Smooth submersion

Let $M$ and $N$ be smooth manifolds of dimensions $m$ and $n$, and let $f : M \to N$ be a smooth map .

Definition

The map $f$ is a smooth submersion if for every $p\in M$, the differential (pushforward)

is surjective, where $T_pM$ is the tangent space at $p$.

Equivalently, $\operatorname{rank}(d f_p)=n$ for all $p\in M$. In particular, if a submersion $M^m \to N^n$ exists then $m \ge n$.

Local normal form (Submersion theorem)

If $f : M^m \to N^n$ is a smooth submersion and $p\in M$, then there exist smooth charts $(U,\varphi)$ around $p$ and $(V,\psi)$ around $f(p)$ (see coordinate charts ) such that, in these coordinates,

So a submersion is locally just a projection map.

A useful viewpoint is that if $f$ is a smooth submersion, then every $y\in N$ is a regular value of $f$. Consequently, each fiber $f^{-1}(y)$ is (locally) a smooth submanifold of codimension $n$.

Examples

1. Projection from a product. For smooth manifolds $M$ and $F$, the projection

   $$\pi_M : M\times F \to M,\qquad \pi_M(m,f)=m$$

   is a smooth submersion: its differential is surjective at every point. (This is the basic model for bundle projections, and its fibers are the slices $\{m\}\times F$.)

2. Coordinate projection. The map $\mathbb{R}^3\to \mathbb{R}^2$, $(x,y,z)\mapsto (x,y)$, is a smooth submersion since its Jacobian matrix has rank $2$ everywhere.

3. Determinant on $GL(n,\mathbb{R})$. The determinant map

   $$\det:GL(n,\mathbb{R}) \to \mathbb{R}\setminus\{0\}$$

   is a smooth submersion. At $A\in GL(n,\mathbb{R})$, one has

   $$d(\det)_A(B)=\det(A)\,\mathrm{tr}(A^{-1}B),$$

   and since $\det(A)\neq 0$ we can choose $B$ so that $d(\det)_A(B)$ is any prescribed real number, proving surjectivity.