S2, with coordinate system {θ, φ}, and can be seenas the intersection of the two hypersurfaces u = constant and v = constant.In relativity, the null coordinates u(w) are called advanced (retarded) timecoordinates and are physically related to incoming (outgoing) spherical wavestraveling at the speed of light. Suppose α(u) is the null curve representingincoming spherical waves and ℓ(u) any of its null tangent vector field. Then, bydefinition x(u, v) = α(u) + v ℓ(u) ; v ∈ I ⊂ R is a ruled surface (also callednull string where one dimension suppressed) on the null cone of R41. Similarly,one can construct another null string using retarded coordinate w.

Since a ruled surface N has dim(RadTN) = 2 and any globally null mani-fold M has exactly 1-dimensional null distribution, M can not carry any rulednull surface. However, in the following we show that there is a direct interplaybetween ruled null surfaces of M̄ and 4-dimensional globally null manifolds.Let (M, g ) be a class of 4-dimensional globally null manifolds with integrablescreen distribution S(TM). Then, it follows from Theorem 2 that M = L ×M ′

is a global product manifold, where (M ′, g′ ) is a 3-dimensional integral mani-fold of S(TM). We first deal with the geometry of 3 - dimensional M ′ which, byDefinition 1, is a compact Riemannian manifold. Here we follow Yamabe [11]for the existence of constant curvature metrics on (M ′, g′ ). Denote by M thespace of all smooth Riemannian metrics on M ′ and M1 ⊂ M of metrics sat-isfying volg′ = 1. Define the total scalar curvature or Einstein - Hilbert actionS : M ′ → R by

S(g′) = v−1/3∫

M ′SM

′

dVg′ ,

where SM′

is the scalar curvature of M ′, dVg′ is the volume element and v is thevolume of M ′. The critical points of S are Einstein metrics. Moreover, only indimension 3 these Einstein metrics are of constant scalar curvature. There isa well-known procedure to obtain Einstein manifolds. Following Yamabe [11],suppose [g′] is a conformal class of any metric g′ ∈ M1. Then there exists ametric g′ ∈ M1 which achieves its infimum µ[g

′] ≡ S|[g′]∩M1 . Such metricsare called Yamabe metrics. However, there are restrictions on the existence ofYamabe metrics. Denote by σ(M ′) = sup(µ[g′])C1 , where C1 is the subset ofunit volume Yamabe metrics. If σ(M ′) ≤ 0, it has been proved by Besse [3] thatany Yamabe metric g

0∈ C1 such that S

M ′g0

= σ(M ′) is Einstein. Otherwise,this problem still remains open. Under these restrictions, it is reasonable tosay that there exists a 4 - dimensional globally null manifold (M, g) whose 3 -dimensional compact Riemannian hypersurface (M ′, g′) is an Einstein manifold


with a constant curvature, say k, and g′ is a Yamabe metric. Indeed, using theinformation in Section 2, one can construct null manifold (M, g) by gluing theRiemannian metric g′ with the degenerate metric g as follows:

g =

(

O1 , 1 O1 , 3

O3 , 1 g

′

)

Now consider a ruled null surface N of a 4-dimensional spacetime manifoldsM̄ of constant curvature, such that its Cauchy hypersurface, say (Σ, Σḡ), isconformal to M ′, that is, its induced metric Σḡ ∈ [g

′]. With this construction,it follows that (M = L × M ′, g ) is tangent to the null cone ∧M̄ of M̄ , that is,

Tx(M) ∩ Tx(∧M̄ ) = Lx − {0} (28)

for any common point of contact x of the pair (M, M̄).

Definition 3. Let (M, M̄) be a pair of 4-dimensional globally null andspacetime manifolds, satisfying (28) and α : [a, b] → M be a null curve segment.A piecewise smooth variation f , defined by a two parameter function

f : [a, b] × (−ǫ, ǫ ) → M̄

is said to be admissible if all the neighboring curves fv : [a, b] → M̄ , given byfv(u) = f(u, v) are null for each v 6= 0 in (−ǫ, ǫ ) and f0(u) = f(u, 0) = α(u)is the null curve common to M and M̄ for all a ≤ u ≤ b.

