Let $\Omega$ be a smooth bounded open set in a complete Riemannian manifold $(M^n,g)$, $n\geq 2$, and denote by $N$ the outward unit normal for $\partial \Omega$.
Is it true that $\Omega$ is (strictly) convex if and only if the second fundamental form of its boundary with respect to $N$ is nonnegative (positive definite)? If so, do you have a reference for these facts?
Just to recall, we say that $\Omega$ is convex if for any $x,y \in \Omega$ the minimizing geodesic in $M$ that joins these points lies entirely in $\Omega$. It is strictly convex if for any $x,y \in \overline{\Omega}$ the interior of the minimizing geodesic in M joining these points lies in $\Omega$.
Finally, the second fundamental form of $\partial \Omega$ with respect to $N$ at a point $x \in \partial \Omega$ is the symmetric bilinear form $B : T_x \partial \Omega \times T_x \partial \Omega \to \mathbb{R}$ given by
$$ B(v,w) = g( \nabla_v N, w).$$