When a set $F^{-1}(0)$ isn't a smooth submanifold

Question

Let F be a smooth function: $\mathbb{R}^n \to \mathbb{R}$. If the rank of a function is not constant, does it imply that $F^{-1}(0)$ isn't a smooth submanifold. And if it doesn't, is there some way to find out if it's a manifold or not.

Answer 1

In general, if a map does not have constant rank there is no way to conclude if $F^{-1}(0)$ isn't a smooth submanifold without more information.

Consider the map $f: \mathbb{R}\to \mathbb{R}$ given by $f(x)=x^2$. $df:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is given by $df_x(a)=2xa$. This doesn't have constant rank and yet $f^{-1}(0)=\{0\}$ is a smooth submanifold of $\mathbb{R}$. More generally, if I take any two functions $f,g: \mathbb{R}^n\to \mathbb{R}$ with $0$ a regular value and $f^{-1}(0)=g^{-1}(0)$, then $d(fg)=fdg+gdf$ which is identically $0$ along $f^{-1}(0)$.