global function defining a $\mathcal{C}^1$ submanifold

Asked Jan 03 '15 at 01:35

Active Jan 03 '15 at 01:35

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Let $M$ be a $\mathcal{C}^1$ submanifold of dimension $1$ of $\mathbb{R}^2$. Then for each $x\in M$, there is a neighbourhood $U$ of $x$ and a $\mathcal{C}^1$ function $f:U\to \mathbb{R}$ such that $M\cap U=f^{-1}(\{0\})\cap U$ and $\nabla f\neq 0$ on $U$.

Question: Is it possible to just define $f$ globally such that $M=f^{-1}(\{0\})$?

asked Jan 03 '15 at 01:35

kaleidoscop

2

Is $M$ compact? If not, there are counterexamples. See Is every embedded submanifold globally a level set?. – Jan 03 '15 at 04:29
@Behaviour, no there are no counterexamples even in the non compact case because such a global $f$ always exists. See also here. – Georges Elencwajg Jan 03 '15 at 10:27

global function defining a $\mathcal{C}^1$ submanifold

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