A class of Riemannian manifolds with integrable geodesic flows (Q1915541)

In the two-dimensional case \(n= 2\), the main theorem of this paper is easily stated: Theorem. Let \((M^2, {\mathbf g})\) be a two-dimensional Riemannian manifold. Suppose that the geodesic flow of \((M^2, {\mathbf g})\) has a first integral \(f: T(M^2)\to \mathbb{R}\) which can be expressed as \(f(X)= {\mathbf g}(\iota(X), X)\), where \(\iota\) is a symmetric tensor field of type \((1, 1)\) on \(M^2\). Assume that the determinant of \(\iota\) is positive on \(M\), and let \(\widetilde{\mathbf g}= (\text{det } \iota)^{- 1} {\mathbf g}\) be the conformal change of the metric. Then the geodesic flow of \((M^2, \widetilde{\mathbf g})\) has the first integral \(\widetilde f: T(M^2)\to \mathbb{R}\) given by \(\widetilde f(X)= \widetilde{\mathbf g}(\iota^{- 1}(X), X)\). The main theorem for general \(n\) is used for giving a differential-geometric proof of the complete integrability of geodesic flows of ellipsoids.