Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ locally convex if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$.

Is there any known condition that is equivalent to local convexity?

Some special cases are easy to treat. If the second fundamental form with respect to some normal direction is strictly positive definite, then $M$ is locally convex. Vice versa, if the second fundamental forms with respect to all normal directions are indefinite, $M$ is not locally convex.

Also the case of hypersurfaces is solved: $M$ is locally convex if and only if the second fundamental form is semidefinite at every point.