Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Let $x\in \partial M$. Let $v\in T_x(\partial M)$ be a unit vector tangent to the boundary. Assume $$II_{\partial M}(v)<0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\, (1)$$ where $II_{\partial M}$ denotes the second fundamental form of the boundary.
Let $\gamma\colon [0,l]\to \partial M$ be a normalized geodesic on the boundary which starts at $x$ in the direction $v$.
Is it true that if $l$ is sufficeintly small then $\gamma$ is a shortest path in $M$ (rather than in $\partial M$)? A reference would be very helpful.
Here the condition (1) seems to be crutial.
Remark: I think I have a proof of the positive answer to the question when $n=2$.
