Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $M$ be a submanifold of $\mathbb R^n$, then is there an open set $\Omega$ in $\mathbb R^n$ such that function $d(x,M)$ (distance function) is smooth on $\Omega$?

share|improve this question

1 Answer 1

Yes. If you choose $\Omega$ to be a small enough normal neighborhood of $M$, then the distance function will be smooth. For a small normal neighborhood, if $x \in \Omega$, then there is a unique vector normal to $M$ that connects $M$ to $x$ and $d(x,M)$ is the length of this vector. Based on this you can show that something called the normal exponential map is a diffeomorphism onto $\Omega$ and this implies that $d$ is smooth in $\Omega$. A complete proof of this fact is proved in many riemannian geometry books (I would check out Bishop and Crittenden's book "Geometry of Manifolds") in the case where $M$ is a hypersurface, and the case where $M$ is not a hypersurface should be similar.

share|improve this answer

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.