We call u - parameter null curves f (u, v = constant), v - parameter nullcurves f (u = constant, v) and α(v) the longitudinal, the transversal and thebase curves respectively. Thus given a point x ∈ α, we have an admissible netof neighboring longitudinal and transversal curves, all of them belonging to thenull cone ∧M̄ with the single base null curve α(u) common to M . Let {Nα(x)}denote the set of all nets for all points x ∈ α. Now, consider a maximumsequence of globally null manifolds {Mi, gi } such that each Mi satisfies theequation (28) and {g′i} is a maximum sequence of unit volume Yamabe metricson each M ′i . In this way we cover the surface of the cone ∧M̄ with a global netof all its longitudinal and transversal null curves. Let ℓ(u) be a null tangentvector field of α(u). Then as α(u) moves on the null cone ∧M̄ , it will generate aruled null 2-surface N , which establishes a link between the pair (M, M̄) withits common u-parameter curve α(u). Observe that, based on Proposition 3, itis possible to consider a maximal sequence of special parameters {pi} such thateach longitudinal curve is a null geodesic. With this possibility, all longitudinalcurves of the global net are geodesics, which means that f is a 1-parameter

408 K.L. Duggal

family of null geodesics. Consequently, N is a null geodesic string in the senseof Schild [10]. Also, see a recent paper by Low [8] in which he has studied someaspects of the causal structure of 4 - dimensional Lorentz manifold with respectto its space of null geodesics.

5. Spacelike 2-surfaces

Let (Rn+21 , ḡ) be a Minkowski spacetime with Lorentz metric ḡ of signature(− + . . . + ) and local coordinates (x0 , . . . , xn, xn+1 ). Consider a hypersurface(M, g) of Rn+21 , defined by

xA = fA(u0, . . . , un) ; rank

[

∂fA

∂uα

]

= n + 1 ,

where A ∈ {0, . . . , n + 1} , α ∈ {0, . . . , n} and {fA} are smooth functions on acoordinate neighborhood U ⊂ M . Set

DA =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∂f0

∂u0...

∂fA−1

∂u0∂fA+1

∂u0...

∂fn+1

∂u0. . . .

. . . .

. . . .

∂f0

∂un...

∂fA−1

∂un∂fA+1

∂un...

∂fn+1

∂un

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

.

Proposition 6. A hypersurface M of Rn+21 is lightlike, if and only if oneach U , functions {fA} satisfy

(D0)2 =

n+1∑

a = 1

(Da)2 . (29)

In this case, the distribution Rad TM = TM⊥ is spanned by

ξ = D0∂

∂x0+

n+1∑

a=1

(−1)a−1 Da∂

∂xa. (30)

Proof. The natural frames field on U is given by

∂

∂uα=

∂fA

∂uα∂

∂xA, α ∈ {0, . . . , n} .


Then it is easy to check that ξ, given by (30), belongs to Γ(TM⊥|U). Hence, M

is lightlike if and only if ḡ(ξ, ξ) = 0, which is equivalent with (29).

Note that a vector field V of TRn+21 defined by V = −D0 ∂

∂x0, is nowhere

tangent to M , since ḡ(V, ξ) = (D0)2. Choose a null vector field ℓ, given by

ℓ = (D0)−2 {V +1

2ξ }. (31)

It follows that ḡ ( ℓ, ξ ) = 1. Denote by NM the vector bundle spanned by ℓ.Thus we have the following decomposition

T (Rn+21 ) = TM ⊕ NM

= S(TM) ⊕ RadTM ⊕ NM, (32)

where NM is called a null transversal vector bundle over M . At this point weneed the following information on the differential geometry of a hypersurface ofany semi-Riemannian manifold (M̄, ḡ). Let ∇̄ and ∇ be metric and torsion-freelinear connections on M̄ and M respectively. Using the first form of (32), weobtain

∇̄X Y = ∇X Y + H(X, Y ) ,

∇̄X V = −AV X + ∇tX V , (33)

for any X,Y ∈ Γ(TM) and V ∈ Γ(tr(TM)), where ∇X Y and AV X belong toΓ(TM) while H(X, Y ) and ∇tX V belong to Γ(NM)). Define a local symmetricF(U)-bilinear form B and a local 1-form τ on U by

