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- May 12th 2013, 09:34 AMPlato13Geodesics
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- May 13th 2013, 08:37 AMxxp9Re: Geodesics
let be a smooth map from a rectangle range of to M. And denote

and as the coordinate vectors

For each u define

Then

=

=

=

And since we have hence

, hence

When u=0 the curve has unit speed so we have

Note the final result has nothing to do with p, so for any p, the energy function achieves extreme when the curve is a geodesic.