I am now reading the book Calculus of Variations written by Jost and I encountered the following problem (in Theorem 2.3.3.):
Let $M$ be a differentiable submanifold of $\mathbb R^d$ diffeomorphic to $\mathbb S^2$. By the compactness and connectedness of $M$, $\forall\ p, q \in M$ with $p \neq q$, there exists a shortest geodesic connecting $p$ and $q$.
Now, what we want to do is to construct a diffeomorphism $$h_0: \mathbb S^2 \rightarrow M$$ with the following properties:
$$p = h_0(0, 0, 1), q = h_0(0, 0, -1)$$ and a shortest geodesic arc $c: [0, 1] \rightarrow M$ with $c(0) = p, c(1) = q$ is given by $$c(t) = h_0(0, \sin \pi t, \cos \pi t).$$
This problem is intuitively true and I have tried "shifting" the inverse image of a geodesic smoothly to a part of a great circle, but I have yet to write down the proof successfully. Thus, I would really like to know if there is anyone who can help me solving this problem or give me some hints. Thanks in advance!