research article generalized projectively symmetric...

Hindawi Publishing CorporationGeometryVolume 2013 Article ID 292691 5 pageshttpdxdoiorg1011552013292691

Research ArticleGeneralized Projectively Symmetric Spaces

Dariush Latifi1 and Asadollah Razavi2

1 Department of Mathematics University of Mohaghegh Ardabili PO Box 56199-11367 Ardabil Iran2 Faculty of Mathematics and Computer Science Amirkabir University of Technology 424 Hafez AvenuePO Box 15875-4413 Tehran Iran

Correspondence should be addressed to Dariush Latifi dlatifigmailcom

Received 30 October 2012 Accepted 1 January 2013

Academic Editor Salvador Hernandez

Copyright copy 2013 D Latifi and A Razavi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We study generalized projectively symmetric spaces We first study some geometric properties of projectively symmetric spacesand prove that any such space is projectively homogeneous and under certain conditions the projective curvature tensor vanishesThen we prove that given any regular projective s-space (119872 nabla) there exists a projectively related connection nabla such that (119872 nabla) isan affine s-manifold

1 Introduction

Affine and Riemannian s-manifolds were first defined in[1] following the introduction of generalized Riemanniansymmetric spaces in [2] They form a more general classthan the symmetric spaces of E Cartan More details aboutgeneralized symmetric spaces can be found in themonograph[3] Let119872 be a connectedmanifold with an affine connectionnabla and let 119860(119872nabla) be the Lie transformation group of allaffine transformation of119872 An affine transformation 119904

119909will

be called an affine symmetry at a point 119909 if 119909 is an isolatedfixed point of 119904

119909 An affine manifold (119872 nabla) will be called an

affine s-manifold if there is a differentiable mapping 119904 119872 rarr

119860(119872nabla) such that for each 119909 isin 119872 119904119909is an affine symmetry

at 119909In [4] Podesta introduced the notion of a projectively

symmetric space in the following sense Let (119872 nabla) be a con-nected 119862

infin manifold with an affine torsion free connectionnabla on its tangent bundle (119872 nabla) is said to be projectivelysymmetric if for every point 119909 of119872 there is an involutive pro-jective transformation 119904

119909of119872 fixing 119909 and whose differential

at 119909 is minusId The assignment of a symmetry 119904119909at each point 119909

of119872 can be viewed as amap 119904 119872 rarr 119875(119872nabla) and119875(119872nabla)

can be topologised so that it is a Lie transformation group Inthe above definition however no further assumption on 119904 ismade even continuity is not assumed

In this paper we define and state prerequisite results onprojective structures and define projective symmetric spacesdue toPodesta Then we generalize them to define projec-tive s-manifolds as manifolds together with more generalsymmetries and consider the cases where they are essentialor inessential A projective s-manifold is called inessentialif it is projectively equivalent to an affine s-manifold andessential otherwise We prove that these spaces are naturallyhomogeneous and moreover under certain conditions theprojective curvature tensor vanishes Later we define regularprojective s-manifolds and prove that they are inessential

2 Preliminaries

Let 119872 be a connected real 119862infin manifold whose tangentbundle119879119872 is endowed with an affine torsion free connectionnabla We recall that a diffeomorphism 119904 of 119872 is said to beprojective transformation if 119904 maps geodesics into geodesicswhen the parametrization is disregarded [5] equivalently 119904 isprojective if the pull back 119904lowastnabla of the connection is projectivelyrelated tonabla that is if there exists a global 1-form120587 on119872 suchthat

119904lowast

nabla119883119884 = nabla

119883119884 + 120587 (119883)119884 + 120587 (119884)119883 forall119883 119884 isin 120594 (119872) (1)

If the form 120587 vanishes identically on119872 then 119904 is said to be anaffine transformation

2 Geometry

Definition 1 (119872 nabla) is said to be projectively symmetric if forevery point 119909 in119872 there exists a projective transformation 119904

119909

with the following properties(a) 119904119909(119909) = 119909 and 119909 is an isolated fixed point of 119904

119909

(b) 119904119909is involutive

It is easy to see that conditions (a) and (b) imply (119889119904119909)119909=

minusId Moreover we recall that a projective transformation isdetermined if we fix its value at a point its differential and itssecond jet at this point [6] hence a symmetry at 119909 in119872 is notuniquely determined in general by the condition (a) and (b)

Affine symmetric spaces are affine homogeneous butin general projectively symmetric spaces are not projectivehomogeneous for more detail and examples see [4 7] but iffollowing Ledger and Obata define the case of a differentiabledistribution of projective symmetries in an affine manifoldthen this happens

Let (119872 119892) be a connected Riemannian manifold Anisometry 119904

119909of (119872 119892) for which 119909 isin 119872 is an isolated fixed

point will be called a Riemannian symmetry of 119872 at 119909Clearly if 119904

119909is a symmetry of (119872 119892) at 119909 then the tangent

map 119878119909= (119889119904

119909)119909is an orthogonal transformation of 119879

119909119872

having no fixed vectors (with the exception of 0) An s-structure on (119872 119892) is a family 119904

