Hindawi Publishing CorporationGeometryVolume 2013, Article ID 292691, 5 pageshttp://dx.doi.org/10.1155/2013/292691

Research ArticleGeneralized Projectively Symmetric Spaces

Dariush Latifi1 and Asadollah Razavi2

1 Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 56199-11367, Ardabil, Iran2 Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue,P.O. Box 15875-4413, Tehran, Iran

Correspondence should be addressed to Dariush Latifi; dlatifi@gmail.com

Received 30 October 2012; Accepted 1 January 2013

Academic Editor: Salvador Hernandez

Copyright © 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 vanishes.Then we prove that given any regular projective s-space (𝑀, ∇), there exists a projectively related connection ∇, such that (𝑀, ∇) 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]. Let𝑀 be a connectedmanifold with an affine connection∇, and let 𝐴(𝑀,∇) be the Lie transformation group of allaffine transformation of𝑀. An affine transformation 𝑠

𝑥will

be called an affine symmetry at a point 𝑥 if 𝑥 is an isolatedfixed point of 𝑠

𝑥. An affine manifold (𝑀, ∇) will be called an

affine s-manifold if there is a differentiable mapping 𝑠 : 𝑀 →𝐴(𝑀,∇), such that for each 𝑥 ∈ 𝑀, 𝑠

𝑥is an affine symmetry

at 𝑥.In [4] Podestà introduced the notion of a projectively

symmetric space in the following sense. Let (𝑀, ∇) be a con-nected 𝐶∞ manifold with an affine torsion free connection∇ on its tangent bundle; (𝑀, ∇) is said to be projectivelysymmetric if for every point 𝑥 of𝑀 there is an involutive pro-jective transformation 𝑠

𝑥of𝑀 fixing 𝑥 and whose differential

at 𝑥 is −Id. The assignment of a symmetry 𝑠𝑥at each point 𝑥

of𝑀 can be viewed as amap 𝑠 : 𝑀 → 𝑃(𝑀,∇), and𝑃(𝑀,∇)can be topologised, so that it is a Lie transformation group. Inthe above definition, however, no further assumption on 𝑠 ismade; even continuity is not assumed.

In this paper we define and state prerequisite results onprojective structures and define projective symmetric spacesdue toPodestà. 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 𝑀 be a connected real 𝐶∞ manifold whose tangentbundle𝑇𝑀 is endowed with an affine torsion free connection∇. We recall that a diffeomorphism 𝑠 of 𝑀 is said to beprojective transformation if 𝑠 maps geodesics into geodesicswhen the parametrization is disregarded [5]; equivalently 𝑠 isprojective if the pull back 𝑠∗∇ of the connection is projectivelyrelated to∇, that is, if there exists a global 1-form𝜋 on𝑀, suchthat

𝑠∗

∇𝑋𝑌 = ∇

𝑋𝑌 + 𝜋 (𝑋)𝑌 + 𝜋 (𝑌)𝑋 ∀𝑋, 𝑌 ∈ 𝜒 (𝑀). (1)

If the form 𝜋 vanishes identically on𝑀, then 𝑠 is said to be anaffine transformation.

2 Geometry

Definition 1. (𝑀, ∇) is said to be projectively symmetric if forevery point 𝑥 in𝑀 there exists a projective transformation 𝑠

𝑥

with the following properties:(a) 𝑠𝑥(𝑥) = 𝑥 and 𝑥 is an isolated fixed point of 𝑠

𝑥.

(b) 𝑠𝑥is involutive.

It is easy to see that conditions (a) and (b) imply (𝑑𝑠𝑥)𝑥=

−Id. 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 𝑥 in𝑀 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 manifold,then this happens.

