Let $M$ be a smooth, connected and bounded manifold and let $f:M \rightarrow \mathbb{R}$ be a smooth function.
The conditions to ensure that $f$ is a geodesically convex on $M$ is to ensure firstly that $M$ is geodesically convex manifold. To do that Hopf-Rinow theorem states that $M$ must be a complete topological space. Secondly, the covariant derivative of $f$, $D^2 f $ is non-negative (or equivalently its geodesic hessian matrix $H_g f$ is semi-definite positive).
Assume that $D^2 f $ is non-negative on the topological closure of $M$.
My question is : Is $f$ a geodesically convex function on $\overline{M}$?
Thanks in advance?