# Can the level set of a critical value be a regular submanifold?

I know that there is a theorem that a non-empty level set of a regular value of a smooth function $f:M\rightarrow\mathbb{R}$ on a smooth manifold is a regular submanifold (or embedded submanifold) of codimension 1.

Now I wonder if there is also a condition in which the converse holds true. I mean suppose $c\in\mathbb{R}$ is a critical value of $f$ is there a condition under which you know the level set of $c$ is not a regular submanifold.

If $g:M\rightarrow\mathbb{R}$ is defined by mapping all of $M$ to $0$ then it seems to me that $0$ is a critical value of $g$. Now $g^{-1}(0)=M$ so definitely a regular submanifold of $M$. So I know that it is at least not true without an extra condition.

Also if the function is second degree polynomial from $\mathbb{R}$ to $\mathbb{R}$ then the level set of a critical value is just a point which would again be a regular submanifold.

If there is no such general condition on the function $f$ or the critical value then how in general does one go about showing that a critical level set is or is not a regular submanifold?

• By the way, I would love a reference for the statement in the first theorem, or even better, a textbook which I can pass to a student of mine who needs to know these things. Oct 13, 2012 at 23:59
• You probably should say amend the statement of the theorem to say that a non-empty level set of a regular value of a smooth function f:M→ℝ on a smooth manifold is a regular submanifold of codimension one. (That takes care of the problem with f identically zero.) Oct 14, 2012 at 0:16
• The most immediate converse is that a manifold with a trivial normal bundle is the level-set corresponding to a regular value. If you have a non-trivial normal bundle, you will need to allow for more degenerate functions. Bott-type Morse functions are a standard generalization. Oct 14, 2012 at 1:28
• @Andrej If you or your student Google submersion theorem or constant rank theorem you or they should find it. Or look in any basic differential geometry text under those names. Oct 14, 2012 at 12:39
• @Andrej I found this theorem in "An Introduction to Manifolds" by Loring W. Tu. In the capter on submanifolds. The constant rank is indeed a generalization of this theorem. Oct 14, 2012 at 14:23

Suppose that $0$ is a regular value of $f$. Then $0$ is a critical value of $g=f^2$, yet the level set $g^{-1}(0)$ is a regular submanifold. In this case all the points on $g^{-1}(0)$ are critical points of $g$.
The general answer is difficult. You need to assume something about $f$. A natural assumption would be that the critical points of $f$ are isolated and of finite type. Near such points one can find local coordinates so that in these coordinate $f$ looks like a polynomial. (This is a generalization of the classical Morse lemma due to Tougeron.) In such cases you need to understand the zero sets of real polynomials which can be challenging. A nice place to consult for such issues is the book by Arnold, Gussein-Zade and Varchenko on singularities of differentiable mappings.