AFFINE SPACES OF PATHS WHICH ARE SYMMETRIC ABOUT EACH POINT

Abstract

This chapter focuses on the affine spaces of paths, which are symmetric about each point. E. Cartan has shown that a necessary and sufficient condition for the reflexion, about any point in an affine space of paths to be an affine collineation, is that the covariant derivative of the curvature tensor shall vanish. Cartan also showed that any symmetric affine space is equivalent to a totally geodesic sub-space in the manifold of its group of automorphisms. The chapter also explains the alternative proofs of the two theorems referred above in while clearly exhibiting analogies between symmetric affine spaces and flat affine spaces.