119909| 119909 isin 119872 of symmetries of

(119872 119892)A Riemannian s-manifold is a Riemannian manifold

(119872 119892) together with a map 119904 119872 rarr 119868(119872 119892) such that foreach 119909 isin 119872 the image 119904

119909is a Riemannian symmetry at 119909

For any affine manifold (119872 nabla) let 119860(119872nabla) denote theLie group of all affine transformation of (119872 nabla) An affinetransformation 119904

119909isin 119860(119872nabla) for which 119909 isin 119872 is an

isolated fixed point will be called an affine symmetry at 119909 Anaffine s-manifold is an affine manifold (119872 nabla) together with adifferentiable mapping 119904 119872 rarr 119860(119872nabla) such that for each119909 isin 119872 the image 119904

119909is an affine symmetry at 119909

Let119872 be an affine s-manifold Since 119904 119872 rarr 119860(119872nabla)

is assumed to be differentiable the tensor field 119878 of type (11)defined by 119878

119909= (119889119904

119909)119909for each 119909 isin 119872 is differentiable

The tensor field 119878 is defined similarly for a Riemannian s-manifold although it may not be smooth For either affine orRiemannian s-manifolds we call 119878 the symmetry tensor field

Following [3] an s-structure 119904119909 is called regular if for

every pair of points 119909 119910 isin 119872 as follows

119904119909∘ 119904119910= 119904119911∘ 119904119909 where 119911 = 119904

119909(119910) (2)

3 Projective s-Space

Let119872 be a connected manifold with an affine connection nablaand let 119875(119872nabla) bethe group of all projective transformationsof119872

Definition 2 A projective transformation 119904119909will be called a

projective symmetry or simply a symmetry at the point 119909 if119909 is an isolated fixed point of 119904

119909and (119889119904

119909)119909= 119878 does not leave

any nonzero vector fixed

Definition 3 A connected affine manifold (119872 nabla) will becalled a projective s-manifold or simply ps-manifold if for

each 119909 isin 119872 there is a projective symmetry 119904119909 such that the

mapping 119904 119872 rarr 119875(119872nabla) 119909 997891rarr 119904119909is smooth

A symmetry 119904119909will be called a symmetry of order 119896 at 119909

if there exist a positive integer 119896 such that 119904119896119909= Id and 119872

will be called ps-manifold of order 119896 if 119896 is the least positivenumber such that each symmetry is of order 119896 Evidentlyevery ps-manifold of order 2 is a projective symmetric space

Lemma 4 Let 119866 be a topological transformation group actingon a connected topological space119872 if for each point119909 in119872 the119866-orbit of 119909 contains a neighborhood of 119909 then 119866 is transitiveon119872

Proof Since 119866 is transitive on each orbit for each 119909 the 119866-orbit 119866(119909) of 119909 is open by our assumption The complement119862(119909) of 119866(119909) in119872 is also open being a union of orbits Thus119866(119909) is open and closed It is nonempty and therefore coin-cides with the connected space119872 thus 119866 is transitive

Theorem 5 If119872 is a 119901119904-manifold then 119875(119872nabla) is transitiveon119872

Proof We fix a point 1199090isin 119872 and consider the 119862infin map ℎ

119872 rarr 119872 given by ℎ(119909) = 119904119909(1199090) since 119904

119909(119909) = 119909 for every

119909 in 119872 the differential (119889ℎ)1199090

of ℎ at the point 1199090is given

by (119889ℎ)1199090= 119868 minus 119878

1199090 where 119878

1199090is the differential of 119904

1199090at 1199090

(119889ℎ)1199090is nonsingular because no eigenvalue of 119878

1199090is equal to

1 Hence ℎ is a diffeomorphism on some neighborhood 119882

of 1199090in 119872 and ℎ(119882) is a neighborhood of 119909

0contained in

the 119875(119872nabla)-orbit 119875(119872nabla)1199090of 1199090 therefore from the above

lemma 119875(119872nabla) is transitive

Definition 6 Let 119872 be a ps-manifold since 119904 119872 rarr

119875(119872nabla) is assumed to be differentiable the tensor field 119878 oftype (1 1) defined by 119878

119909= (119889119904

119909)119909is differentiable we call 119878

the symmetry tensor field

Lemma 7 If 119904119909is a projective symmetry of (119872 nabla) then there

exists a connection nabla projectively equivalent with nabla which is119904119909-invariant

Proof Since 119904119909is a projective symmetry of (119872 nabla) then there

is a 1-form 120572 on119872 such that

(119904lowast

119909nabla)119883119884 = nabla

119883119884 + 120572 (119883)119884 + 120572 (119884)119883 (3)

We are looking for a connection nabla with the followingproperties


119883119884 + 120587 (119883)119884 + 120587 (119884)119883 (4)

As nabla should be 119904119909-invariant we need

(119904lowast

119909nabla)119883

119884 = nabla119883119884 (5)

Geometry 3

that is 119904119909is an affine transformation of (119872 nabla) We have

(119904lowast

119909nabla)119883

119884 = 119904minus1

119909lowast

nabla119904119909lowast119883119904119909lowast119884

= 119904minus1

119909lowast

(nabla119904119909lowast119883119904119909lowast119884 + 120587 (119904

119909lowast119883) 119904119909lowast119884

+120587 (119904119909lowast119884) 119904119909lowast119883)

