Let $M$ be a complete Riemannian manifold and $N\subset M$ a closed submanifold. If codimension of $N$ is $0$ take $q\in\partial N$ and $v\in T_qN$, where $\partial N$ is the boundary of $N$ as a subset of $M$ and if codimension of $N$ is bigger than $0$, take $q\in N$ and $v\in T_qN$. Let $\gamma:(-\epsilon,\epsilon)\rightarrow M$ be a differentuable curve such that $\gamma(0)=q$ and $\gamma'(0)=v$. Let $d$ denote the geodesic distance in $M$.
How can one show that $d(\gamma(t),N)=o(t)$ for small $t$?