Riemannian 4-symmetric spaces

Abstract

The main purpose of this paper is to classify the compact simply connected Riemannian 4 4 -symmetric spaces. As homogeneous manifolds, these spaces are of the form G / L G/L where G G is a connected compact semisimple Lie group with an automorphism σ \sigma of order four whose fixed point set is (essentially) L L . Geometrically, they can be regarded as fiber bundles over Riemannian 2 2 -symmetric spaces with totally geodesic fibers isometric to a Riemannian 2 2 -symmetric space. A detailed description of these fibrations is also given. A compact simply connected Riemannian 4 4 -symmetric space decomposes as a product M 1 × … × M r {M_1} \times \ldots \times {M_r} where each irreducible factor is: (i) a Riemannian 2 2 -symmetric space, (ii) a space of the form { U × U × U × U } / Δ U \{ U \times U \times U \times U\} /\Delta U with U U a compact simply connected simple Lie group, Δ U = \Delta U = diagonal inclusion of U U , (iii) { U × U } / Δ U θ \{ U \times U\} /\Delta {U^\theta } with U U as in (ii) and U θ {U^\theta } the fixed point set of an involution θ \theta of U U , and (iv) U / K U/K with U U as in (ii) and K K the fixed point set of an automorphism of order four of U U . The core of the paper is the classification of the spaces in (iv). This is accomplished by first classifying the pairs ( g , σ ) (\mathfrak {g},\,\sigma ) with g \mathfrak {g} a compact simple Lie algebra and σ \sigma an automorphism of order four of g \mathfrak {g} . Tables are drawn listing all the possibilities for both the Lie algebras and the corresponding spaces. For U U "classical," the automorphisms σ \sigma are explicitly constructed using their matrix representations. The idea of duality for 2 2 -symmetric spaces is extended to 4 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 K the centralizer of a torus.