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

SIGMA 3 (2007), 024, 9 pages      math.DG/0702383
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

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.

pdf (182 kb)   ps (138 kb)   tex (12 kb)


  1. Bao D., Chern S.-S., Shen Z., An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, Vol. 200, Springer-Verlag, New York, 2000.
  2. Benenti S., Special symmetric two-tensors, equivalent dynamical systems, cofactor and bi-cofactor systems, Acta Appl. Math. 87 (2005), 33-91.
  3. Crampin M., Sarlet W., Thompson G., Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors, J. Phys. A: Math. Gen. 33 (2000), 8755-8770.
  4. Sarlet W., Vermeire F., A class of Poisson-Nijenhuis structures on a tangent bundle, J. Phys. A: Math. Gen. 37 (2004), 6319-6336, math.DG/0402076.
  5. Topalov P., Matveev V.S., Geodesic equivalence via integrability, Geom. Dedic. 96 (2003), 91-115.
  6. Vermeire F., Sarlet W., Crampin M., A class of recursion operators on a tangent bundle, J. Phys. A: Math. Gen. 39 (2006), 7319-7340.

Previous article   Next article   Contents of Volume 3 (2007)