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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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