Sub-semi-Riemannian geometry on $H$-type groups

TL;DR

This paper explores sub-semi-Riemannian geometry on H-type groups with indefinite metrics, deriving explicit formulas for causal geodesics, expanding understanding beyond traditional sub-Riemannian frameworks.

Contribution

It introduces a novel study of sub-semi-Riemannian structures on H-type groups with indefinite metrics and provides explicit geodesic formulas.

Findings

Explicit formulas for causal geodesics derived

Extension of sub-Riemannian results to indefinite metrics

Analysis of geometric properties with nondegenerate indefinite metrics

Abstract

We consider $H$(eisenberg)-type groups whose law of left translation gives rise to a bracket generating distribution of step 2. In the contrast with sub-Riemannian studies we furnish the horizontal distribution with a nondegenerate indefinite metric of arbitrary index and investigate the problem concerning causal geodesics on underlying manifolds. The exact formulae for geodesics are obtained.