Confusion about whether or not an embedded submanifold can be written as $f^{-1}(0)$ for some $f:M\rightarrow \mathbb R$

Asked May 10 '24 at 19:42

Active May 10 '24 at 19:42

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Let $M$ be a smooth manifold, then this post here seems to indicate that embedded submanifolds (that is not an open submanifold) need not be $f^{-1}(0)$ for some smooth function $f:M\rightarrow \mathbb R$, however, this seems to contradict the following statement theorem from Lee:

Let $M$ be a smooth manifold, then for any closed $K\subset M$ there is a smooth nonnegative function such that $f^{-1}(0)=K$.

What is going on here? Am I missing something?

asked May 10 '24 at 19:42

Chris

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Note that 0 is not required to be a regular value for $f$, in contrast to the linked question. – Ben Steffan May 10 '24 at 19:51
Ok so every submanifold can be written as $f^{-1}(0)$ for some smooth function? – Chris May 10 '24 at 19:55
That's what Lee's proposition says, yes. – Ben Steffan May 10 '24 at 19:56
Every closed subset is the zero level set of some smooth function. The closedness assumption is obviously necessary. – Moishe Kohan May 10 '24 at 20:36

Confusion about whether or not an embedded submanifold can be written as $f^{-1}(0)$ for some $f:M\rightarrow \mathbb R$

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