# Geodesic exponential map distance

1. Oct 13, 2014

### ireallymetal

1. The problem statement, all variables and given/known data
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: $d(exp_p(tv), exp_p(tw)) = |t||v-w| + O(t^2 )$

2. Relevant 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.

3. 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

2. Oct 18, 2014