Totally convex, convex and locally convex sets

Question

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

Answer 1

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 $M$.