# Definition:Immersion

## Open Sets in $\R^n$

Let $m, n \ge 1$ be natural numbers.

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

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

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

### Rank

The rank of an immersion is the rank of its differential at any point.

## Manifolds

An immersion is a map $f: X \to Y$ from one smooth manifold $X$ to another $Y$ such that:

$d f_x: \map {T_x} X \to \map {T_{\map f x} } Y$

is injective.