I was just sitting in my office and thinking about spheres. And then I thought, hey, intuitively on a sphere for each two points, you have two interesting geodesics connecting two points. By interesting I mean the following.
Given two points $p_1$ and $p_2$ in $\mathbb{S}^2$, with $p_1 \neq p_2.$
Now let $P =\{ \gamma: (0, T)\rightarrow \mathbb{S}^2: \gamma(0)=p1, \gamma(T)=p2 \}$ be the set of geodesics from $p_1$ to $p_2.$
Define on $P$ the following equivalency relation:
$\gamma_1 \sim \gamma_2 : \Longleftrightarrow \exists \ \ \mathrm{ geodesic} \ \ \Gamma:(0, T) \rightarrow \mathbb{S}^2$ such that $\Gamma(0) = \Gamma(T)$ and $\gamma_1 +\Gamma = \gamma_2$ or $\gamma_2 + \Gamma.$
By the plus sign, I mean that the geodesics are concatenated. and then parametrized such that it is again a geodesic from $0$ to $T.$
It should be clear that this is an equivalency class relation. And I think there should be exactly 2 equivalncy classes. The interesting geodesics I was talking about are the the shortest representative of each one of these classes.
Now, going from a sphere to an open semi sphere and assuming that both points are lying on the half. I think there should be exactly one of the two also entirely on the semi sphere. In fact I am pretty sure that there exist exactly one geodesic that connects the points. But I am only sure from an intuitive perspective.
Now my Question:
How can I show this ?
Thanks in advance