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.

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.

For a given space X, the existence of an embedding X → Y is a topological invariant of X. This allows two spaces to be distinguished if one is able to be embedded into a space which the other is not.

An embedding is proper if it behaves well w.r.t. boundaries: one requires the map $f: X \rightarrow Y$ to be such that

$f(\partial X) = f(X) \cap \partial Y$ , and
$f(X)$ is transversal to $\partial Y$ in any point of $f(\partial X)$ .

The first condition is equivalent to having $f(\partial X) \subseteq \partial Y$ and $f(X \setminus \partial X) \subseteq Y \setminus \partial Y$ . The second condition, roughly speaking, says that f(X) is not tangent to the boundary of Y.

Differential geometry

In differential geometry: Let M and N be smooth manifolds and $f:M\to N$ be a smooth map, it is called an immersion if for any point $x\in M$ the differential $d_xf:T_x(M)\to T_{f(x)}(N)$ is injective (here $T_x(M)$ denotes tangent space of $M$ at $x$ ). 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 $x\in M$ there is a neighborhood $x\in U\subset M$ such that $f:U\to N$ 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. The Roman surface fails to be an immersion as it contains cross-caps.

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

v,w\in T_x(M)

we have

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. Furthermore, it is a well-known property of fields that their only ideals are the zero ideal and the whole field itself. Therefore, the kernel is 0, so 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 $F$ in the function space →Y such that

$\forall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow F(x_1)\leq F(x_2)$ and
$\forall y\in Y:\{x: F(x)\leq y\}$ is directed.

Based on an article from FOLDOC, Foldoc license.

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