Topological entropy and integrable geodesic flows
by
Alexei V. Bolsinov
MSU
Coauthors: I.A. Taimanov

We discuss topological obstructions to the existence of integrable geodesic flows on Riemannian manifolds and give a simple example of the geodesic flow on a three-dimensional manifolds M with the following properties:

1. the geodesic flow is real-analytic,
2. the geodesic flow is integrable by means of C^\infty-integrals and does not admit two independent real analytic integrals,
3. the topological entropy of the flow is positive,
4. the growth of the fundamental group \pi₁(M) is exponential.

This example shows that the topological entropy cannot be considered as some characteristic of non-integrability. Besides, it demonstrates a new mechanism for constructing new examples of integrable geodesic flows on Riemannian manifolds.

Date received: June 29, 1999