Let (𝑀, 𝑔) be a connected Riemannian manifold. Anisometry 𝑠

𝑥of (𝑀, 𝑔) for which 𝑥 ∈ 𝑀 is an isolated fixed

point will be called a Riemannian symmetry of 𝑀 at 𝑥.Clearly, if 𝑠

𝑥is a symmetry of (𝑀, 𝑔) at 𝑥, then the tangent

map 𝑆𝑥= (𝑑𝑠

𝑥)𝑥is an orthogonal transformation of 𝑇

𝑥𝑀

having no fixed vectors (with the exception of 0). An s-structure on (𝑀, 𝑔) is a family {𝑠

𝑥| 𝑥 ∈ 𝑀} of symmetries of

(𝑀, 𝑔).A Riemannian s-manifold is a Riemannian manifold

(𝑀, 𝑔) together with a map 𝑠 : 𝑀 → 𝐼(𝑀, 𝑔), such that foreach 𝑥 ∈ 𝑀 the image 𝑠

𝑥is a Riemannian symmetry at 𝑥.

For any affine manifold (𝑀, ∇) let 𝐴(𝑀,∇) denote theLie group of all affine transformation of (𝑀, ∇). An affinetransformation 𝑠

𝑥∈ 𝐴(𝑀,∇) for which 𝑥 ∈ 𝑀 is an

isolated fixed point will be called an affine symmetry at 𝑥. Anaffine s-manifold is an affine manifold (𝑀, ∇) together with adifferentiable mapping 𝑠 : 𝑀 → 𝐴(𝑀,∇), such that for each𝑥 ∈ 𝑀, the image 𝑠

𝑥is an affine symmetry at 𝑥.

Let𝑀 be an affine s-manifold. Since 𝑠 : 𝑀 → 𝐴(𝑀,∇)is assumed to be differentiable, the tensor field 𝑆 of type (1,1)defined by 𝑆

𝑥= (𝑑𝑠

𝑥)𝑥for each 𝑥 ∈ 𝑀 is differentiable.

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

Following [3] an s-structure {𝑠𝑥} is called regular if for

every pair of points 𝑥, 𝑦 ∈ 𝑀 as follows:

𝑠𝑥∘ 𝑠𝑦= 𝑠𝑧∘ 𝑠𝑥, where 𝑧 = 𝑠

𝑥(𝑦). (2)

3. Projective s-Space

Let𝑀 be a connected manifold with an affine connection ∇,and let 𝑃(𝑀,∇) bethe group of all projective transformationsof𝑀.

Definition 2. A projective transformation 𝑠𝑥will be called a

projective symmetry or simply a symmetry at the point 𝑥, if𝑥 is an isolated fixed point of 𝑠

𝑥and (𝑑𝑠

𝑥)𝑥= 𝑆 does not leave

any nonzero vector fixed.

Definition 3. A connected affine manifold (𝑀, ∇) will becalled a projective s-manifold or simply ps-manifold if for

each 𝑥 ∈ 𝑀 there is a projective symmetry 𝑠𝑥, such that the

mapping 𝑠 : 𝑀 → 𝑃(𝑀,∇), 𝑥 → 𝑠𝑥is smooth.

A symmetry 𝑠𝑥will be called a symmetry of order 𝑘 at 𝑥,

if there exist a positive integer 𝑘, such that 𝑠𝑘𝑥= Id, and 𝑀

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

Lemma 4. Let 𝐺 be a topological transformation group actingon a connected topological space𝑀, if for each point𝑥 in𝑀, the𝐺-orbit of 𝑥 contains a neighborhood of 𝑥, then 𝐺 is transitiveon𝑀.

Proof. Since 𝐺 is transitive on each orbit, for each 𝑥 the 𝐺-orbit 𝐺(𝑥) of 𝑥 is open by our assumption. The complement𝐶(𝑥) of 𝐺(𝑥) in𝑀 is also open, being a union of orbits. Thus𝐺(𝑥) is open and closed. It is nonempty and therefore coin-cides with the connected space𝑀, thus 𝐺 is transitive.

Theorem 5. If𝑀 is a 𝑝𝑠-manifold, then 𝑃(𝑀,∇) is transitiveon𝑀.

