Geodesic exponential map distance

  • #1

Homework Statement


Hi all. For some reason I have been having a lot of difficulty with this problem in Peter Petersen's text. The problem is
Prove: ##d(exp_p(tv), exp_p(tw)) = |t||v-w| + O(t^2 )##

Homework Equations


The exponential map is the usual geodesic exponential map. And ##d(p,q)## is the infinum of the lengths of all curves starting at ##p## and ending at ##q##. ##|v-w|## is in the norm of the metric of the manifold.

The Attempt at a Solution


I am aware that for small time the geodesics are length minimizing at that ##d(exp_p(v),exp_p(tv) = \int_t ^1|\gamma'|ds = \int_t^1|v| = (1-t)|v|## and that I can probably make a similar argument to get that locally ##d(exp_p(tv), exp_p(tw)) = |t||v-w| ##.

I'm having a lot of trouble with the explicit computation. I think intuitively I sort of understand what this equation is telling me but I'm having trouble with the proof. I have no idea how to work in the ##O(t^2 )##. I don't see any Taylor expansions or where non-linear terms would come in. Thank you
 

Answers and Replies

  • #2
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Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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