Symmetric spaces of order k

In chapter I we give an account of the important theorems and results needed in the subsequent work, we also include all the references from which a general point of view can be obtained. Chapter II is a continuation of Chapter I, where we consider only one specific concept Lie groups and homogeneous spaces. Chapter III deals with Riemannian (locally) symmetric manifolds. The theorems and results in this chapter are included with their proofs, since both are very relevant for the work in the coming chapters. The main original contributions of this thesis are presented in Chapters IV and V. In Chapter IV, Riemannian s-manifolds, and Riemannian-symmetric spaces, in the sense of A.J. Ledger, are defined. We also define Riemannian s-regular manifolds, and Riemannian k-regular symmetric spaces. We discuss in detail the case when k is an odd positive integer, and we establish some results concerning this case. The whole of Chapter V is concerned with Riemannian (locally) 5 - (regular) symmetric manifolds. Our treatment of these manifolds is in some way similar to that adopted by Gray [s] for 3 - (regular) symmetric manifolds. We will also show that Riemannian (locally) 5 - (regular) symmetric manifolds diverge from Riemmannian (locally) 3 - (regular) symmetric manifolds. Finally, the appendix contains calculations needed in Chapter V, section 5.3