Let $(M,g)$ be a riemannian manifold. Let $\gamma : [0,1] \to M$ be a constant speed minimizing geodesic in $M$. Is there an $U \subset M$ normal neighborhood containing $\gamma$?
My main interest is actually another question: is there an $U \subset M$ open set, such that $U$ contains $\gamma$ and $r$, the radial distance function from $\gamma(0)$, is smooth in $U$?
Edit: I changed the desired property of the radial function.
For me, a normal neighborhood of a point $p \in M$ is an open set $U \subset M$ containing $p$ such that the exponential map at $p$ is a diffeomorphism (that is, there is a $V$ in $T_pM$ with \exp_p(V) = $U$, and $V$ is a diffeomorphism there).
The whole point is to define the smooth field $\partial_r$ in such a way that $\dot{\gamma}$ is an extendible vector field along $\gamma$.