Proof. We fix a point 𝑥0∈ 𝑀 and consider the 𝐶∞ map ℎ :

𝑀 → 𝑀 given by ℎ(𝑥) = 𝑠𝑥(𝑥0); since 𝑠

𝑥(𝑥) = 𝑥 for every

𝑥 in 𝑀, the differential (𝑑ℎ)𝑥0

of ℎ at the point 𝑥0is given

by (𝑑ℎ)𝑥0= 𝐼 − 𝑆

𝑥0, where 𝑆

𝑥0is the differential of 𝑠

𝑥0at 𝑥0.

(𝑑ℎ)𝑥0is nonsingular because no eigenvalue of 𝑆

𝑥0is equal to

1. Hence ℎ is a diffeomorphism on some neighborhood 𝑊of 𝑥0in 𝑀, and ℎ(𝑊) is a neighborhood of 𝑥

0contained in

the 𝑃(𝑀,∇)-orbit 𝑃(𝑀,∇)𝑥0of 𝑥0, therefore from the above

lemma 𝑃(𝑀,∇) is transitive.

Definition 6. Let 𝑀 be a ps-manifold; since 𝑠 : 𝑀 →𝑃(𝑀,∇) is assumed to be differentiable, the tensor field 𝑆 oftype (1, 1) defined by 𝑆

𝑥= (𝑑𝑠

𝑥)𝑥is differentiable, we call 𝑆

the symmetry tensor field.

Lemma 7. If 𝑠𝑥is a projective symmetry of (𝑀, ∇) then there

exists a connection ∇ projectively equivalent with ∇ which is𝑠𝑥-invariant.

Proof. Since 𝑠𝑥is a projective symmetry of (𝑀, ∇) then there

is a 1-form 𝛼 on𝑀, such that

(𝑠∗

𝑥∇)𝑋𝑌 = ∇

𝑋𝑌 + 𝛼 (𝑋)𝑌 + 𝛼 (𝑌)𝑋. (3)

We are looking for a connection ∇ with the followingproperties:

∇𝑋𝑌 = ∇

𝑋𝑌 + 𝜋 (𝑋)𝑌 + 𝜋 (𝑌)𝑋. (4)

As ∇ should be 𝑠𝑥-invariant we need

(𝑠∗

𝑥∇)𝑋

𝑌 = ∇𝑋𝑌 (5)

Geometry 3

that is, 𝑠𝑥is an affine transformation of (𝑀, ∇). We have

(𝑠∗

𝑥∇)𝑋

𝑌 = 𝑠−1

𝑥∗

∇𝑠𝑥∗𝑋𝑠𝑥∗𝑌

= 𝑠−1

𝑥∗

(∇𝑠𝑥∗𝑋𝑠𝑥∗𝑌 + 𝜋 (𝑠

𝑥∗𝑋) 𝑠𝑥∗𝑌

+𝜋 (𝑠𝑥∗𝑌) 𝑠𝑥∗𝑋)

= (𝑠∗

𝑥∇)𝑋𝑌 + 𝜋 (𝑠

𝑥∗𝑋)𝑌 + 𝜋 (𝑠

𝑥∗𝑌)𝑋.

(6)

It follows from (3)

(𝑠∗

𝑥∇)𝑋

𝑌 = ∇𝑋𝑌 + 𝛼 (𝑋)𝑌 + 𝛼 (𝑌)𝑋

+ 𝜋 (𝑠𝑥∗𝑋)𝑌 + 𝜋 (𝑠

𝑥∗𝑌)𝑋.

