Riemannian $4$-symmetric spaces

Abstract:

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 $\sigma$ 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 ${M_1} \times \ldots \times {M_r}$ where each irreducible factor is: (i) a Riemannian $2$-symmetric space, (ii) a space of the form $\{ U \times U \times U \times U\} /\Delta U$ with $U$ a compact simply connected simple Lie group, $\Delta U =$ diagonal inclusion of $U$, (iii) $\{ U \times U\} /\Delta {U^\theta }$ with $U$ as in (ii) and ${U^\theta }$ the fixed point set of an involution $\theta$ 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 $(\mathfrak {g}, \sigma )$ with $\mathfrak {g}$ a compact simple Lie algebra and $\sigma$ an automorphism of order four of $\mathfrak {g}$. Tables are drawn listing all the possibilities for both the Lie algebras and the corresponding spaces. For $U$ "classical," the automorphisms $\sigma$ 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.