Geodesic orbit pseudo-Riemannian H-type nilmanifolds: case of minimal admissible Clifford modules

TL;DR

This paper extends the understanding of geodesic orbit properties to pseudo-Riemannian H-type nilmanifolds, focusing on those built from minimal admissible Clifford modules, thus broadening the classification of such geometric structures.

Contribution

It provides a complete characterization of geodesic orbit property for pseudo-Riemannian H-type Lie groups derived from minimal admissible Clifford modules, extending previous Riemannian results.

Findings

Characterization of geodesic orbit property for pseudo-Riemannian H-type nilmanifolds

Extension of Riehm's Riemannian results to pseudo-Riemannian case

Identification of conditions based on Clifford modules for geodesic orbit property

Abstract

We investigate the geodesic orbit property of pseudo-Riemannian nilmanifolds, specifically those known in the literature as pseudo $H$-type Lie groups -- i.e., 2-step nilpotent Lie groups of Heisenberg type equipped with a left invariant pseudo-Riemannian metric. The study of homogeneous geodesics on Riemannian $H$-type Lie groups was completed by C.~Riehm in 1984. In this work, we extend these results to the pseudo-Riemannian $H$-type Lie groups and provide a complete characterization of the geodesic orbit property for the case where the underlying Lie algebras are constructed from the admissible Clifford modules of minimal dimension.