Let $(M,g)$ be a Riemannian Manifold and let $p\in M$. I was wondering if $c_1, c_2$ are two geodesics in a geodesic ball say $B_c(p)$, (where $B_c(p)$ is defined to be the image of a ball in $T_pM$ under $exp_p$) with the same end points and having as starting point $p$ then they are identical (where identical means $c_1(t)=c_2(t) $ for every $t$).
Any help? (At least if is it true)