Given a subset $S$ of a [Hadamard manifold](http://en.wikipedia.org/wiki/Hadamard_manifold) $M$. Is there a curvature criterion (for $\partial S$) to decide whether $S$ is convex.


I am looking for a ganeralization of the following statement:


A connected subset of $\mathbb{R}^2$ bounded by a smooth curve $\gamma$ with $||\dot{\gamma}||=1$ is convex, if $\gamma$ "always turns in the same direction" - more formally if $t\mapsto det(\dot{\gamma}(t),\ddot{\gamma}(t))$ never changes sign. 



Convex subsets of Hadamard manifolds
Thank you. After some googling I found the paper "Locally convex hypersurfaces of negatively curved spaces" by S. Alexander (ams.org/journals/proc/1977-064-02/…), which adresses precisely this question.