= (119904lowast

119909nabla)119883119884 + 120587 (119904

119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

(6)

It follows from (3)

(119904lowast

119909nabla)119883

119884 = nabla119883119884 + 120572 (119883)119884 + 120572 (119884)119883

+ 120587 (119904119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

(7)

From (5) we have

nabla119883119884 + 120572 (119883)119884 + 120572 (119884)119883 + 120587 (119904

119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

= nabla119883119884 + 120587 (119883)119884 + 120587 (119884)119883

(8)

thus it is enough to have for every vector field 119885 as follows

120572 (119885) + 120587 (119904119909lowast119885) = 120587 (119885) (9)

which is equivalent to

120587 (119885 minus 119904119909lowast119885) = 120572 (119885) (10)

or simply

120587 ∘ (119868 minus 119904119909lowast)119885 = 120572 (119885) (11)

since 119904119909is symmetry then 119868minus119904

119909is invertible hence we obtain

120587 = 120572 ∘ (119868 minus 119904119909lowast)minus1 (12)

thus if we choose 120587 as (12) then (4) and (5) are true and nabla isthe required connection

So it would be convenient to introduce the followingdefinition for connection nabla and 1-form 120587 = 120572 ∘ (119868 minus 119904

119909lowast)minus1

Definition 8 Let (119872 nabla) be a ps-manifold and let 119904119909be the

projective symmetry at the point 119909Then we call the associateconnection nabla the fundamental connection of 119904

119909 Also the 1-

form


will be called the fundamental 1-form of 119904119909 where 120572 is the

1-form on119872 such that

119904lowast


119883119884 + 120587 (119883)119884 + 120587 (119884)119883 forall119883 119884 isin 120594 (119872) (14)

Definition 9 The projective curvature tensor of (119872 nabla) isdefined as follows [5 8]

119882119894

119895119896119897= Π119894

119895119896119897minus

1

119899 minus 1(120575119894

119896Π119895119897minus 120575119894

119897Π119895119896) (15)

where

Π119894

119895119896= Γ119894

119895119896minus

2

119899 + 1120575119894

(119895Γ119897

119896)119897

Π119894

119895119896119897= 120597119896Π119894

119895119897minus 120597119897Π119894

119895119896+ Πℎ

119895119897Π119894

ℎ119896minus Πℎ

119895119896Π119894

ℎ119897 Π119895119896= Πℎ

119895ℎ119896

(16)

The projective curvature tensor119882 is invariant with respect toprojective transformations [5 8]

Theorem 10 In a ps-manifold (119872 nabla) let 119904119909be a symmetry

and let nabla be the fundamental connection of 119904119909 if nabla119878 = 0 that

is ((nabla119883119878)(119884) = 120587(119884)119878(119883) minus 120587(119878(119884))119883) then (nabla119882)

119909= 0

Proof Let 119904 119872 rarr 119875(119872nabla) be the 119901119904-structure andnabla119878 = 0Let 119883119884 119885 isin 119879

119909119872 be tangent vectors and let 120596 isin 119879

lowast

119909119872

be a covector at 119909 isin 119872 By parallel translation along eachgeodesics through 119909 119883 119884 119885 and 120596 can be extended tolocal vector fields 119883 119885 and with vanishing covariantderivative with respect to nabla at 119909 Because 119878 is parallel thelocal vector fields 119878119883 119878 119878119885 and 119878lowast have also vanishingcovariant derivative at 119909 (Here 119878

lowast denotes the transposemap to 119878) As 119882 is invariant with respect to the projectivetransformation 119904

119901 119901 isin 119872 we have

119882(119878lowast

119883 119885) = 119882( 119878119883 119878 119878119885) (17)

Now we show that nabla119882(119878lowast

119883 119885 119880) and nabla119882( 119878119883

119878 119878119885 119878119880) are equal at 119909 These are equal if and only if(119878lowast

120596)(nabla119880119882(119883 119885)) and 120596(nabla

119878119880119882(119878119883 119878 119878119885)) are equal

which follows from the assumption on nabla That is

nabla119882(119878lowast

119909120596119883 119884 119885 119880) = nabla119882(120596 119878

119909119883 119878119909119884 119878119909119885 119878119909119880) (18)

or

nabla119882(120596119883 119884 119885 119880) = nabla119882(119878lowastminus1

120596 119878119883 119878119884 119878119885 119878119880) (19)

Differentiating covariantly (17) with respect to nabla in thedirection of 119878119880 at 119909 and using (19) we get

nabla119882(120596119883 119884 119885 119878119880) = nabla119882(120596119883 119884 119885 119880) (20)

thus

(nabla119882)119909

(120596119883 119884 119885 (119868 minus 119878)119880) = 0 (21)

for all 119883119884 119885119880 isin 119879119909119872 and 120596 isin 119879

lowast

119909119872 and because (119868 minus 119878)

119909

is a nonsingular transformation we obtain

(nabla119882)119909

= 0 (22)

