Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
Would the statement be true if we assume that the intersections are transversal and replace Lipschitz by locally Lipschitz? –  Christian Aug 29 '11 at 21:17

1 Answer 1

up vote 4 down vote accepted

No, take $N = S^1$, $U = \mathbb{R}^2$ and a suitable collection of line-segments parallel to an axis (for example).

Edit: To respond to the question in the comment: even if we assume the intersections to be transversal we don't get a locally Lipschitz map. Consider for example the following open curve to be our manifold $N$ and the blue (and one red) vertical line-segments to indicate the set of geodesics. Then there is even a discontinuity for our mapping at the red line.

(The open ends of the curve are needed here for the example to work. A good starting point for proving something positive could be a closed curve $N$.)

share|improve this answer
    
Ok, :-) thank you –  Christian Aug 30 '11 at 7:00

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.