Definition:Embedding (Differential Geometry)

This page is about embedding in the context of differential geometry. For other uses, see embedding.

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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.

Also see

• Definition:Immersion
• Definition:Submersion