Theorem 11 Let (119872 nabla) be a 119901119904-manifold of dimension 119899 gt

2 if there exist two different projective symmetries 1205901 1205902

at a point 119902 of 119872 such that 1205901lowast119902

= 1205902lowast119902

and nabla119878 = 0where nabla is the fundamental connection corresponding to 120590

1

then the projective curvature tensor 119882 vanishes that is 119872 isprojectively flat

4 Geometry

Proof By a similar method used in Proposition 11 of [7] theproof follows from Lemma 7 andTheorem 10

Corollary 12 If (119872 nabla) is a ps-manifold of order 2 and twodifferent projective symmetry 120590

1 1205902can be defined at a point

119902 then119872 is projectively flat

Proof It is evident from the fact that 1205901lowast119902

= 1205902lowast119902

Proposition 13 Let (119872 nabla) be ps-manifold such that at everypoint 119909 of119872 the projective symmetry is uniquely determinedThen the linear isotropy representation 120588 119875(119872 nabla)

119909rarr

119866119871(119899 119877) is faithful for every 119909 isin 119872

Proof Since 119904119909and 119904minus1

119909both are projective symmetry at 119909

then we have 119904119909

= 119904minus1

119909 that is 1199042

119909= Id thus (119872 nabla) is

a ps-manifold of order 2 Now our assertion follows fromTheorem 11 of [7]

4 Regular Projective s-Space

Definition 14 A 119901119904-manifold (119872 nabla) is called regular 119901119904-manifold or simply 119903119901119904-manifold if for all 119901 119902 isin 119872 119904

119901∘ 119904119902=

119904119911∘ 119904119901 where 119911 = 119904

119901(119902)

Lemma 15 Let (119872 nabla) be a regular ps-manifold then the (1 1)tensor field 119878 is invariant under all symmetries 119904

119909 that is

119889119904119909(119878119883) = 119878 (119889119904

119909119883) (23)

for all119883 isin 120594(119872)

Proof Since119872 is regular ps-manifold then for all 119883 isin 119879119910119872

we have 119889(119904119909∘119904119910)119883 = 119889(119904

119911∘119904119909)119883 and so 119889119904

119909(119878119883) = 119878

119911(119889119904119909119883)

Thus 119878 is 119904119909invariant for all 119904

119909

Lemma 16 Let (119872 nabla) be a connected ps-manifold such thatat every point 119909 of 119872 the projective symmetry is uniquelydetermined then (119872 nabla) is rps-manifold

Proof Suppose 119901 119902 isin 119872 and 119911 = 119904119901(119902) then from the

uniqueness of the projective symmetry we have 119904119901∘119904119902= 119904119911∘119904119901

so (119872 nabla) is regular ps-manifold

Remark 17 A general question is to find condition underwhich given a ps-manifold (119872 nabla) there exists a projectivelyrelated connectionnabla such that (119872 nabla) is an affine s-manifoldwe shall call such spaces inessential ps-manifold and essentialotherwise

Definition 18 A ps-manifold (119872 nabla) is called inessential ps-manifold if there exists a projectively related connection nabla

such that (119872 nabla) is an affine s-manifold

Let us denote by Φℎthe 1-form corresponding to an

element ℎ of 119875(119872nabla) We want to see when (119872 nabla) isinessential in order to show that (119872 nabla) is inessential wemust find a connection nabla which is projectively related to nabla

and is invariant under all symmetries Let 119904119902be a symmetry

at 119902 we must find a one-form 120587 such that


119883119884 + 120587 (119883)119884 + 120587 (119884)119883 (24)

As 119904119902is a projective transformation for (119872 nabla) and leaves the

connection nabla invariant we find that

Φ119904119902(119883) + 120587 (119904

119902lowast119883) = 120587 (119883) forall119883 isin 120594 (119872) (25)

and hence at 119902 we have

120587|119902(119868 minus (119904

119902)lowast119902

)119883 = Φ119904119902(119883) (26)

So we define a 1-form 120587 through the following formula

120587|119909(119883) = Φ

119904119909|119909∘ (119868 minus (119904

119909)lowast119909)minus1

119883 forall119883 isin 119879119909119872 (27)

Theorem 19 Let (119872 nabla) be an rps-manifold then (119872 nabla) isinessential

Proof We define a torsion free affine connection nabla projec-tively related to nabla through the fundamental 1-form of 119904

119909 120587 as

follows

120587|119909(119883) = Φ

119904119909|119909∘ (119868 minus (119904


119883 forall119883 isin 119879119909119872 (28)

and prove that the connection nabla is invariant under all thesymmetries of119872

Let 119904119902be a symmetry at 119902 of 119872 the condition that nabla is

invariant under 119904119902is equivalent to

120587 (119883) minus 120587 (119904119902lowast119883) = Φ

119904119902(119883) (29)

We verify (29) at a point 119901 of119872 we have to prove that by (28)

Φ119904119901|119901

∘ (119868 minus (119904119901)lowast119901

)

minus1

119883 minus 120587119904119902(119901)

