## Geodesics are locally length-minimizingNovember 13, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: , ,

Fix a Riemannian manifold with metric ${g}$ and Levi-Civita connection ${\nabla}$. Then we can talk about geodesics on ${M}$ with respect to ${\nabla}$. We can also talk about the length of a piecewise smooth curve ${c: I \rightarrow M}$ as

$\displaystyle l(c) := \int g(c'(t),c'(t))^{1/2} dt .$

Our main goal today is:

Theorem 1 Given ${p \in M}$, there is a neighborhood ${U}$ containing ${p}$ such that geodesics from ${p}$ to every point of ${U}$ exist and also such that given a path ${c}$ inside ${U}$ from ${p}$ to ${q}$, we have

$\displaystyle l(\gamma_{pq}) \leq l(c)$

with equality holding if and only if ${c}$ is a reparametrization of ${\gamma_{pq}}$.

In other words, geodesics are locally path-minimizing.   Not necessarily globally–a great circle is a geodesic on a sphere with the Riemannian metric coming from the embedding in $\mathbb{R}^3$, but it need not be the shortest path between two points. (more…)