(7)

From (5) we have

∇𝑋𝑌 + 𝛼 (𝑋)𝑌 + 𝛼 (𝑌)𝑋 + 𝜋 (𝑠

𝑥∗𝑋)𝑌 + 𝜋 (𝑠

𝑥∗𝑌)𝑋

= ∇𝑋𝑌 + 𝜋 (𝑋)𝑌 + 𝜋 (𝑌)𝑋,

(8)

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

𝛼 (𝑍) + 𝜋 (𝑠𝑥∗𝑍) = 𝜋 (𝑍) (9)

which is equivalent to

𝜋 (𝑍 − 𝑠𝑥∗𝑍) = 𝛼 (𝑍) (10)

or simply

𝜋 ∘ (𝐼 − 𝑠𝑥∗)𝑍 = 𝛼 (𝑍) (11)

since 𝑠𝑥is symmetry, then 𝐼−𝑠

𝑥is invertible; hence we obtain

𝜋 = 𝛼 ∘ (𝐼 − 𝑠𝑥∗)−1 (12)

thus if we choose 𝜋 as (12), then (4) and (5) are true, and ∇ isthe required connection.

So it would be convenient to introduce the followingdefinition for connection ∇ and 1-form 𝜋 = 𝛼 ∘ (𝐼 − 𝑠

𝑥∗)−1.

Definition 8. Let (𝑀, ∇) be a ps-manifold, and let 𝑠𝑥be the

projective symmetry at the point 𝑥.Then we call the associateconnection ∇ the fundamental connection of 𝑠

𝑥. Also the 1-

form

𝜋 = 𝛼 ∘ (𝐼 − 𝑠𝑥∗)−1 (13)

will be called the fundamental 1-form of 𝑠𝑥, where 𝛼 is the

1-form on𝑀, such that

𝑠∗

∇𝑋𝑌 = ∇

𝑋𝑌 + 𝜋 (𝑋)𝑌 + 𝜋 (𝑌)𝑋 ∀𝑋, 𝑌 ∈ 𝜒 (𝑀). (14)

Definition 9. The projective curvature tensor of (𝑀, ∇) isdefined as follows [5, 8]:

𝑊𝑖

𝑗𝑘𝑙= Π𝑖

𝑗𝑘𝑙−

1

𝑛 − 1(𝛿𝑖

𝑘Π𝑗𝑙− 𝛿𝑖

𝑙Π𝑗𝑘), (15)

where

Π𝑖

𝑗𝑘= Γ𝑖

𝑗𝑘−

2

𝑛 + 1𝛿𝑖

(𝑗Γ𝑙

𝑘)𝑙,

Π𝑖

𝑗𝑘𝑙= 𝜕𝑘Π𝑖

𝑗𝑙− 𝜕𝑙Π𝑖

𝑗𝑘+ Πℎ

𝑗𝑙Π𝑖

ℎ𝑘− Πℎ

𝑗𝑘Π𝑖

ℎ𝑙, Π𝑗𝑘= Πℎ

𝑗ℎ𝑘.

(16)

The projective curvature tensor𝑊 is invariant with respect toprojective transformations [5, 8].

Theorem 10. In a ps-manifold (𝑀, ∇), let 𝑠𝑥be a symmetry,

and let ∇ be the fundamental connection of 𝑠𝑥, if ∇𝑆 = 0; that

is, ((∇𝑋𝑆)(𝑌) = 𝜋(𝑌)𝑆(𝑋) − 𝜋(𝑆(𝑌))𝑋), then (∇𝑊)

𝑥= 0.

Proof. Let 𝑠 : 𝑀 → 𝑃(𝑀,∇) be the 𝑝𝑠-structure and∇𝑆 = 0.Let 𝑋,𝑌, 𝑍 ∈ 𝑇

𝑥𝑀 be tangent vectors, and let 𝜔 ∈ 𝑇∗

𝑥𝑀

be a covector at 𝑥 ∈ 𝑀. By parallel translation along eachgeodesics through 𝑥, 𝑋, 𝑌, 𝑍, and 𝜔 can be extended tolocal vector fields 𝑋, �̃�, 𝑍, and �̃� with vanishing covariantderivative with respect to ∇ at 𝑥. Because 𝑆 is parallel, thelocal vector fields 𝑆𝑋, 𝑆�̃�, 𝑆𝑍, and 𝑆∗�̃� have also vanishingcovariant derivative at 𝑥. (Here 𝑆∗ denotes the transposemap to 𝑆.) As 𝑊 is invariant with respect to the projectivetransformation 𝑠