B(X, Y ) = ḡ(H(X, Y ), ξ), τ(X) = ḡ(∇tXℓ, ξ), A

H(X, Y ) = B(X, Y )ℓ , ∇tX ℓ = τ(X) ℓ

for all X,Y ∈ Γ(TM|U ). Hence, on U , the equations (33) become

∇̄X Y = ∇X Y + B(X, Y )ℓ ,

∇̄X ℓ = −Aℓ X + τ(X)ℓ , (34)

where we call H and B global and local second fundamental forms of M and(33)-(34) the global and local Gauss and Weingarten equations respectively.Next, if P denotes the projection morphism of TM on S(TM) with respectto the decomposition (2) of the Section 2, then a procedure similar to aboveprovides the following equations

∇X PY =∗∇X PY + B

′(X, PY )ξ ,

∇X ξ = −∗Aξ X − τ(X) ξ , (35)

410 K.L. Duggal

respectively, which are local Gauss and Weingarten equations for S(TM),

where B′ denotes its local second fundamental form,∗∇ is linear connection on

S(TM) and∗Aξ X ∈ Γ(S(TM)). �

Theorem 4. The screen distribution S(TM) on any lightlike hypersurfaceM of a Minkowski spacetime Rn+21 is integrable.

Proof. Consider a screen distribution S(TM) on M and take X,Y ∈Γ(S(TM)). Then, taking into account that ∇̄ is the Levi-Civita connectionon Rn+21 , using Gauss equation (34) and (31) we obtain

ḡ([X,Y ], ℓ) = (D0)−1 ḡ

(

∇̄XY − ∇̄Y X,∂

∂x0

)

= −(D0)−1{

ḡ

(

X, ∇̄Y∂

∂x0

)

− ḡ

(

Y, ∇̄X∂

∂x0

)}

= 0.

Hence, [X,Y ] ∈ Γ(S(TM)), that is, S(TM) is integrable. �

Note. Since Theorem 4 holds for any lightlike manifold, we conclude thatthere always exists an integrable screen distribution S(TM) of a globally nullmanifold M ⊂ Rn+21 , which we now assume.

Suppose (M ′, g′) is a 2-surface of M , where g′ is the induced tensor field onM ′ of g, i.e., g′ (X, Y ) = g(X, Y ), for all X,Y ∈ Γ(TM ′). It follows fromSection 2 that M ′ is a spacelike 2-surface of M if

TM ′ ⊆ S(TM) ⇐⇒ RadTM ∩ TM ′ = ∅

which we assume. At this point, for physical reasons, we restrict to n = 2 sothat dim(M) = 3 and TM ′ = S(TM). This means that M ′ is a leaf of anintegrable screen distribution S(TM) of M .

Theorem 5. Suppose (M ′, g′) is a spacelike 2-surface of a 3-dimensionalglobally null manifold M ⊂ R41. Then, the geometry of M essentially reducesto the Riemannian geometry of its 2-surface M ′.

Proof. Since M has an integrable screen distribution S(TM), based onRemark 1 (Section 3) the degenerate metric g can be identified with the Rie-mannian metric g′ of its 2-surface M ′. Therefore, M and M ′ have the same


first fundamental forms. For a relation between their second fundamental forms,using the second set of (34) and (31), for X ∈ Γ(TM), we get

τ(X) = ḡ(∇̄X ℓ, ξ) =1

2(D0)−2 ḡ(∇̄X ξ, ξ) = 0 . (36)

Hence, the two Weingarten equations of (34) and (35) reduce to

∇̄X ℓ = −Aℓ X , ∇X ξ = −∗Aξ ,

respectively. Also, it follows from (33) and (34) that

B(X, ξ) = ḡ(∇̄X

ξ, ξ) = 0.

Using this and (31) we obtain

∇̄X ℓ =1

2∇̄X ξ =

1

2∇X ξ = −

1

2

∗Aξ X .

Now using above reduced Weingarten equations we get

Aℓ =1

2

∗Aξ .

