Commutative conservation laws for geodesic flows of metrics admitting projective symmetry. Approximation spaces. (English) Zbl 1330.53113

Topalov, Peter J.

Bolsinov, A. V. et al., Topological methods in the theory of integrable systems. Cambridge: Cambridge Scientific Publishers (ISBN 978-1-904868-42-2/hbk). 271-287 (2006).

Let \(X\) be a vector field on a Riemannian manifold \((M,g)\). It is a projective symmetry if the local flow maps geodesics of \(g\) to geodesics, as unparameterised curves. The author defines, from successive derivatives of \(g\) by \(X\), an increasing family of vector spaces of morphisms of \(TM\), which are called approximation spaces. It is proven that the dimensions of these spaces give lower bounds on the number of independent, involutive integrals of the geodesic flow of \(g\). Some examples are reviewed in the final section.
For the entire collection see [Zbl 1142.37001].

Reviewer: Leo T. Butler (MR 2010c:53125)

MSC:

53D25	Geodesic flows in symplectic geometry and contact geometry
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H06	Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics