Embedding

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In mathematics, an embedding is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.

Table of contents

1 Topology/Geometry

1.1 General topology
1.2 Differential geometry
1.3 Riemannian geometry

2 Domain theory

In general topology: an embedding is a homeomorphism onto its image. In other words it can be thought of as an inclusion map on a space X that is considered as a subspace of Y, such that the topology on X is the same as the subspace topology it has in Y.

Differential geometry

In differential geometry: Let M and N be smooth manifolds and be a smooth map, it is called an immersion if for any point the differential is injective (here denotes tangent space of at ). An embedding, or smooth embedding, is an injective immersion.

In other words embedding is diffeomorphism to its image (in particular image of embedding is a submanifold). Immersion is a local embedding (i.e. for any point there is a naighborhood such that is an embedding.)

An important case is N=Rⁿ. The interest here is in how large n must be, in terms of the dimension m of M. The Whitney embedding theorem states that n = 2m is enough. For example the real projective plane of dimension 2 requires n = 4 for an embedding. The less restrictive condition of immersion applies to the Boy's surface - which has self-intersections.

Riemannian geometry

In Riemannian geometry: Let (M,g) and (N,h) be Riemannian manifolds. An isometric embedding is a smooth embedding which preserves the Riemannian metric, i.e. for any two tangent vectors

we have

Equivalently, an isometric embedding is a smooth embedding which preserves length of curves (cf. Nash embedding theorem).

Domain theory

In domain theory, an embedding of partial orders is F in the function space [X → Y] such that

For all x₁, x₂ in X, x₁ ≤ x₂ if and only if F (x₁) ≤ F(x₂) and
For all y in Y, {x : F (x) ≤ y } is directed.

Based on an article from FOLDOC, used by permission.