# Definition:Embedding (Differential Geometry)

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## Definition

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 $C^k$-**embedding** if and only if $f$ is:

- injective
- a $C^k$-immersion
- a homeomorphism on its image

### Rank

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

### Smooth Embedding

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