# Definition:Submersion

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## Open Sets in $\R^n$

Let $k, m, n \ge 1$ 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 $\d f$ 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 $\map f x = y$.

Let $\d f_x: \map {T_x} X \to \map {T_y} Y$ be a surjection.

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