#### Abstract

We study generalized projectively symmetric spaces. We first study some geometric properties of projectively symmetric spaces and 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 (, ), there exists a projectively related connection , such that (, ) is an affine s-manifold.

#### 1. Introduction

Affine and Riemannian s-manifolds were first defined in [1] following the introduction of generalized Riemannian symmetric spaces in [2]. They form a more general class than the symmetric spaces of E. Cartan. More details about generalized symmetric spaces can be found in the monograph [3]. Let be a connected manifold with an affine connection , and let be the Lie transformation group of all affine transformation of . An affine transformation will be called an affine symmetry at a point if is an isolated fixed point of . An affine manifold will be called an affine s-manifold if there is a differentiable mapping , such that for each , is an affine symmetry at .

In [4] Podestà introduced the notion of a projectively symmetric space in the following sense. Let be a connected manifold with an affine torsion free connection on its tangent bundle; is said to be projectively symmetric if for every point of there is an involutive projective transformation of fixing and whose differential at is . The assignment of a symmetry at each point of can be viewed as a map , and can be topologised, so that it is a Lie transformation group. In the above definition, however, no further assumption on is made; even continuity is not assumed.

In this paper we define and state prerequisite results on projective structures and define projective symmetric spaces due toPodestà. Then we generalize them to define projective s-manifolds as manifolds together with more general symmetries and consider the cases where they are essential or inessential. A projective s-manifold is called inessential if it is projectively equivalent to an affine s-manifold and essential otherwise. We prove that these spaces are naturally homogeneous, and moreover under certain conditions the projective curvature tensor vanishes. Later we define regular projective s-manifolds and prove that they are inessential.

#### 2. Preliminaries

Let be a connected real manifold whose tangent bundle is endowed with an affine torsion free connection . We recall that a diffeomorphism of is said to be projective transformation if maps geodesics into geodesics when the parametrization is disregarded [5]; equivalently is projective if the pull back of the connection is projectively related to , that is, if there exists a global 1-form on , such that If the form vanishes identically on , then is said to be an affine transformation.

*Definition 1. * is said to be projectively symmetric if for every point in there exists a projective transformation with the following properties: (a) and is an isolated fixed point of . (b) is involutive.

It is easy to see that conditions (a) and (b) imply . Moreover we recall that a projective transformation is determined if we fix its value at a point, its differential, and its second jet at this point [6], hence a symmetry at in is not uniquely determined in general by the condition (a) and (b).

Affine symmetric spaces are affine homogeneous, but in general projectively symmetric spaces are not projective homogeneous; for more detail and examples see [4, 7], but if following Ledger and Obata define the case of a differentiable distribution of projective symmetries in an affine manifold, then this happens.

Let be a connected Riemannian manifold. An isometry of for which is an isolated fixed point will be called a Riemannian symmetry of at . Clearly, if is a symmetry of at , then the tangent map is an orthogonal transformation of having no fixed vectors (with the exception of 0). An s-structure on is a family of symmetries of .

A Riemannian s-manifold is a Riemannian manifold together with a map , such that for each the image is a Riemannian symmetry at .

For any affine manifold let denote the Lie group of all affine transformation of . An affine transformation for which is an isolated fixed point will be called an affine symmetry at . An affine s-manifold is an affine manifold together with a differentiable mapping , such that for each , the image is an affine symmetry at .

Let be an affine s-manifold. Since is assumed to be differentiable, the tensor field of type (1,1) defined by for each is differentiable. The tensor field is defined similarly for a Riemannian s-manifold, although it may not be smooth. For either affine or Riemannian s-manifolds we call the symmetry tensor field.

Following [3] an s-structure is called regular if for every pair of points as follows:

#### 3. Projective s-Space

Let be a connected manifold with an affine connection , and let bethe group of all projective transformations of .

*Definition 2. *A projective transformation will be called a projective symmetry or simply a symmetry at the point , if is an isolated fixed point of and does not leave any nonzero vector fixed.

*Definition 3. *A connected affine manifold will be called a projective s-manifold or simply ps-manifold if for each there is a projective symmetry , such that the mapping , is smooth.

A symmetry will be called a symmetry of order at , if there exist a positive integer , such that , and will be called ps-manifold of order , if is the least positive number such that each symmetry is of order . Evidently every ps-manifold of order 2 is a projective symmetric space.

Lemma 4. *Let be a topological transformation group acting on a connected topological space , if for each point in , the -orbit of contains a neighborhood of , then is transitive on . *

* Proof. *Since is transitive on each orbit, for each the -orbit of is open by our assumption. The complement of in is also open, being a union of orbits. Thus is open and closed. It is nonempty and therefore coincides with the connected space , thus is transitive.

Theorem 5. *If is a -manifold, then is transitive on . *

* Proof. *We fix a point and consider the map given by ; since for every in , the differential of at the point is given by , where is the differential of at . is nonsingular because no eigenvalue of is equal to . Hence is a diffeomorphism on some neighborhood of in , and is a neighborhood of contained in the -orbit of , therefore from the above lemma is transitive.

