Let $H\in M$ be a convex hypersurface, where $M$ is a complete Riemannian manifold and $H$ is an embedded (complete as a induced metric space) hyper surface without boundary and with positive definite second fundamental form. Is it true that $H$ bound a convex domain $D$ in $M$? i.e. any two point $x, y \in D$ can be connected by a minimal geodesic lies in $D$ and $\partial D=H$.
I suspect this is not true in general, but can anyone provide a countexample or any references?