(119904119902lowast|119901

119883) = Φ119904119902|119901

(119883) (30)

so if we put 119911 = 119904119902(119901) (30) reduces to

Φ119904119901|119901

∘ (119868 minus (119904119901)lowast119901)minus1

119883 minus Φ119904119911|119911


(119904119902lowast|119901

119883)

= Φ119904119902|119901

(119883)

(31)

But since 119904119902∘ 119904119901= 119904119911∘ 119904119902 we have

Φ119904119901(119884) + Φ

119904119902(119904119901lowast119884) = Φ

119904119902(119884) + Φ

119904119911(119904119902lowast119884) (32)

Now evaluate (32) at 119901 and let 119884 = (119868 minus (119904119901)lowast119901

)minus1

119883 then as119904119902∘ 119904119901= 119904119911∘ 119904119902 we have (31) and we are done

Remark 20 The authors have studied Finsler homogeneousand symmetric spaces [9] recently Habibi and the secondauthor generalized them to Finsler s-manifolds and weaklyFinsler symmetric spaces [10 11] Therefore these conceptscan bemixed and findmore generalizations which will be thecontent of other papers

Geometry 5

Acknowledgment

The authors would like to thank the anonymous referees fortheir suggestions and comments which helped in improvingthe paper

References

[1] A J Ledger and M Obata ldquoAffine and Riemannian 119904-manifoldsrdquo Journal of Differential Geometry vol 2 pp 451ndash4591968

[2] A J Ledger ldquoEspaces de Riemann symetriques generalisesrdquoComptes Rendus de lrsquoAcademie des Sciences vol 264 pp A947ndashA948 1967

[3] O Kowalski Generalized Symmetric Spaces vol 805 of LectureNotes in Mathematics Springer Berlin Germany 1980

[4] F Podesta ldquoProjectively symmetric spacesrdquoAnnali di Matemat-ica Pura ed Applicata Serie Quarta vol 154 pp 371ndash383 1989

[5] A V Aminova ldquoProjective transformations of pseudo-Riemannian manifoldsrdquo Journal of Mathematical Sciences vol113 no 3 pp 367ndash470 2003

[6] S Kobayashi Transformation Groups in Differential GeometrySpringer Berlin Germany 1980

[7] F Podesta ldquoA class of symmetric spacesrdquo Bulletin de la SocieteMathematique de France vol 117 no 3 pp 343ndash360 1989

[8] L P Eisenhart Non-Riemannian Geometry vol 8 of Ameri-can Mathematical Society Colloquium Publications AmericanMathematical Society 1927

[9] D Latifi and A Razavi ldquoOn homogeneous Finsler spacesrdquoReports on Mathematical Physics vol 57 no 3 pp 357ndash3662006

[10] P Habibi and A Razavi ldquoOn generalized symmetric Finslerspacesrdquo Geometriae Dedicata vol 149 pp 121ndash127 2010

[11] P Habibi and A Razavi ldquoOn weakly symmetric Finsler spacesrdquoJournal of Geometry and Physics vol 60 no 4 pp 570ndash5732010

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2 Geometry

Definition 1 (119872 nabla) is said to be projectively symmetric if forevery point 119909 in119872 there exists a projective transformation 119904

119909

with the following properties(a) 119904119909(119909) = 119909 and 119909 is an isolated fixed point of 119904

119909

(b) 119904119909is involutive

It is easy to see that conditions (a) and (b) imply (119889119904119909)119909=

minusId Moreover we recall that a projective transformation isdetermined if we fix its value at a point its differential and itssecond jet at this point [6] hence a symmetry at 119909 in119872 is notuniquely determined in general by the condition (a) and (b)

Affine symmetric spaces are affine homogeneous butin general projectively symmetric spaces are not projectivehomogeneous for more detail and examples see [4 7] but iffollowing Ledger and Obata define the case of a differentiabledistribution of projective symmetries in an affine manifoldthen this happens

Let (119872 119892) be a connected Riemannian manifold Anisometry 119904

119909of (119872 119892) for which 119909 isin 119872 is an isolated fixed

point will be called a Riemannian symmetry of 119872 at 119909Clearly if 119904

119909is a symmetry of (119872 119892) at 119909 then the tangent

map 119878119909= (119889119904

119909)119909is an orthogonal transformation of 119879

119909119872

having no fixed vectors (with the exception of 0) An s-structure on (119872 119892) is a family 119904

119909| 119909 isin 119872 of symmetries of

(119872 119892)A Riemannian s-manifold is a Riemannian manifold

(119872 119892) together with a map 119904 119872 rarr 119868(119872 119892) such that foreach 119909 isin 119872 the image 119904

119909is a Riemannian symmetry at 119909

For any affine manifold (119872 nabla) let 119860(119872nabla) denote theLie group of all affine transformation of (119872 nabla) An affinetransformation 119904

119909isin 119860(119872nabla) for which 119909 isin 119872 is an

isolated fixed point will be called an affine symmetry at 119909 Anaffine s-manifold is an affine manifold (119872 nabla) together with adifferentiable mapping 119904 119872 rarr 119860(119872nabla) such that for each119909 isin 119872 the image 119904

