Sub-semi-Riemannian geometry of general $H$-type groups

TL;DR

This paper generalizes $H$-type groups by introducing sub-semi-Riemannian geometry with arbitrary scalar products, deriving geodesic equations, and analyzing curvature properties of these nilpotent Lie groups.

Contribution

It extends the concept of $H$-type groups to sub-semi-Riemannian settings with arbitrary scalar products, providing explicit geodesic solutions and curvature analysis.

Findings

Derived geodesic equations for general $H$-type groups

Solved geodesic equations explicitly in certain cases

Analyzed sectional curvature and Ricci tensor properties

Abstract

We introduce a special class of nilpotent Lie groups of step 2, that generalizes the so called $H$(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms. We present the geodesic equation for sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of general $H$-type groups. We also present some results on sectional curvature and the Ricci tensor of general $H$-type groups.