Definition:Immersion

Open Sets in $\R^n$
Let $m,n\geq1$ be natural numbers.

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

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

Then $f$ is a $C^k$-immersion $f$ 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:
 * $d f_x: T_x \left({X}\right) \to T_{f \left({x}\right)} \left({Y}\right)$

is injective.

Also see

 * Definition:Immersed Submanifold
 * Definition:Embedding (Differential Geometry)
 * Definition:Submersion