Suppose we have a manifold $M$, with a metric(or connection). Suppose $p$ is a point and $v \in T_p(M)$. Then there is a geodesic $\gamma$ starting at $p$ such that $\gamma'(0) = v$.
Now imagine $\gamma(t) = q$ and $q \in U$.
The question is whether for $v'$ very close to $v$ and $p'$ very close to $p$ the geodesic starting at $p'$ with velocity $v'$ will pass through $U$ or not.
Thanks.
Yes, this is true.
The standard existence and uniqueness theorem for systems of linear ODE's (which is what one uses to construct geodesics to begin with) has a strong version in which the conclusion says that integral curves depend continuously on initial conditions. In the application of that theorem to construction of geodesics, the initial conditions are precisely $p$ and $\gamma'(0)$.