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

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

2007-02-13

arxiv.org/abs/math/0702383v1

SIGMA 3 (2007), 024, 9 pages

Mathematics

Differential Geometry

This is a contribution to the Proc. of workshop on Geometric Aspects of Integrable Systems (July 17-19, 2006; Coimbra, Portuga

Scientific paper

10.3842/SIGMA.2007.024

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 $TQ$. 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 $TQ$. It is shown that when a symmetric type (1,1) tensor field $K$ along the tangent bundle projection $\tau: TQ\to Q$ 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.

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