Consider the following definitions : 


$C\subset M$ is convex if any $p,\ q\in C$ *all minimizing
 geodesic* between $p$ and $q$ are in $C$ 


$C$ is totally convex if for $p,\ q\in C$, every geodesic between $p$ and $q$ are in
$C$ 


$C$ is locally convex if for $p\in C$ there is open set $U$ around $p$ s.t. $U\cap C$ is convex 


So on $S^2(1)$ closed hemisphere $C$ is locally convex but not convex And $C$ is not totally geodesic 


And on a suitable sphere which has no constant curvature there is a convex set which is not totally geodesic 


Hence these are distinct 


But are there connected locally convex set $C$ s.t. in $C\subset C\_1\subset C\_2$ where $C\_1$ is smallest convex set containing $C$ and $C\_2$ is smallest totally geodesic set containing $C\_1$, inclusions are strict? 


If $C$ is not connected, then we have an example 


Thank you in advance 



Totally convex, convex and locally convex sets
Let $M$ be the cylinder $\{x^2+y^2=1,\ |z| < 2\}$ plus hemispherical caps on both ends.


Let $C$ be the set $\{x < \frac{1}{2},\ x^2+y^2=1,\ |z| < 1\}$. $C$ is locally convex.


Then $C\_1$ is the cylinder $\{x^2+y^2=1,\ |z| < 1\}$, with all shortest geodesics between points in $C$.


And $C\_2$, with all geodesics between points in $C$, is all of $M$.


