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