119909is an affine symmetry at 119909

Let119872 be an affine s-manifold Since 119904 119872 rarr 119860(119872nabla)

is assumed to be differentiable the tensor field 119878 of type (11)defined by 119878

119909= (119889119904

119909)119909for each 119909 isin 119872 is differentiable

The tensor field 119878 is defined similarly for a Riemannian s-manifold although it may not be smooth For either affine orRiemannian s-manifolds we call 119878 the symmetry tensor field

Following [3] an s-structure 119904119909 is called regular if for

every pair of points 119909 119910 isin 119872 as follows

119904119909∘ 119904119910= 119904119911∘ 119904119909 where 119911 = 119904

119909(119910) (2)

3 Projective s-Space

Let119872 be a connected manifold with an affine connection nablaand let 119875(119872nabla) bethe group of all projective transformationsof119872

Definition 2 A projective transformation 119904119909will be called a

projective symmetry or simply a symmetry at the point 119909 if119909 is an isolated fixed point of 119904

119909and (119889119904

119909)119909= 119878 does not leave

any nonzero vector fixed

Definition 3 A connected affine manifold (119872 nabla) will becalled a projective s-manifold or simply ps-manifold if for

each 119909 isin 119872 there is a projective symmetry 119904119909 such that the

mapping 119904 119872 rarr 119875(119872nabla) 119909 997891rarr 119904119909is smooth

A symmetry 119904119909will be called a symmetry of order 119896 at 119909

if there exist a positive integer 119896 such that 119904119896119909= Id and 119872

will be called ps-manifold of order 119896 if 119896 is the least positivenumber such that each symmetry is of order 119896 Evidentlyevery ps-manifold of order 2 is a projective symmetric space

Lemma 4 Let 119866 be a topological transformation group actingon a connected topological space119872 if for each point119909 in119872 the119866-orbit of 119909 contains a neighborhood of 119909 then 119866 is transitiveon119872

Proof Since 119866 is transitive on each orbit for each 119909 the 119866-orbit 119866(119909) of 119909 is open by our assumption The complement119862(119909) of 119866(119909) in119872 is also open being a union of orbits Thus119866(119909) is open and closed It is nonempty and therefore coin-cides with the connected space119872 thus 119866 is transitive

Theorem 5 If119872 is a 119901119904-manifold then 119875(119872nabla) is transitiveon119872

Proof We fix a point 1199090isin 119872 and consider the 119862infin map ℎ

119872 rarr 119872 given by ℎ(119909) = 119904119909(1199090) since 119904

119909(119909) = 119909 for every

119909 in 119872 the differential (119889ℎ)1199090

of ℎ at the point 1199090is given

by (119889ℎ)1199090= 119868 minus 119878

1199090 where 119878

1199090is the differential of 119904

1199090at 1199090

(119889ℎ)1199090is nonsingular because no eigenvalue of 119878

1199090is equal to

1 Hence ℎ is a diffeomorphism on some neighborhood 119882

of 1199090in 119872 and ℎ(119882) is a neighborhood of 119909

0contained in

the 119875(119872nabla)-orbit 119875(119872nabla)1199090of 1199090 therefore from the above

lemma 119875(119872nabla) is transitive

Definition 6 Let 119872 be a ps-manifold since 119904 119872 rarr

119875(119872nabla) is assumed to be differentiable the tensor field 119878 oftype (1 1) defined by 119878

119909= (119889119904

119909)119909is differentiable we call 119878

the symmetry tensor field

Lemma 7 If 119904119909is a projective symmetry of (119872 nabla) then there

exists a connection nabla projectively equivalent with nabla which is119904119909-invariant

Proof Since 119904119909is a projective symmetry of (119872 nabla) then there

is a 1-form 120572 on119872 such that

(119904lowast

119909nabla)119883119884 = nabla

119883119884 + 120572 (119883)119884 + 120572 (119884)119883 (3)

We are looking for a connection nabla with the followingproperties


119883119884 + 120587 (119883)119884 + 120587 (119884)119883 (4)

As nabla should be 119904119909-invariant we need

(119904lowast

119909nabla)119883

119884 = nabla119883119884 (5)

Geometry 3


(119904lowast

119909nabla)119883

119884 = 119904minus1

119909lowast


= 119904minus1

119909lowast


119909lowast119883) 119904119909lowast119884

+120587 (119904119909lowast119884) 119904119909lowast119883)

= (119904lowast

119909nabla)119883119884 + 120587 (119904

119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

(6)

It follows from (3)

(119904lowast

119909nabla)119883

119884 = nabla119883119884 + 120572 (119883)119884 + 120572 (119884)119883

+ 120587 (119904119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

(7)

From (5) we have

nabla119883119884 + 120572 (119883)119884 + 120572 (119884)119883 + 120587 (119904

119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

= nabla119883119884 + 120587 (119883)119884 + 120587 (119884)119883

(8)


120572 (119885) + 120587 (119904119909lowast119885) = 120587 (119885) (9)


120587 (119885 minus 119904119909lowast119885) = 120572 (119885) (10)

or simply

120587 ∘ (119868 minus 119904119909lowast)119885 = 120572 (119885) (11)






