Stability analysis for the pseudo-Riemannian geodesic flows of step-two nilpotent Lie groups

TL;DR

This paper investigates the stability of geodesic flows on step-two nilpotent Lie groups with pseudo-Riemannian metrics, using Lie-Poisson equations and Williamson types to classify equilibrium stability.

Contribution

It introduces a stability analysis framework for geodesic flows on step-two nilpotent Lie groups using the $j$-mapping and Williamson classification.

Findings

Stability characterized by Williamson types.

Equilibrium points' stability depends on the $j$-mapping.

Provides criteria for stability in pseudo-Riemannian settings.

Abstract

The present paper deals with the stability analysis for the geodesic flow of a step-two nilpotent Lie group equipped with a left-invariant pseudo-Riemannian metric. The Lie-Poisson equation can be described in terms of the so-called $j$-mapping, a linear operator associated to the step-two nilpotent Lie algebras equipped with the induced scalar product. The stability of equilibrium points for the Hamilton equation is determined in terms of their Williamson types.