Let $M$ be a Riemannian manifold of dimension $n$. Let $N\subset M$ be a subset with smooth boundary $\Sigma=\partial N$. If one assume the second fundamental form $II$ with respect to inner normal direction of $\Sigma$ is nonnegative. (here we use the convention that the second fundamental form of round sphere in $\mathbb R^n$ is positive respective to inner normal.) It is easy to show that for any point $x\in \Sigma$ there exists a neighborhood $U$ around $x$ in $M$ such that $U\cap N$ is convex in the sense that $p, q\in U$ then any short geodesic connecting $p$ and $q$ remains in $U\cap N$.
Is it true that for any two point $p, q\in N$ any short geodesic(realize the distance) connecting them remains in $N$?