Rank of smooth map

Definition of Rank

For a smooth map $F:M\to N$ on smooth manifolds with charts (centered at $x$ and $F(x)$) like listed above, the ${\text{rank}}_x(F)$ is the rank of the Jacobian matrix $D_{\varphi (x)}(\psi \circ F\circ \varphi ):{\mathbb{R}}^m\to {\mathbb{R}}^n.$

Set of max rank

The set of points in $M$ that has maximum rank is open. Also, the rank operator $rank:M\to \mathbb{Z}$ that sends $x\mapsto {\text{rank}}_x(F)$ is lower semi-continuous. That means that there is an open neighborhood $U$ around any point $x_0$ such that ${\text{rank}}_x(F)\ge {\text{rank}}_{x_0}(F)$ for $x\in U$.

Note the max rank is either a submersion or immersion depending on the dimensions of the domain or codomain.

Proof

Let ${\text{rank}}_{x_0}(F)=k$ then the Jacobian $D_{\varphi (x_0)}$ has rank $k$ so there is some $k\times k$ sub matrix (or some $k\times k$ minor) that is invertible. Thus, by continuity of determinant, there is an open neighborhood of the matrix that is also invertible and an open neighborhood $V\subset {\mathbb{R}}^n$ such that the Jacobian in invertible. So, following this back up the chart (which also is continuous) we can find the open set $U={\varphi }^{-1}(V)$.

Notes

We can define rank using the differential which I find to be a little easier to understand.