119909lowast)minus1



119909 Also the 1-

form




119904lowast


119883119884 + 120587 (119883)119884 + 120587 (119884)119883 forall119883 119884 isin 120594 (119872) (14)


119882119894

119895119896119897= Π119894

119895119896119897minus

1

119899 minus 1(120575119894

119896Π119895119897minus 120575119894

119897Π119895119896) (15)

where

Π119894

119895119896= Γ119894

119895119896minus

2

119899 + 1120575119894

(119895Γ119897

119896)119897

Π119894

119895119896119897= 120597119896Π119894

119895119897minus 120597119897Π119894

119895119896+ Πℎ

119895119897Π119894


119895119896Π119894

ℎ119897 Π119895119896= Πℎ

119895ℎ119896

(16)





119909= 0



lowast

119909119872



119901 119901 isin 119872 we have

119882(119878lowast

119883 119885) = 119882( 119878119883 119878 119878119885) (17)


119883 119885 119880) and nabla119882( 119878119883


120596)(nabla119880119882(119883 119885)) and 120596(nabla

119878119880119882(119878119883 119878 119878119885)) are equal



119909120596119883 119884 119885 119880) = nabla119882(120596 119878

119909119883 119878119909119884 119878119909119885 119878119909119880) (18)

or


120596 119878119883 119878119884 119878119885 119878119880) (19)


nabla119882(120596119883 119884 119885 119878119880) = nabla119882(120596119883 119884 119885 119880) (20)

thus

(nabla119882)119909

(120596119883 119884 119885 (119868 minus 119878)119880) = 0 (21)


lowast


119909


(nabla119882)119909

= 0 (22)




= 1205902lowast119902


1


4 Geometry






= 1205902lowast119902


119909rarr





= 119904minus1

119909 that is 1199042

119909= Id thus (119872 nabla) is




119901∘ 119904119902=

119904119911∘ 119904119901 where 119911 = 119904

119901(119902)


119909 that is

119889119904119909(119878119883) = 119878 (119889119904

119909119883) (23)

for all119883 isin 120594(119872)


we have 119889(119904119909∘119904119910)119883 = 119889(119904

119911∘119904119909)119883 and so 119889119904

119909(119878119883) = 119878

119911(119889119904119909119883)


119909













119883119884 + 120587 (119883)119884 + 120587 (119884)119883 (24)



Φ119904119902(119883) + 120587 (119904



120587|119902(119868 minus (119904

119902)lowast119902

)119883 = Φ119904119902(119883) (26)


120587|119909(119883) = Φ

119904119909|119909∘ (119868 minus (119904


119883 forall119883 isin 119879119909119872 (27)



119909 120587 as

follows

120587|119909(119883) = Φ

119904119909|119909∘ (119868 minus (119904


119883 forall119883 isin 119879119909119872 (28)




120587 (119883) minus 120587 (119904119902lowast119883) = Φ

119904119902(119883) (29)


Φ119904119901|119901

∘ (119868 minus (119904119901)lowast119901

)

minus1

119883 minus 120587119904119902(119901)

(119904119902lowast|119901

119883) = Φ119904119902|119901

(119883) (30)


Φ119904119901|119901


119883 minus Φ119904119911|119911


(119904119902lowast|119901

119883)

= Φ119904119902|119901

(119883)

(31)

But since 119904119902∘ 119904119901= 119904119911∘ 119904119902 we have

Φ119904119901(119884) + Φ

119904119902(119904119901lowast119884) = Φ

119904119902(119884) + Φ

119904119911(119904119902lowast119884) (32)


)minus1



Geometry 5

Acknowledgment


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 3


(119904lowast

119909nabla)119883

119884 = 119904minus1

119909lowast


= 119904minus1

119909lowast


119909lowast119883) 119904119909lowast119884

+120587 (119904119909lowast119884) 119904119909lowast119883)

= (119904lowast

119909nabla)119883119884 + 120587 (119904

119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

(6)

It follows from (3)

(119904lowast

119909nabla)119883

119884 = nabla119883119884 + 120572 (119883)119884 + 120572 (119884)119883

+ 120587 (119904119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

(7)

From (5) we have

nabla119883119884 + 120572 (119883)119884 + 120572 (119884)119883 + 120587 (119904

119909lowast119883)119884 + 120587 (119904

119909lowast119884)119883

= nabla119883119884 + 120587 (119883)119884 + 120587 (119884)119883

(8)


120572 (119885) + 120587 (119904119909lowast119885) = 120587 (119885) (9)


120587 (119885 minus 119904119909lowast119885) = 120572 (119885) (10)

or simply

120587 ∘ (119868 minus 119904119909lowast)119885 = 120572 (119885) (11)






119909lowast)minus1



119909 Also the 1-

form




119904lowast


119883119884 + 120587 (119883)119884 + 120587 (119884)119883 forall119883 119884 isin 120594 (119872) (14)