As a consequence of above relation, we see that the second fundamental formsB and B′ of M and S(TM), respectively, are related to

B′(X, PY ) =1

2B(X, PY ) for all X,Y ∈ Γ(TM) . (37)

Denote by H ′ the global second fundamental form of M ′. Then,

∇̄X Y = ∇′X Y + H

′(X, Y ) for all X,Y ∈ Γ(TM ′) .

where ∇′ is a torsion-free linear connection on M ′. By using Gauss equations(34)-(35) and (37), we obtain

∇̄X Y = ∇′X Y + B(X, Y )U ,

where U = {1

2ξ + ℓ } is a spacelike unit vector field in (TM ′)⊥. Hence,

H ′(X, Y ) = B(X, Y )U, X, Y ∈ Γ(TM ′) (38)

is a relation between the second fundamental forms B and H ′ of M and M ′,respectively. Since τ vanishes and ξ (and therefore, ℓ) is a global vector field

412 K.L. Duggal

(see Definition 1) on M (which means that U is also globally defined), (38) is aglobal relation. This result and identical first fundamental forms of M and M ′

imply that the geometry of M reduces to the geometry of M ′. �

Remark 3. The results can be generalized for higher dimensions. If theMinkowski space is replaced by a semi-Euclidean space Rn+2q , the proofs ofTheorems 4 and 5 are expected to be quite involved and difficult. On the otherhand, if the Minkowski space is replaced by an arbitrary Lorentz manifold, ofdimension 4 or higher, then there is no guarantee for the existence of integrabledistribution of its lightlike hypersurface. Thus, the results do not hold for anarbitrary ambient Lorentz manifold. However, in particular, we do have severalphysical spacetimes of general relativity for which Theorems 4 and 5 hold. Herewe mention some of them.

Physical Examples. De Sitter spacetime, Schwarzchild and Robertson -Walker spacetimes [6] all have lightlike hypersurface with an integrable 2 -dimensional screen distribution whose integral manifold is a 2-sphere S2. Also,consider the following global wave solution of a spacetime, with global coordi-nate system (u, v, x, y ) and metric

ds2 = 2 du dv + [(x2 − y2)A(u) − 2xy B(u)] du2 + dx2 + dy2 ,

where A(u) and B(u) are arbitrary functions. This spacetime admits plane wavesolutions of the empty Einstein field equations [6, page 178]. It has a familyof lightlike hypersurfaces defined by u = constant. Let one such hypersurfacebe denoted by (M, g). It follows from the above metric that M has a 2 -dimensional oriented plane, with metric dΩ2 = dx2+dy2, as a leaf of a integrablescreen distribution S(TM) of M .

Finally, we propose the following problem:

Suppose (M, g) is an (n + 1)-dimensional globally null hypersurface of an(n + 2)-dimensional Lorentz manifold (M̄, ḡ). Suppose M has an integrablescreen distribution S(TM), whose leaf is an n-dimensional Riemannian hyper-surface M ′ of M . Find a class of Lorentz manifolds M̄ such that the nullgeometry of M , embedded in M̄ , is same as the Riemannian geometry of M ′.

References

[1] J. Beem, P.E. Ehrlich, K.L. Easley, Global Lorentzian Geometry, MarcelDekker, Inc. New York, Second Edition (1996).


[2] Marcel Berger, Riemannian Geometry During the Second Half of the Twen-tieth Century, University Lecture Series, 17, Amer. Math. Soc. (2000).

[3] A. Besse, Einstein Manifolds, Springer Verlag, New York (1987).

[4] K.L. Duggal, Warped product of lightlike manifolds, Nonlinear Analysis:Theory, Methods and Applications, 47, No. 5 (2001), 3061-3072.

[5] K.L. Duggal, Constant scalar curvature and warped product globally nullmanifolds, Journal of Geometry and Physics, 809 (2002), 1-14.

[6] S.W. Hawking, C.F.R. Ellis, The Large Scale Structure of Spacetime, Cam-bridge University Press, Cambridge (1973).

[7] K. Ilyenko, Twistor representation of null 2-surfaces, Preprint, LNL archivNo. hep-th/0109036 (2001).

[8] R.J. Low, The space of null geodesics, Nonlinear Analysis: Theory, Meth-ods and Applications, 47, No. 5, (2001), 3005-3017.

[9] B. O’Neill, Semi -Riemannian Geometry with Applications to Relativity,Academic Press, New York (1983).

[10] A. Schild, Classical null strings, Phys. Rev. D, 16, No. 2, (1977), 1722-1726.

[11] H. Yamabe, On a deformation of Riemannian structures on compact man-ifolds, Osaka Math. J., 12 (1960), 21 - 37.

international journal of pure and applied mathematicstheorem 1. (duggal, [4]) let (m ,g) be an...

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