*Definition 6. *Let be a ps-manifold;since is assumed to be differentiable, the tensor field of type defined by is differentiable, we call the symmetry tensor field.

Lemma 7. *If is a projective symmetry of then there exists a connection projectively equivalent with which is -invariant. *

* Proof. *Since is a projective symmetry of then there is a 1-form on , such that
We are looking for a connection with the following properties:
As should be -invariant we need
that is, is an affine transformation of . We have
It follows from (3)
From (5) we have
thus it is enough to have for every vector field as follows:
which is equivalent to
or simply
since is symmetry, then is invertible; hence we obtain
thus if we choose as (12), then (4) and (5) are true, and is the required connection.

So it would be convenient to introduce the following definition for connection and 1-form .

*Definition 8. *Let be a ps-manifold, and let be the projective symmetry at the point . Then we call the associate connection the fundamental connection of . Also the 1-form
will be called the fundamental 1-form of , where is the 1-form on , such that

*Definition 9. *The projective curvature tensor of is defined as follows [5, 8]:
where
The projective curvature tensor is invariant with respect to projective transformations [5, 8].

Theorem 10. *In a ps-manifold , let be a symmetry, and let be the fundamental connection of , if ; that is, , then .*

* Proof. * Let be the -structure and . Let be tangent vectors, and let be a covector at . By parallel translation along each geodesics through , , and can be extended to local vector fields , and with vanishing covariant derivative with respect to at . Because is parallel, the local vector fields and have also vanishing covariant derivative at . (Here denotes the transpose map to .) As is invariant with respect to the projective transformation , we have
Now we show that and are equal at . These are equal if and only if and are equal, which follows from the assumption on . That is
or
Differentiating covariantly (17) with respect to in the direction of at and using (19) we get
thus
for all and , and because is a nonsingular transformation we obtain

Theorem 11. *Let be a -manifold of dimension ; if there exist two different projective symmetries at a point of , such that and , where is the fundamental connection corresponding to , then the projective curvature tensor ; vanishes that is, is projectively flat. *

* Proof. *By a similar method used in Proposition 1.1 of [7] the proof follows from Lemma 7 and Theorem 10.

Corollary 12. *If is a ps-manifold of order 2, and two different projective symmetry can be defined at a point , then is projectively flat. *

* Proof. * It is evident from the fact that .

Proposition 13. *Let be ps-manifold, such that at every point of the projective symmetry is uniquely determined. Then the linear isotropy representation is faithful for every . *

* Proof. *Since and both are projective symmetry at , then we have ; that is, , thus is a ps-manifold of order 2. Now, our assertion follows from Theorem 1.1 of [7].

#### 4. Regular Projective s-Space

*Definition 14. *A -manifold is called regular -manifold or simply -manifold if for all , , where .

Lemma 15. *Let be a regular ps-manifold, then the tensor field is invariant under all symmetries ; that is
**
for all . *

* Proof. *Since is regular ps-manifold, then for all we have and so . Thus is invariant for all .

Lemma 16. *Let be a connected ps-manifold, such that at every point of the projective symmetry is uniquely determined, then is rps-manifold. *

* Proof. *Suppose and ; then from the uniqueness of the projective symmetry, we have , so is regular ps-manifold.

*Remark 17. *A general question is to find condition under which, given a ps-manifold , there exists a projectively related connection , such that is an affine s-manifold; we shall call such spaces inessential ps-manifold and essential otherwise.

*Definition 18. *A ps-manifold is called inessential ps-manifold if there exists a projectively related connection such that is an affine s-manifold.

Let us denote by the 1-form corresponding to an element of . We want to see when is inessential, in order to show that is inessential, we must find a connection which is projectively related to and is invariant under all symmetries. Let be a symmetry at , we must find a one-form , such that As is a projective transformation for and leaves the connection invariant, we find that and hence at we have So we define a 1-form through the following formula:

Theorem 19. *Let be an rps-manifold, then is inessential. *

* Proof. *We define a torsion free affine connection projectively related to through the fundamental 1-form of , as follows:
and prove that the connection is invariant under all the symmetries of .

Let be a symmetry at of , the condition that is invariant under is equivalent to
We verify (29) at a point of , we have to prove that by (28)
so if we put , (30) reduces to
But since , we have
Now evaluate (32) at , and let ; then as , we have (31), and we are done.

*Remark 20. *The authors have studied Finsler homogeneous and symmetric spaces [9]; recently Habibi and the second author generalized them to Finsler s-manifolds and weakly Finsler symmetric spaces [10, 11]. Therefore these concepts can be mixed and find more generalizations which will be the content of other papers.

#### Acknowledgment

The author would like to thank the anonymous referees for their suggestions and comments, which helped in improving the paper.