I am studying Sharafutdinov's Convex sets in a manifold on nonnegative curvature.
I have found the following statement, $S_0$ being a totally geodesic submanifold of an open Riemannian variety $M$ and $h_t$ a map from $S_0$ onto $S_t$ that respects the Riemannian distance (i.e. a metric isometry of $S_0$ onto $S_t$).
Since $S_0$ is a totally geodesic submanifold and $h_t$ is an isometry of $S_0$ onto $S_t$, it follows easily that $S_t$ is a totally geodesic submanifold in $M$ and the mapping $h_t$ is smooth.
I have trouble in understanding why is true. How precisely do I convert a statement on distances (the fact that $h_t$ is a metric isometry) in the fact that $S_t$ is also a submanifold?
EDIT: The part that $h_t$ is smooth follows then from Myers-Steenrod.
Thank you in advance.