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.

I have the following problem. Let $M$ be a compact $C^r$-manifold (with $r>1$) of dimension $m$ embedded in an euclidean space of dimension $k$. I am told that then there is some $\epsilon >0$ such that no $(k-1)$-dimensional sphere of radius $\epsilon$ tangent to $M$ intersects $M$ other than in its point of tangency.

I want to know how can I prove this and, moreover, where enters the condition $C^r$ with $r>1$ (in particular, why the result is false for $C^1$-manifolds).

Thanks in advance.

share|improve this question
add comment

1 Answer

This has to do with curvature, so you need the manifold to admit a notion of curvature, so that in particular $M$ would have to be at least $C^2$. In fact, once you understand the 1-dimensional situation, you will understand the general situation. Given a compact $C^2$ curve, by compactness its curvature admits an upper bound $K$, and then any circle of radius smaller than $1/K$ that has a point of tangency with the curve $M$, will not meet it in any nearby point. Also compactness guarantees that the circle will not meet $M$ in "faraway" points if the radius is small enough. In general you will need to work with the shape operator (Weingarten map) of $M$ and its eigenvalues, and proceed similarly.

share|improve this answer
add comment

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.