# Embedding

*For other uses of this term, see Embedded (disambiguation).*

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

## Contents

## Topology/Geometry

### General topology

In general topology, an embedding is a homeomorphism onto its image. More explicitly, a map *f* : *X* → *Y* between topological spaces *X* and *Y* is an embedding if *f* yields a homeomorphism between *X* and *f*(*X*) (where *f*(*X*) carries the subspace topology inherited from *Y*). Intuitively then, the embedding *f* : *X* → *Y* lets us treat *X* as a subspace of *Y*. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image *f*(*X*) is neither an open set nor a closed set in *Y*.

### Differential geometry

In differential geometry:
Let *M* and *N* be smooth manifolds and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:M\to N}**
be a smooth map, it is called an
**immersion** if for any point **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in M}**
the differential **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_xf:T_x(M)\to T_{f(x)}(N)}**
is injective (here **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_x(M)}**
denotes tangent space of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M}**
at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}**
).
Then an **embedding**, or a **smooth embedding**, is defined to be an immersion which is an embedding in the above sense (i.e. homeomorphism onto its image).
When the manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.

In other words, an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is a local embedding (i.e. for any point **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in M}**
there is a neighborhood **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in U\subset M}**
such that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:U\to N}**
is an embedding.)

An important case is *N*=**R**^{n}. 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* = 2*m* 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 *f* : *M* → *N* which preserves the metric in the sense that *g* is equal to the pullback of *h* by *f*, i.e. *g* = *f***h*. Explicitly, for any two tangent vectors

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v,w\in T_x(M)}**

we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(v,w)=h(df(v),df(w))}**.

Analogously, **isometric immersion** is an immersion between Riemannian manifolds which preserves the Riemannian metrics.

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

## Algebra

### Field theory

In field theory, an **embedding** of a field *E* in a field *F* is a ring homomorphism σ : *E* → *F*.

The kernel of σ is an ideal of *E* which cannot be the whole field *E*, because of the condition σ(1)=1. Therefore the kernel is 0 and thus any embedding of fields is a monomorphism. Moreover, *E* is isomorphic to the subfield σ(*E*) of *F*. This justifies the name *embedding* for an arbitrary homomorphism of fields.

## Domain theory

In domain theory, an **embedding** of partial orders is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F}**
in the function space [X →Y] such that

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow F(x_1)\leq F(x_2)}**and**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall y\in Y:\{x: F(x)\leq y\}}**is directed.

*Based on an article from FOLDOC, used by permission.*