Convex subsets of pinched Hadamard manifolds

Question

Let $X$ be a pinched Hadamard manifold (in my particular case, $X=\mathbb H^n$ is the $n$-dim. hyperbolic space) and $N$ be a closed (edit : open) convex subset of $X$.

Is it true that $N$ is also a pinched Hadamard manifold ?

Answer 1

Yes, it is a Riemannian manifold with the same curvature bounds as the original one; in the case of interest, of constant curvature $-1$. It is also dimply connected. However, most (if not all) definitions require Hadamard manifolds to be complete and in your case open convex subsets are hardly ever complete.