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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2007
• Main Author: Sarlet, W.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2007
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/147793
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold / W. Sarlet // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 6 назв. — англ.

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. 
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