Convexity and Strong convexity of subsets of Surfaces

Question

In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly convex according to the following definitions (cited from the book):

Let $M$ be a complete Riemannian manifold, and $A \subset M$. $A$ is:

1. convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq} \subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$.
2. strongly convex if for any $p,q\in A$ there exists a geodesic $\gamma_{pq}\subset A$ such that $\gamma_{pq}$ is the unique minimizer in $M$ connecting $p$ to $q$, and $\gamma_{pq}$ is the only geodesic contained in $A$ joining $p$ and $q$.

I believe that this picture gives an example. This is not an accurate figure! The set I'm referring to is the one centered at around a geodesic segment with end points conjugate one to the other along it, and bounded symmetrically by two close non minimal geodesics.

Is my example correct? If not, then can someone provide a valid example?

Edit: Caution I know of something like 7-8 different definitions of convexity in the case of a Riemannian manifold. Try to refer to the definitions I gave above.

Answer 1

I believe your example is correct. As you are allowing $A$ closed the following is pretty similar. Choose a real number $ N > 1.$ Take the smooth and analytic curve (write it as a level curve and check the gradient) $$ y^2 = x^3 - 3 N^2 x^2 + 3 N^4 x = \left( \frac{x}{4} \right) \left( (2 x - 3 N^2)^2 + 3 N^4 \right) $$ Note that $$ 2 y y' = 3 x^2 - 6 N^2 x + 3 N^4 = 3 (x - N^2)^2.$$ Also $$ 4 y^3 y'' = 3 (x^4 - 4 N^2 x^3 + 6 N^4 x^2 - 3 N^8 ) = 3\left((x-N^2)^4 + 4N^6(x-N^2)\right). $$ Furthermore, when $$x = N^2, y = \pm N^3.$$ Revolve this around the $x$-axis, making a simply connected surface. There is now a closed geodesic along $ x = N^2,$ of circumference $ 2 \pi N^3.$ The minimizing geodesic between $( N^2, N^3, 0)$ and $( N^2, -N^3, 0)$ is the original curve in the plane $z=0,$ of length no larger than $ 2 N^2 + 2 N^3.$ The length of half the closed geodesic is $ \pi N^3,$ which is larger for large enough $N.$ So, as I did not say, we are taking $N > 2$ and $$ A = \left\{(x,y,z) \in \mathbb R^3 : y^2 + z^2 = x^3 - 3 N^2 x^2 + 3 N^4 x \; \mbox{and} \; x \leq N^2.\right\} $$