𝑝, 𝑝 ∈ 𝑀, we have

𝑊(𝑆∗

�̃�, 𝑋, �̃�, 𝑍) = 𝑊(�̃�, 𝑆𝑋, 𝑆�̃�, 𝑆𝑍). (17)

Now we show that ∇𝑊(𝑆∗�̃�, 𝑋, �̃�, 𝑍, 𝑈) and ∇𝑊(�̃�, 𝑆𝑋,𝑆�̃�, 𝑆𝑍, 𝑆𝑈) are equal at 𝑥. These are equal if and only if(𝑆∗

𝜔)(∇𝑈𝑊(𝑋, �̃�, 𝑍)) and 𝜔(∇

𝑆𝑈𝑊(𝑆𝑋, 𝑆�̃�, 𝑆𝑍)) are equal,

which follows from the assumption on ∇. That is

∇𝑊(𝑆∗

𝑥𝜔,𝑋, 𝑌, 𝑍, 𝑈) = ∇𝑊(𝜔, 𝑆

𝑥𝑋, 𝑆𝑥𝑌, 𝑆𝑥𝑍, 𝑆𝑥𝑈) (18)

or

∇𝑊(𝜔,𝑋, 𝑌, 𝑍, 𝑈) = ∇𝑊(𝑆∗−1

𝜔, 𝑆𝑋, 𝑆𝑌, 𝑆𝑍, 𝑆𝑈). (19)

Differentiating covariantly (17) with respect to ∇ in thedirection of 𝑆𝑈 at 𝑥 and using (19) we get

∇𝑊(𝜔,𝑋, 𝑌, 𝑍, 𝑆𝑈) = ∇𝑊(𝜔,𝑋, 𝑌, 𝑍, 𝑈), (20)

thus

(∇𝑊)𝑥

(𝜔,𝑋, 𝑌, 𝑍, (𝐼 − 𝑆)𝑈) = 0, (21)

for all 𝑋,𝑌, 𝑍,𝑈 ∈ 𝑇𝑥𝑀 and 𝜔 ∈ 𝑇∗

𝑥𝑀, and because (𝐼 − 𝑆)

𝑥

is a nonsingular transformation we obtain

(∇𝑊)𝑥

= 0. (22)

Theorem 11. Let (𝑀, ∇) be a 𝑝𝑠-manifold of dimension 𝑛 >2; if there exist two different projective symmetries 𝜎

1, 𝜎2

at a point 𝑞 of 𝑀, such that 𝜎1∗𝑞

= 𝜎2∗𝑞

and ∇𝑆 = 0,where ∇ is the fundamental connection corresponding to 𝜎

1,

then the projective curvature tensor 𝑊; vanishes that is, 𝑀 isprojectively flat.

4 Geometry

Proof. By a similar method used in Proposition 1.1 of [7] theproof follows from Lemma 7 andTheorem 10.

Corollary 12. If (𝑀, ∇) is a ps-manifold of order 2, and twodifferent projective symmetry 𝜎

1, 𝜎2can be defined at a point

𝑞, then𝑀 is projectively flat.

Proof. It is evident from the fact that 𝜎1∗𝑞

= 𝜎2∗𝑞

.

Proposition 13. Let (𝑀, ∇) be ps-manifold, such that at everypoint 𝑥 of𝑀 the projective symmetry is uniquely determined.Then the linear isotropy representation 𝜌 : 𝑃(𝑀, ∇)

𝑥→

𝐺𝐿(𝑛, 𝑅) is faithful for every 𝑥 ∈ 𝑀.

