Integrable geodesic flows with wild first integrals: the case of two-step nilmanifolds

2003/06/01 by LEO BUTLER, Léo T. Butler · 3 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Quantum chaos and dynamical systems

paper · doi:10.1017/s0143385702001517

openalex publication_date 2003/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26

Abstract

This paper has four main results: (i) it shows that left-invariant geodesic flows on a broad class of two-step nilmanifolds—which are dubbed almost non-singular—are integrable in the non-commutative sense of Nehoros˘ev; (ii) the left-invariant geodesic flows on all Heisenberg–Reiter nilmanifolds are Liouville integrable; (iii) the topological entropy of a left-invariant geodesic flow on a two-step nilmanifold vanishes; (iv) there exist two-step nilmanifolds with non-integrable left-invariant geodesic flows. It is also shown that for each of the integrable Hamiltonians investigated here, there is a C²-open neighbourhood in C²(T^* M) such that every integrable Hamiltonian vector field in this neighbourhood must have wild first integrals.

Cited by

• Symmetric Killing tensors on almost abelian Lie groups
• On first integrals of the geodesic flow on Heisenberg nilmanifolds
• Stability analysis for the pseudo-Riemannian geodesic flows of step-two nilpotent Lie groups

Related

• The geodesic flow on nilmanifolds associated to graphs
• Non-commutative Integrability, Moment Map and Geodesic Flows
• Examples of integrable sub-Riemannian geodesic flows
• Integrability of the sub-Riemannian geodesic flow of the left-invariant metric on the Heisenberg group
• Geodesic Conjugacy in two-step nilmanifolds