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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 euklidian 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?
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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 euklidian 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?
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My Question/Problem is:

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. 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 that associates to each geodesic $\gamma$ the point $\gamma(\tau(\gamma))$, lipschitz in some sense\gamma(0)\mapsto\gamma(\tau(\gamma))$ Lipschitz?I would expect yes, but I was not able to find the precise estimate.
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