Proof. Since 𝑠𝑥and 𝑠−1𝑥

both are projective symmetry at 𝑥,then we have 𝑠

𝑥= 𝑠−1

𝑥; that is, 𝑠2

𝑥= Id, thus (𝑀, ∇) is

a ps-manifold of order 2. Now, our assertion follows fromTheorem 1.1 of [7].

4. Regular Projective s-Space

Definition 14. A 𝑝𝑠-manifold (𝑀, ∇) is called regular 𝑝𝑠-manifold or simply 𝑟𝑝𝑠-manifold if for all 𝑝, 𝑞 ∈ 𝑀, 𝑠

𝑝∘ 𝑠𝑞=

𝑠𝑧∘ 𝑠𝑝, where 𝑧 = 𝑠

𝑝(𝑞).

Lemma 15. Let (𝑀, ∇) be a regular ps-manifold, then the (1, 1)tensor field 𝑆 is invariant under all symmetries 𝑠

𝑥; that is

𝑑𝑠𝑥(𝑆𝑋) = 𝑆 (𝑑𝑠

𝑥𝑋), (23)

for all𝑋 ∈ 𝜒(𝑀).

Proof. Since𝑀 is regular ps-manifold, then for all 𝑋 ∈ 𝑇𝑦𝑀

we have 𝑑(𝑠𝑥∘𝑠𝑦)𝑋 = 𝑑(𝑠

𝑧∘𝑠𝑥)𝑋 and so 𝑑𝑠

𝑥(𝑆𝑋) = 𝑆

𝑧(𝑑𝑠𝑥𝑋).

Thus 𝑆 is 𝑠𝑥invariant for all 𝑠

𝑥.

Lemma 16. Let (𝑀, ∇) be a connected ps-manifold, such thatat every point 𝑥 of 𝑀 the projective symmetry is uniquelydetermined, then (𝑀, ∇) is rps-manifold.

Proof. Suppose 𝑝, 𝑞 ∈ 𝑀 and 𝑧 = 𝑠𝑝(𝑞); then from the

uniqueness of the projective symmetry, we have 𝑠𝑝∘𝑠𝑞= 𝑠𝑧∘𝑠𝑝,

so (𝑀, ∇) is regular ps-manifold.

Remark 17. A general question is to find condition underwhich, given a ps-manifold (𝑀, ∇), there exists a projectivelyrelated connection∇, such that (𝑀, ∇) is an affine s-manifold;we shall call such spaces inessential ps-manifold and essentialotherwise.

Definition 18. A ps-manifold (𝑀, ∇) is called inessential ps-manifold if there exists a projectively related connection ∇such that (𝑀, ∇) is an affine s-manifold.

Let us denote by Φℎthe 1-form corresponding to an

element ℎ of 𝑃(𝑀,∇). We want to see when (𝑀, ∇) isinessential, in order to show that (𝑀, ∇) is inessential, wemust find a connection ∇ which is projectively related to ∇

and is invariant under all symmetries. Let 𝑠𝑞be a symmetry

at 𝑞, we must find a one-form 𝜋, such that

∇𝑋𝑌 = ∇

𝑋𝑌 + 𝜋 (𝑋)𝑌 + 𝜋 (𝑌)𝑋. (24)

As 𝑠𝑞is a projective transformation for (𝑀, ∇) and leaves the

connection ∇ invariant, we find that

Φ𝑠𝑞(𝑋) + 𝜋 (𝑠

𝑞∗𝑋) = 𝜋 (𝑋) ∀𝑋 ∈ 𝜒 (𝑀) (25)

and hence at 𝑞 we have

𝜋|𝑞(𝐼 − (𝑠

𝑞)∗𝑞

)𝑋 = Φ𝑠𝑞(𝑋). (26)

So we define a 1-form 𝜋 through the following formula:

𝜋|𝑥(𝑋) = Φ

𝑠𝑥|𝑥∘ (𝐼 − (𝑠

𝑥)∗𝑥)−1

𝑋 ∀𝑋 ∈ 𝑇𝑥𝑀. (27)

Theorem 19. Let (𝑀, ∇) be an rps-manifold, then (𝑀, ∇) isinessential.

