Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 024, 9 pages      math.DG/0702383      https://doi.org/10.3842/SIGMA.2007.024
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

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

Willy Sarlet
Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B-9000 Ghent, Belgium

Received October 30, 2006, in final form January 17, 2007; Published online February 13, 2007

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 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 τ: TQ ® 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.

Key words: special conformal Killing tensors; Finsler spaces.

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References

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