The main purpose of this paper is to classify the compact simply connected Riemannian 4-symmetric spaces. As homogeneous manifolds, these spaces are of the form G/L where G is a connected compact semisimple Lie group with an automorphism a of order four whose fixed point set is (essentially) L. Geometrically, they can be regarded as fiber bundles over Riemannian 2-symmetric spaces with totally geodesic fibers isometric to a Riemannian 2-symmetric space. A detailed description of these fibrations is also given. A compact simply connected Riemannian 4-symmetric space decomposes as a product M1 x • • • x Mr where each irreducible factor is: (i) a Riemannian 2-symmetric space, (ii) a space of the form (U x U x U x U)/DU with U a compact simply connected simple Lie group, DU = diagonal inclusion of U, (iii) (U x U)/DUθwith U as in (ii) and Uθthe fixed point set of an involution θ of U, and (iv) U/K with U as in (ii) and K the fixed point set of an automorphism of order four of U. The core of the paper is the classification of the spaces in (iv). This is accomplished by first classifying the pairs (g, σ) with g a compact simple Lie algebra and a an automorphism of order four of g. Tables are drawn listing all the possibilities for both the Lie algebras and the corresponding spaces. For U “classical, ” the automorphisms a are explicitly constructed using their matrix representations. The idea of duality for 2-symmetric spaces is extended to 4-symmetric spaces and the duals are determined. Finally, those spaces that admit invariant almost complex structures are also determined: they are the spaces whose factors belong to the class (iv) with K the centralizer of a torus.
All Science Journal Classification (ASJC) codes
- Applied Mathematics