# Converse of the regular value theorem

Given a smooth surjective map between two manifolds $$M,N$$, $$f:M\to N$$ with $$\dim M=:m>\dim N=:n$$. Suppose $$f^{-1}(y)$$ is a smooth manifold, how do I show $$y$$ is a regular value?

I am trying to show the Jacobian $$(\frac{\partial f_i}{\partial x_j})_{m\times n}$$ at each $$x\in f^{-1}(y)$$ has maximal rank $$n$$. And I am thinking maybe I can apply the inverse function in the form of: a set of smooth functions $$g_1,...,g_k$$ of $$K$$ ($$\dim K=k$$) with local coordinate charts $$(t_1,...,t_k)$$ of a point $$h\in K$$ forms a coordinate system about $$h$$ iff $$\det (\frac{\partial g_i}{\partial t_j})$$ is non-zero. So far I am stuck on this.

• This is unfortunately false; consider the smooth map $f:\mathbb{R}^3\to\mathbb{R}^2$ given by $f(x,y,z)=(x,0)$. Then $f^{-1}(0,0)=\{0\}\times\mathbb{R}^2$ is an embedded submanifold of $\mathbb{R}^3$, but $(0,0)$ is not a regular value of $f$.
– Ken
Commented Mar 3, 2020 at 4:33
• In general, the mere condition that a map has a constant rank guarantees that the inverse image is an embedded submanifold. (See Theorem 5.12 of J.Lee's Introduction to Smooth Manifolds, for example.)
– Ken
Commented Mar 3, 2020 at 4:38
• @Ken what if $f$ is surjective? I will add onto, thanks for your answering though
– user318946
Commented Mar 3, 2020 at 5:21
• Here's how bad things can be working over $\Bbb R$ (but less so over $\Bbb C$). Suppose $f\colon M\to\Bbb R$ has $0$ as a regular value, so that $f^{-1}(0)$ is a lovely submanifold of $M$. Now take the function $f^2$ instead. $(f^2)^{-1}(0)=M$, but $0$ is a critical value. Commented Mar 3, 2020 at 6:54

This is false. For instance, consider $$f:\mathbb{R}^2\to\mathbb{R}$$ given by $$f(x,y)=x^3-x$$. Then every fiber of $$f$$ is a finite union of vertical lines and in particular is a smooth manifold, but the derivative of $$f$$ vanishes when $$x=\pm1/\sqrt{3}$$ so not every value is a regular value.

• Is that any chance to add some conditions to make that (modified) claim true?
– user318946
Commented Mar 3, 2020 at 6:15
• Well, if you add strong enough conditions it will eventually become true, but nothing at a high level of generality comes to mind. Commented Mar 3, 2020 at 6:18