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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.

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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.

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