On the integrability of geodesic flows of submersion metrics

Abstract

Suppose we are given a compact Riemannian manifold (Q, g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by isometrics. The natural question arises: will the geodesic flow on Q/G equipped with the submersion metric be integrable? Under one natural assumption, we prove that the answer is affirmative. New examples of manifolds with completely integrable geodesic flows are obtained.