I'm currently working through some notes on surfaces in $\mathbb{R}^3$ and their geodesics, with the following definitions (it's kind of lengthy, I will highlight the key parts):
- For a sufficiently smooth surface $S$ parametrized by $\varphi : D \to \mathbb{R}^3$, a geodesic is a curve $\alpha : I \to S$ to have acceleration orthogonal to the tangent spaces, i.e. $\alpha$ is a geodesic if and only if
$$ \alpha''(t) \perp T_{\alpha(t)}S \quad (\forall t \in I). $$
It is proven that fixing $p \in S$ and $v \in T_pS$ there is a unique geodesic $\gamma_{p,v} : I_{p,v} \to S$ such that $\gamma(0) = p$ and $\gamma'(0) = v$, with $I_{p,v}$ the maximal interval of definition. That is, geodesics that stem from $p$ with velocity $v$ are (locally) unique, because two different ones give the same solution to an ODE in a sufficiently small interval of $p$. It is also claimed that, since these curves and their definition intervals depend smoothly on $p$ and $v$ and $\gamma_{p,0}$ is defined on $\mathbb{R}$, by continuity there exists some open ball $B_R(0_p) \subset T_pS$ where $1 \in I_{p,v}$ for $v \in B_R(0_p)$. Hence it is possible to define the Riemannian exponential as
$$ \begin{align} \exp_p : &B_R(0_p) \to S \\ & v \longmapsto \gamma_{p,v}(1) \end{align} $$
Shortly after it is proved that $\exp_p$ sends lines through $0$ to geodesics, since $\gamma_{s,v}(t) = \gamma_{s,tv}(1)$, and that $D(\exp_p)_0 = Id$ which says that in a neighbourhood of $0$, the exponential is a diffeomorphism. We assume from now on that $R$ is small enough for this to hold.
From the previous definitions and results, it is then claimed that one can see that if $q = \exp_p(v) \in \exp(B_R(0_p))$ then there is a unique geodesic that joins $p$ and $q$, namely $$ \gamma(t) := \exp_p(tv) $$ with $[0,1] \subset Dom(\gamma)$.
I do not see why local uniqueness is derived just from the previous results. How can this be proven?
I am aware that (given sufficient regularity hypotheses), any curve $\alpha \subset S$ in $\exp_p(B_R(0_p))$ is a lift $\alpha(t) = \exp_p(\beta(t))$ with $\beta$ a curve in $B_R(0_p)$ (that is, in it's identification with a ball of the plane) but I haven't been able to prove much with that. I also presume a strong usage of locality is needed, as for example in the sphere any two points there are two geodesics joining them.