
SIGMA 3 (2007), 024, 9 pages math.DG/0702383
http://dx.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, B9000 Ghent, Belgium
Received October 30, 2006, in final form January 17, 2007; Published online February 13, 2007
Abstract
We review properties of socalled special conformal
Killing tensors on a Riemannian manifold (Q,g) and the way they
give rise to a PoissonNijenhuis 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)
References
 Bao D., Chern S.S., Shen Z., An introduction to RiemannFinsler
geometry, Graduate Texts in Mathematics, Vol. 200,
SpringerVerlag, New York, 2000.
 Benenti S., Special symmetric twotensors, equivalent dynamical
systems, cofactor and bicofactor systems, Acta Appl.
Math. 87 (2005), 3391.
 Crampin M., Sarlet W., Thompson G., Bidifferential calculi,
biHamiltonian systems and conformal Killing tensors, J.
Phys. A: Math. Gen. 33 (2000), 87558770.
 Sarlet W., Vermeire F., A class of PoissonNijenhuis structures
on a tangent bundle, J. Phys. A: Math. Gen. 37
(2004), 63196336,
math.DG/0402076.
 Topalov P., Matveev V.S., Geodesic equivalence via integrability,
Geom. Dedic. 96 (2003), 91115.
 Vermeire F., Sarlet W., Crampin M., A class of recursion operators
on a tangent bundle, J. Phys. A: Math. Gen. 39
(2006), 73197340.

