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.