Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

2008/03/31 by Alexey V. Bolsinov, Vladimir S. Matveev, Giuseppe Pucacco · 26 citations
Mathematics · Physics and Astronomy · #Computer science #Construct (python library) #Geodesic #Geodesic map #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Integrable system #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Quadratic equation #Riemannian geometry #Solving the geodesic equations #math-ph #math.MP #msc:35Q72 #msc:37J35 #msc:53A20 #msc:53B80 #msc:58F07 #nlin.SI

paper · pdf · doi:10.1016/j.geomphys.2009.04.010

published in Journal of Geometry and Physics 59(7), 1048-1062 (Elsevier BV)

arxiv created 2009/04/08 · openalex publication_date 2009/04/24 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: 1) they admit geodesically equivalent metrics; 2) one can use them to construct a big family of natural systems admitting integrals quadratic in momenta; 3) the integrability of such systems can be generalized to the quantum setting; 4) these natural systems are integrable by quadratures.

Citations

• Complex variables for separation of the Hamilton-Jacobi equation on real pseudo-Riemannian manifolds
• Duality between integrable Stäckel systems
• Reciprocal transformations and local Hamiltonian structures of hydrodynamic-type systems
• Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems
• Riemannian Geometry