# Definition:Submersion

## Open Sets in $\R^n$

Let $k, m, n\geq1$ be natural numbers.

Let $U\subset \R^n$ be open.

Let $f : U \to \R^m$ be a mapping.

Then $f$ is a $C^k$-**submersion** if and only if $f$ is of class $C^k$ and its differential $df$ is surjective at every point of $U$.

### Rank

The **rank** of a submersion is the rank of its differential at any point.

### Smooth Submersion

## Manifolds

Let $X$ and $Y$ be manifolds, with $\dim X \ge \dim Y$.

Let $f: X \to Y$ be smooth and $f \left({x}\right) = y$.

Let $df_x: T_x \left({X}\right) \to T_y \left({Y}\right)$ be a surjection.

Then $f$ is a **submersion** of $X$ on $Y$.