I am looking for an example of a smooth manifold of dimension $> 0$ that arises as the preimage of a critical value of a smooth function $F : \mathbb{R}^n \to \mathbb{R}$. The first comment to the question below (How to show a level set isn't a regular submanifold) gives an argument as to why no such creature exists that seems very plausible to me, but I might be missing something.
Thoughts?
Keep it simple. The preimage of zero under the zero map from $\mathbb{R}^n\to \mathbb{R}$ is all of $\mathbb{R}^n$, which is of course a smooth manifold. The preimage of a critical value can fail to be a manifold, but it doesn't have to.
Consider $F: \mathbb{R}^2\to \mathbb{R}$ given by $F(x,y)=x^2$. Then each $(0,y)$ is a critical point, and so $0$ is a critical value. Further, $F^{-1}(\{0\})$ is a line.
