Submersion

Remarks

Equivalently, the rank of $df_p$ equals $\dim N$ at every point $p$; when $M$ is not empty this forces $\dim M \ge \dim N$. A point $p$ where $df_p$ is surjective is a regular point, so a submersion is a smooth map all of whose points are regular. Example: the projection $\pi \colon \R^3 \to \R^2$, $\pi(x, y, z) = (x, y)$, is a submersion. Its differential at every point is the linear map $\pi$ itself, which is onto. Non-example: $f \colon \R^2 \to \R$, $f(x, y) = x^2 + y^2$, is not a submersion. At the origin $df_0 = 0$, which is not surjective onto $\R$. By the constant rank theorem, near each point a submersion has the local form $(x_1, \dots, x_m) \mapsto (x_1, \dots, x_n)$ in suitable charts.