Geodesic spheres and limit.

Question

I want to prove the following limit:

Let $M$ be a manifold. Given $\epsilon>0$ there exist some $\delta>0$ such that

$$\frac{d(\exp_p(v),\exp_p(w))}{||v-w||}=1\pm o(\epsilon^2)$$

for every $u,v\in B_\delta(p)$.

I tried hard to prove that equation based on the following equation $$g_{ij}=\delta_{ij}+\frac{1}{3}R_{kilj}w^kw^l s^2 + o(s^2) $$ but I just get stuck.

Any hint to make this worḳ?

Answer 1

If $c(t)=(1-t)v+tw$ then $c$ has a length $| v-w|$. Here tangent is $c'(t)=w-v$.

Then $$ g(c(t))(c',c')= |w-v|^2+ \frac{1}{2} R_{kilj}(p) ((1-t)v+tw)_k( (1-t)v+tw)_l (w-v)_i(w-v)_j + o(|c(t)|^2) $$ so that $$ | g(c(t))(c',c') - |w-v|^2 | = O(\delta^2)$$ That is, $| \sqrt{ g(c(t))(c',c') } - |w-v|| =O(\delta^2)$.

That is, ${\rm length}$ of $ \exp_p\ c(t)$ is $|w-v|$ up to $O(\delta^2) $.