Regular value theorem

Statement

For a smooth map $F:M\to N$ on smooth manifolds with $\dim M=k$ and $\dim N=l$, let $y\in N$ be a regular value. The level set $F^{-1}(\{y\})$ is an embedded submanifold with codimension is equal to the dimension of the codomain (in other words $\dim F^{-1}(\{y\})=k-l$).

Proof

Let $y\in N$ be a regular value. We know we can find charts $\psi$ and $\varphi$ centered at $y$ and $x\in F^{-1}(y)$ such that for open sets $U\subset M$, $V\subset N$, $U^{\prime }\subset {\mathbb{R}}^k$, and $V^{\prime }\subset {\mathbb{R}}^l$

$$\begin{array}{rl}\psi \circ F\circ {\varphi }^{-1}:U^{\prime }& ⟶V^{\prime }\\ (x_1,\dots ,x_k)& ⟼(x_1,\dots ,x_l)\end{array}$$

This can be seen in the following diagram:

Thus, we can take the level set $W=F^{-1}(y)\subset U$, and this image under $\varphi$ (call it $W^{\prime }$) would all be in the kernel of ${\varphi }^{-1}\circ F\circ \psi$. Thus, $({\varphi }^{-1}\circ F\circ \psi )(W^{\prime })=\{0\}$ So $W^{\prime }=\{(x_1,\dots ,x_n)\in U^{\prime }|x_1=\cdots =x_l=0\}=\{0\}\times {\mathbb{R}}^{(k-l)}$ This satisfies the definition of an submanifold. (Note that in the last equation, it should be just on an open set.)

Uses/ why we care

This is a very easy way to find submanifolds for some smooth manifold.