119882119894

119895119896119897= Π119894

119895119896119897minus

1

119899 minus 1(120575119894

119896Π119895119897minus 120575119894

119897Π119895119896) (15)

where

Π119894

119895119896= Γ119894

119895119896minus

2

119899 + 1120575119894

(119895Γ119897

119896)119897

Π119894

119895119896119897= 120597119896Π119894

119895119897minus 120597119897Π119894

119895119896+ Πℎ

119895119897Π119894


119895119896Π119894

ℎ119897 Π119895119896= Πℎ

119895ℎ119896

(16)





119909= 0



lowast

119909119872



119901 119901 isin 119872 we have

119882(119878lowast

119883 119885) = 119882( 119878119883 119878 119878119885) (17)


119883 119885 119880) and nabla119882( 119878119883


120596)(nabla119880119882(119883 119885)) and 120596(nabla

119878119880119882(119878119883 119878 119878119885)) are equal



119909120596119883 119884 119885 119880) = nabla119882(120596 119878

119909119883 119878119909119884 119878119909119885 119878119909119880) (18)

or


120596 119878119883 119878119884 119878119885 119878119880) (19)


nabla119882(120596119883 119884 119885 119878119880) = nabla119882(120596119883 119884 119885 119880) (20)

thus

(nabla119882)119909

(120596119883 119884 119885 (119868 minus 119878)119880) = 0 (21)


lowast


119909


(nabla119882)119909

= 0 (22)




= 1205902lowast119902


1


4 Geometry






= 1205902lowast119902


119909rarr





= 119904minus1

119909 that is 1199042

119909= Id thus (119872 nabla) is




119901∘ 119904119902=

119904119911∘ 119904119901 where 119911 = 119904

119901(119902)


119909 that is

119889119904119909(119878119883) = 119878 (119889119904

119909119883) (23)

for all119883 isin 120594(119872)


we have 119889(119904119909∘119904119910)119883 = 119889(119904

119911∘119904119909)119883 and so 119889119904

119909(119878119883) = 119878

119911(119889119904119909119883)


119909













119883119884 + 120587 (119883)119884 + 120587 (119884)119883 (24)



Φ119904119902(119883) + 120587 (119904



120587|119902(119868 minus (119904

119902)lowast119902

)119883 = Φ119904119902(119883) (26)


120587|119909(119883) = Φ

119904119909|119909∘ (119868 minus (119904


119883 forall119883 isin 119879119909119872 (27)



119909 120587 as

follows

120587|119909(119883) = Φ

119904119909|119909∘ (119868 minus (119904


119883 forall119883 isin 119879119909119872 (28)




120587 (119883) minus 120587 (119904119902lowast119883) = Φ

119904119902(119883) (29)


Φ119904119901|119901

∘ (119868 minus (119904119901)lowast119901

)

minus1

119883 minus 120587119904119902(119901)

(119904119902lowast|119901

119883) = Φ119904119902|119901

(119883) (30)


Φ119904119901|119901


119883 minus Φ119904119911|119911


(119904119902lowast|119901

119883)

= Φ119904119902|119901

(119883)

(31)

But since 119904119902∘ 119904119901= 119904119911∘ 119904119902 we have

Φ119904119901(119884) + Φ

119904119902(119904119901lowast119884) = Φ

119904119902(119884) + Φ

119904119911(119904119902lowast119884) (32)


)minus1



Geometry 5

Acknowledgment


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 Geometry






= 1205902lowast119902


119909rarr





= 119904minus1

119909 that is 1199042

119909= Id thus (119872 nabla) is




119901∘ 119904119902=

119904119911∘ 119904119901 where 119911 = 119904

119901(119902)


119909 that is

119889119904119909(119878119883) = 119878 (119889119904

119909119883) (23)

for all119883 isin 120594(119872)


we have 119889(119904119909∘119904119910)119883 = 119889(119904

119911∘119904119909)119883 and so 119889119904

119909(119878119883) = 119878

119911(119889119904119909119883)


119909













119883119884 + 120587 (119883)119884 + 120587 (119884)119883 (24)



Φ119904119902(119883) + 120587 (119904



120587|119902(119868 minus (119904

119902)lowast119902

)119883 = Φ119904119902(119883) (26)


120587|119909(119883) = Φ

119904119909|119909∘ (119868 minus (119904


119883 forall119883 isin 119879119909119872 (27)



119909 120587 as

follows

120587|119909(119883) = Φ

119904119909|119909∘ (119868 minus (119904


119883 forall119883 isin 119879119909119872 (28)




120587 (119883) minus 120587 (119904119902lowast119883) = Φ

119904119902(119883) (29)


Φ119904119901|119901

∘ (119868 minus (119904119901)lowast119901

)

minus1

119883 minus 120587119904119902(119901)

(119904119902lowast|119901

119883) = Φ119904119902|119901

(119883) (30)


Φ119904119901|119901


119883 minus Φ119904119911|119911


(119904119902lowast|119901

119883)

= Φ119904119902|119901

(119883)

(31)

But since 119904119902∘ 119904119901= 119904119911∘ 119904119902 we have

Φ119904119901(119884) + Φ

119904119902(119904119901lowast119884) = Φ

119904119902(119884) + Φ

119904119911(119904119902lowast119884) (32)


)minus1



Geometry 5

Acknowledgment


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Geometry 5

Acknowledgment


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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