Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds

Abstract

Two homogeneous pseudo-Riemannian manifolds $(G/H, \textrm{d}s^2)$ and $(G'/H', \textrm{d}s'^2)$ belong to the same real form family if their complexifications $(G_\mathbb {C}/H_\mathbb {C}, \textrm{d}s_\mathbb {C}^2)$ and $(G_\mathbb {C}'/H_\mathbb {C}', \textrm{d}s_\mathbb {C}'^2)$ are isometric. The point is that in many cases, a particular space $(G/H, \textrm{d}s^2)$ has interesting properties, and those properties hold for the spaces in its real form family. Here we prove that if $(G/H, \textrm{d}s^2)$ is a geodesic orbit space with a reductive decomposition $\mathfrak {g}= \mathfrak {h}\oplus \mathfrak {m}$, 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.