Let $U$ be an open subset in $R^n$ and let $N$ be a $C^1$-submanifold. We have a family of geodesics $\gamma:[0,1]\rightarrow U$ in U with respect to the euclidian metric. Each geodesic is parametrized with constant speed and and intersects with $N$ in exactly one point $\gamma(\tau(\gamma))$. We have $sup_{t\in [0,1]}|\gamma(t),\tilde{\gamma}(t)|\leq C|\gamma(0),\tilde{\gamma}(0)|$ for all $\gamma$ and $\tilde{\gamma}$ and C is a constant.

Is the map $\gamma(0)\mapsto\gamma(\tau(\gamma))$ Lipschitz?