Let $H^n$ be the hyperbolic space. Then the claim is: Every closed and totally geodesic submanifold $M^k$ of $H^n$ is isometric to $H^k$, $k \le n.$
This problem comes from do Carmo's book of Riemannian geometry. It supposed to be easy under the assumption that $M^k$ is simply connected. The result would come from the combination of Hadamard theorem, implying that the exponential map is a diffeo and from the fact that since $M^k$ is closed, it is also complete (since $H^n$ is) and once $M^k$ is totally geodesic, its curvature is $-1$.
We know yet that the universal covering of $M^k$ is isometric to $H^k$ and we can also speculate that $M^k$ cannot be bounded, since if it was, this would imply compactness. So if we stated that $M^k$ is simply connected it would be a contradiction.
How to solve it?
Thanks a lot