Riemannian 4-symmetric spaces

Abstract

This thesis studies the theory of Riemannian 4-symmetric spaces. It follows the methods first introduced by E. Cartan to study ordinary symmetric spaces, and extended by J. Wolf and A. Gray and by Kac. The theory of generalized n-symmetric spaces was initiated by A. Ledger in 1967, and 2- and 3-symmetric spaces have already been classified. The theory of generalized n-symmetric spaces is completely new. The thesis naturally divides into two chapters. The first chapter treats the geometry of the spaces. Their homogeneous structure and their invariant connections are studied. The existence of a canonical invariant almost product structure is pointed out. A fibration over 2-symmetric spaces with 2-symmetric fibers is obtained. Root systems are used to obtain geometric invariants. Finally a local characterization in terms of curvature is obtained. Chapter II centres on the problems of classification. A local classification is given for the compact spaces in terms of simple Lie algebras. A global formulation is given for the compact classical simple Lie algebras. A final section is devoted to invariant almost complex structures. A characterization is given in terms of their homogeneous structure. It is shown that they can bear both Hodge and non-Kahler structures.