Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds

TL;DR

This paper explores how properties of geodesic orbit pseudo-Riemannian manifolds are preserved within their real form families, extending understanding from Riemannian to pseudo-Riemannian cases.

Contribution

It proves that geodesic orbit, naturally reductive, commutative, and weakly symmetric properties are maintained across real form families of homogeneous pseudo-Riemannian manifolds.

Findings

Properties of geodesic orbit spaces are preserved in real form families.

Results extend known Riemannian properties to pseudo-Riemannian manifolds.

Discussion of inclusions, D'Atri spaces, and open problems included.

Abstract

Two homogeneous pseudo-riemannian manifolds $(G/H, ds^2)$ and $(G'/H', ds'^2)$ belong to the same {\it real form family} if their complexifications $(G_{\mathbb C}/H_{\mathbb C}, ds_{\mathbb C}^2)$ and $(G'_{\mathbb C}/H'_{\mathbb C}, ds'^2_{\mathbb C})$ are isometric. The point is that in many cases a particular space $(G/H, ds^2)$ has interesting properties, and those properties hold for the spaces in its real form family. Here we prove that if $(G/H, ds^2)$ 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.