I understand what a geodesic is, but I'm struggling to understand the meaning of the geodesic flow (as defined e.g. by Do Carmo, Riemannian Geometry, page 63).
I can state my confusion in two different ways:
1)
Do Carmo writes:
Why does a geodesic $\gamma$ uniquely define a vector field on an open subset? In other words, why are the values of the vector fields uniquely defined on those points that are not on the geodesic $\gamma$?
2)
In local coordinates, the geodesic flow is defined as the solution to the ordinary differential equation
$$ \tag{1}\frac{d^2 x_k}{dt^2}+\sum_{i,j}\Gamma^k_{ij}\frac{dx_i}{dt}\frac{dx_j}{dt}=0 $$
For the solution to be unique on $TM$ (or on an open subset), we need some boundary condition. The only boundary condition I can see is a given geodesic $\gamma(t)$.
What are the boundary conditions for this ODE?
The basic idea behind the approach Do Carmo takes is to interpret a curve $(\gamma(t), \gamma'(t))$ where $\gamma \colon [a,b] \rightarrow M$ is a geodesic in $M$ as an integral curve of a globally defined vector field $G$ on $TM$. Once you do that, in order to show the existence of a geodesic $\gamma$ with $\gamma(0) = p$ and $\gamma'(0) = v$ (with $v \in T_pM$) you can take an integral curve $\beta \colon (a,b) \rightarrow TM$ of $G$ with $\beta(0) = (p,v)$ and then the projection $\gamma := \pi \circ \beta$ of $\beta$ to $M$ will be a geodesic which satisfies the initial conditions $\gamma(0) = p$ and $\gamma'(0) = v$.
The uniqueness of geodesics will follow from the uniqueness of integral curves of the vector field $G$, once a starting point $(p,v) \in TM$ is specified. There is no need to impose boundary conditions (and in general, an ODE with boundary conditions might have no solutions satisfying arbitrary boundary conditions), only initial conditions.