Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds

Abstract

Two homogeneous pseudo-riemannian manifolds $(G/H,d{s}^2)$ and $(G^{\prime }/H^{\prime },d{s}^{\prime 2})$ belong to the same {\it real form family} if their complexifications $(G_{\mathbb{C}}/H_{\mathbb{C}},d{s}_{\mathbb{C}}^2)$ and $(G_{\mathbb{C}}^{\prime }/H_{\mathbb{C}}^{\prime },d{s}_{\mathbb{C}}^{\prime 2})$ are isometric. The point is that in many cases a particular space $(G/H,d{s}^2)$ has interesting properties, and those properties hold for the spaces in its real form family. Here we prove that if $(G/H,d{s}^2)$ is a geodesic orbit space with a reductive decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$, then the same holds all the members of its real form family. In particular our understanding of compact geodesic orbit riemannian manifolds gives information on geodesic orbit pseudo-riemannian manifolds. We also prove similar results for naturally reductive spaces, for commutative spaces, and in most cases for weakly symmetric spaces. We end with a discussion of inclusions of these real form families, a discussion of D'Atri spaces, and a number of open problems.

This note complements earlier work of Zhiqi Chen and the author