Topological Obstructions to Integrability of Geodesic Flows on Non-Simply-Connected Manifolds

Integrable geodesic flows with positive topological entropy

“…This shows that some of the results of [8,9] are not generalized for the C ∞ case. Note that the topological entropy vanishes for Butler's examples.…”

“…For higher-dimensional manifolds, obstructions to integrability were found in [8,9] where it was proved that analytic integrability of the geodesic flow on a manifold M n implies that 1) the fundamental group π 1 (M n ) of M n is almost commutative, i.e., contains a commutative subgroup of finite index;…”

“…It follows from the results of [8] (also exposed in Section 1) that if this flow is analytically integrable, then π 1 (M A ) is almost commutative and, therefore, has a polynomial growth. This contradiction establishes the corollary.…”

Topology and Integrability in Lagrangian Mechanics

“…Tăĭmanov tells us that if the Tonelli Hamiltonian is completely integrable with real-analytic first integrals, then the three-dimensional configuration space ∑ has a finite coveringp : ∑ ! ∑ such that the fundamental group π 1 ð∑ Þ is abelian and of rank at most 3 [41][42][43]. Based on the resolution of the Poincaré conjecture, this result implies that, up to finite covering the only such configuration spaces are…”

“…It is clear that the general solution of (42b), without the compatibility condition (42c), is obtained via repeated quadratures of products of s and S. The compatibility condition distinguishes those solutions which may arise from (41). The behaviour of s at r ¼AET ultimately determines whether the solution obtained arises from a T 1 -invariant Riemannian Hamiltonian H and an independent first integral F on T Ã S 2 .…”