  Let $Q$ be a compact, connected $n$-dimensional Riemannian manifold, and
assume that the geodesic flow is toric integrable. If $n \neq 3$ is odd, or if
$\pi_1(Q)$ is infinite, we show that the cosphere bundle of $Q$ is
equivariantly contactomorphic to the cosphere bundle of the torus $\T^n$. As a
consequence, $Q$ is homeomorphic to $\T^n$.
