The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta

TL;DR

This paper demonstrates that for a generic metric, the geodesic flow does not possess nontrivial polynomial or analytic integrals in momenta, indicating a lack of local integrability in such cases.

Contribution

It establishes that generically, smooth geodesic flows lack polynomial or analytic integrals, contrasting with the local integrability of smooth integrals.

Findings

Generic metrics do not admit polynomial integrals in momenta.

Real-analytic metrics typically lack real-analytic integrals.

Most geodesic flows are not locally integrable with polynomial or analytic integrals.

Abstract

Any smooth geodesic flow is locally integrable with smooth integrals. We show that generically this fails if we require, in addition, that the integrals are polynomial (or, more generally, analytic) in momenta. Consequently we obtain that a generic real-analytic metric does not admit, even locally, a real-analytic integral.