Characterizations of simply connected rotationally symmetric manifolds

Abstract:

We prove that a simply connected, complete Riemannian manifold $M$ is rotationally symmetric at $p$ if and only if the exponential image of every linear subspace of ${M_p}$ is a smooth, closed, totally geodesic submanifold of $M$. This result is in essence Schur’s theorem at one point $p$, as it becomes apparent in the proof.