# Geodesic exponential map distance

## 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