Embedding

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

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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.

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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.

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Differential geometry

In differential geometry:

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Let M and N be smooth manifolds and f:M o N be a smooth map, it is called an

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immersion if for any point xin M the differential d_f:T_x(M) o T_{f(x)}(N) is injective (here T_x(M) denotes tangent space of M at x).

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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).

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When the manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.

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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 xin M there is a neighborhood xin Usubset M such that f:U o N is an embedding.)

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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.

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Riemannian geometry

In Riemannian geometry:

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Let (M,g) and (N,h) be Riemannian manifolds.

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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

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:v,win T_x(M)

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we have

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:g(v,w)=h(df(v),df(w)).

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

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

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Group: The term group can refer to several concepts:...

That: The word that is used in the English language for several grammatical purposes:...

Subgroup: In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation on H....