Proof. We define a torsion free affine connection ∇ projec-tively related to ∇ through the fundamental 1-form of 𝑠

𝑥, 𝜋 as

follows:

𝜋|𝑥(𝑋) = Φ

𝑠𝑥|𝑥∘ (𝐼 − (𝑠

𝑥)∗𝑥)−1

𝑋 ∀𝑋 ∈ 𝑇𝑥𝑀 (28)

and prove that the connection ∇ is invariant under all thesymmetries of𝑀.

Let 𝑠𝑞be a symmetry at 𝑞 of 𝑀, the condition that ∇ is

invariant under 𝑠𝑞is equivalent to

𝜋 (𝑋) − 𝜋 (𝑠𝑞∗𝑋) = Φ

𝑠𝑞(𝑋) . (29)

We verify (29) at a point 𝑝 of𝑀, we have to prove that by (28)

Φ𝑠𝑝|𝑝

∘ (𝐼 − (𝑠𝑝)∗𝑝

)

−1

𝑋 − 𝜋𝑠𝑞(𝑝)

(𝑠𝑞∗|𝑝

𝑋) = Φ𝑠𝑞|𝑝

(𝑋) (30)

so if we put 𝑧 = 𝑠𝑞(𝑝), (30) reduces to

Φ𝑠𝑝|𝑝

∘ (𝐼 − (𝑠𝑝)∗𝑝)−1

𝑋 − Φ𝑠𝑧|𝑧

∘ (𝐼 − (𝑠𝑧)∗𝑧)−1

(𝑠𝑞∗|𝑝

𝑋)

= Φ𝑠𝑞|𝑝

(𝑋).

(31)

But since 𝑠𝑞∘ 𝑠𝑝= 𝑠𝑧∘ 𝑠𝑞, we have

Φ𝑠𝑝(𝑌) + Φ

𝑠𝑞(𝑠𝑝∗𝑌) = Φ

𝑠𝑞(𝑌) + Φ

𝑠𝑧(𝑠𝑞∗𝑌). (32)

Now evaluate (32) at 𝑝, and let 𝑌 = (𝐼 − (𝑠𝑝)∗𝑝

)−1

𝑋; then as𝑠𝑞∘ 𝑠𝑝= 𝑠𝑧∘ 𝑠𝑞, 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, “Affine and Riemannian 𝑠-manifolds,” Journal of Differential Geometry, vol. 2, pp. 451–459,1968.

[2] A. J. Ledger, “Espaces de Riemann symétriques généralisés,”Comptes Rendus de l’Académie des Sciences, vol. 264, pp. A947–A948, 1967.

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

[4] F. Podestà, “Projectively symmetric spaces,”Annali di Matemat-ica Pura ed Applicata. Serie Quarta, vol. 154, pp. 371–383, 1989.

[5] A. V. Aminova, “Projective transformations of pseudo-Riemannian manifolds,” Journal of Mathematical Sciences, vol.113, no. 3, pp. 367–470, 2003.

[6] S. Kobayashi, Transformation Groups in Differential Geometry,Springer, Berlin, Germany, 1980.

[7] F. Podestà, “A class of symmetric spaces,” Bulletin de la SociétéMathématique de France, vol. 117, no. 3, pp. 343–360, 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, “On homogeneous Finsler spaces,”Reports on Mathematical Physics, vol. 57, no. 3, pp. 357–366,2006.

[10] P. Habibi and A. Razavi, “On generalized symmetric Finslerspaces,” Geometriae Dedicata, vol. 149, pp. 121–127, 2010.

[11] P. Habibi and A. Razavi, “On weakly symmetric Finsler spaces,”Journal of Geometry and Physics, vol. 60, no. 4, pp. 570–573,2010.

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