Existence of Geodesics

Question

Consider a unit sphere in $\mathbb{R}^3$. How can we prove that any two points on the sphere can be joined by a minimizing geodesic?

More generally, under what conditions does the same statement hold for a given surface in $\mathbb{R}^3$?

Answer 1

To answer your second question, there is a general theorem in Riemannian geometry that if $M$ is a compact, connected Riemannian manifold then any two points $x,y \in M$ can be connected by a minimizing geodesic. That theorem applies to your answer your question affirmatively for any smooth surface in $\mathbb R^3$ which is closed and bounded (hence compact) and connected.

The proof of that theorem is an application of the Ascoli-Arzela theorem, applied to any sequence of smooth paths $\gamma_i$ in $M$ connecting $x$ to $y$ such that $\text{Length}(\gamma_i)$ approaches the infimum of the lengths of all such paths. Compactness is used for obtaining a subsequence of $\gamma_i$ that is convergent in the sense specified by the Ascoli-Arzela theorem.

There is also a noncompact version called the the Hopf-Rinow Theorem, which requires a connected, smooth Riemannian manifold that satisfies the additional hypothesis of geodesic completeness. The Hopf-Rinow Theorem again lets you conclude that any two points can be connected by a minimizing geodesic. The proof is similar to the outline above in the compact case, although one needs to use geodesic completeness cleverly to get around the possible lack of compactness.

For surfaces in $\mathbb R^3$, geodesic completeness will be satisfied as long as the surface is a closed subset of $\mathbb R^3$. Examples include planes, infinite cylinders, and things like the Loch Ness Monster surface.