Stability analysis for the pseudo-Riemannian geodesic flows of step-two nilpotent Lie groups
Genki Ishikawa, Daisuke Tarama
TL;DR
The paper develops a unified stability analysis for left-invariant geodesic flows on step-two nilpotent Lie groups with pseudo-Riemannian metrics by recasting the dynamics as Lie-Poisson equations via the $j$-mapping. It reduces the problem to linearization on coadjoint orbits and classifies equilibrium stability through Williamson types, leveraging the classification of Cartan subalgebras in real forms of $\mathfrak{so}(p,q)$. The authors provide explicit Williamson-type formulas and stability criteria for several concrete classes—Heisenberg, Carnot, Métivier, $H$-type and pseudo-$H$-type, Heisenberg-Reiter, and semi-simple module–associated groups—highlighting how the center metric sign and rank conditions govern stability. The results extend stability analysis beyond semi-simple groups, offering practical criteria and symbolic descriptions that can be applied to a broad family of nilpotent Lie groups in geometric mechanics and control theory.
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.