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Geodesics are locally length-minimizing November 13, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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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…)