Generalised symmetric spaces

Abstract

This thesis treats the symmetric spaces (originally studied by E. Cartan) and their various generalisations. Chapter I presents the necessary fundamental definitions and results. Chapter II describes the historical background to the subject: in Part A the relevant aspects of the theory of symmetric spaces are reviewed, the notion of k-symmetric space (due to A.J.Ledger) is introduced and various results (in particular those due to Gray for 3-symmetric spaces) are noted; in Part B the theory of Jordan algebras is summarised (as needed in this thesis) and the intimate relationship between Jordan algebras and symmetric spaces is discussed. Chapter III contains largely original results on a class of manifolds, the symmetric spaces of order k (a generalisation of the symmetric spaces made in the spirit of the "algebraic" approach to symmetric spaces developed by 0. Loos). A symmetric space of order k is a differentiable manifold M together with a smooth multiplication μ : M X M → M satisfying certain properties. The main result is that on such a manifold M an affine connexion V may be defined in terms of the multiplication u M may then be shown to be a reductive homogeneous space and ▼ the (complete) canonical affine connexion of the second kind. Chapter IV presents two original observations concerning the relationship between Jordan algebras and symmetric spaces (of order 2).Chapter V contains a summary of results and various suggestions for further research. A bibliography follows Chapter V.