Generalized Projectively Symmetric Spaces

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];