A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold

Abstract

Abstract. We review properties of so-called special conformal Killing tensors on a Riemannian manifold (Q, g) and the way they give rise to a Poisson-Nijenhuis structure on the tangent bundle T Q. We then address the question of generalizing this concept to a Finsler space, where the metric tensor field comes from a regular Lagrangian function E, homogeneous of degree two in the fibre coordinates on T Q. It is shown that when a symmetric type (1,1) tensor field K along the tangent bundle projection τ : T Q → Q satisfies a differential condition which is similar to the defining relation of special conformal Killing tensors, there exists a direct recursive scheme again for first integrals of the geodesic spray. Involutivity of such integrals, unfortunately, remains an open problem.

A class of Finsler metrics admitting first integrals

“…In Riemannian geometry, Topalov and Matveev [21, Theorem 1] associate to each pair of geodesically equivalent metrics a set of n first integrals. An extension of this result, to the Finslerian setting, has been proposed by Sarlet in [16] and his Ph.D student Vermeire [23].…”

“…Existence of first integrals is of great importance, they provide a lot of information about the corresponding geometry, including some rigidity results, [6], [7], [21] In Riemannian geometry, Topalov and Matveev obtained in [21], for two projectively equivalent metrics on an n-dimensional manifold, a set of n first integrals. An extension of this result to the Finslerian context has been proposed by Sarlet in [16]. In [7], Foulon and Ruggiero have shown the existence of a first integral for k-basic (of isotropic curvature) Finsler surfaces.…”

“…The first two assumptions of Theorem 1.1 tell us that S is a Finsler metric projectively related to F . One can use this and [21, Theorem 1] and [16,Theorem 2] to provide a set of n first integrals for Finsler metric with vanishing χ-curvature and mean Berwald curvature of maximal rank.…”

Invariant volume forms and first integrals for geodesically equivalent Finsler metrics

“…This result was extended to arbitrary dimension in [3], by providing a class of Finsler manifolds that admit a first integral. A different approach for obtaining first integrals in Finsler geometry is due to Sarlet, who provides in [10] a recursive scheme of first integrals of the geodesic flow of a Finsler manifold.…”