The geodesic flow on nilmanifolds

TL;DR

This paper investigates the geodesic flow on nilmanifolds with invariant metrics, deriving formulas for Poisson involution, and demonstrating complete integrability on certain quotients and metrics.

Contribution

It provides explicit formulas for the Poisson involution and shows how to identify integrals of motion for geodesic flows on nilmanifolds, including the Heisenberg group.

Findings

Derived general formulas for Poisson involution on nilmanifolds

Established complete integrability for geodesic flows on compact quotients

Applied results specifically to the Heisenberg Lie group

Abstract

In this paper we study the geodesic flow on nilmanifolds equipped with a left-invariant metric. We write the underlying definitions and find general formulas for the Poisson involution. As an example we develop the Heisenberg Lie group equipped with its canonical metric. We prove that a family of first integrals giving the complete integrability can be read off at the Lie algebra of the isometry group. We also explain the complete integrability on compact quotients and for any invariant metric.