In mathematics, an **embedding** (or **imbedding**^{ [1] }) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

- Topology and geometry
- General topology
- Differential topology
- Riemannian geometry
- Algebra
- Field theory
- Universal algebra and model theory
- Order theory and domain theory
- Metric spaces
- Normed spaces
- Category theory
- See also
- Notes
- References
- External links

When some object *X* is said to be embedded in another object *Y*, the embedding is given by some injective and structure-preserving map *f* : *X* → *Y*. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which *X* and *Y* are instances. In the terminology of category theory, a structure-preserving map is called a morphism.

The fact that a map *f* : *X* → *Y* is an embedding is often indicated by the use of a "hooked arrow" ( U+21AA↪RIGHTWARDS ARROW WITH HOOK);^{ [2] } thus: (On the other hand, this notation is sometimes reserved for inclusion maps.)

Given *X* and *Y*, several different embeddings of *X* in *Y* may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain *X* with its image *f*(*X*) contained in *Y*, so that *f*(*X*) ⊆ *Y*.

In general topology, an embedding is a homeomorphism onto its image.^{ [3] } More explicitly, an injective continuous map between topological spaces and is a **topological embedding** if yields a homeomorphism between and (where carries the subspace topology inherited from ). Intuitively then, the embedding lets us treat as a subspace of . 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 is neither an open set nor a closed set in .

For a given space , the existence of an embedding is a topological invariant of . This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.

In differential topology: Let and be smooth manifolds and be a smooth map. Then is called an immersion if its derivative is everywhere injective. An **embedding**, or a **smooth embedding**, is defined to be an injective immersion which is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image).^{ [4] }

In other words, the domain of 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 there is a neighborhood such that is an embedding.)

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

An important case is . The interest here is in how large must be for an embedding, in terms of the dimension of . The Whitney embedding theorem ^{ [5] } states that is enough, and is the best possible linear bound. For example, the real projective space **RP**^{m} of dimension , where is a power of two, requires for an embedding. However, this does not apply to immersions; for instance, **RP**^{2} can be immersed in as is explicitly shown by Boy's surface —which has self-intersections. The Roman surface fails to be an immersion as it contains cross-caps.

An embedding is **proper** if it behaves well with respect to boundaries: one requires the map to be such that

- , and
- is transverse to in any point of .

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

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 we have

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).^{ [6] }

In general, for an algebraic category *C*, an embedding between two *C*-algebraic structures *X* and *Y* is a *C*-morphism *e* : *X* → *Y* that is injective.

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. Hence, *E* is isomorphic to the subfield *σ*(*E*) of *F*. This justifies the name *embedding* for an arbitrary homomorphism of fields.

If σ is a signature and are σ-structures (also called σ-algebras in universal algebra or models in model theory), then a map is a σ-embedding iff all of the following hold:

- is injective,
- for every -ary function symbol and we have ,
- for every -ary relation symbol and we have iff

Here is a model theoretical notation equivalent to . In model theory there is also a stronger notion of elementary embedding.

In order theory, an embedding of partially ordered sets is a function *F* between partially ordered sets *X* and *Y* such that

Injectivity of *F* follows quickly from this definition. In domain theory, an additional requirement is that

- is directed.

A mapping of metric spaces is called an *embedding* (with distortion ) if

for some constant .

An important special case is that of normed spaces; in this case it is natural to consider linear embeddings.

One of the basic questions that can be asked about a finite-dimensional normed space is, *what is the maximal dimension such that the Hilbert space can be linearly embedded into with constant distortion?*

The answer is given by Dvoretzky's theorem.

In category theory, there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullbacks.

Ideally the class of all embedded subobjects of a given object, up to isomorphism, should also be small, and thus an ordered set. In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operator).

In a concrete category, an **embedding** is a morphism *ƒ*: *A* → *B* which is an injective function from the underlying set of *A* to the underlying set of *B* and is also an **initial morphism** in the following sense: If *g* is a function from the underlying set of an object *C* to the underlying set of *A*, and if its composition with *ƒ* is a morphism *ƒg*: *C* → *B*, then *g* itself is a morphism.

A factorization system for a category also gives rise to a notion of embedding. If (*E*, *M*) is a factorization system, then the morphisms in *M* may be regarded as the embeddings, especially when the category is well powered with respect to *M*. Concrete theories often have a factorization system in which *M* consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article.

As usual in category theory, there is a dual concept, known as quotient. All the preceding properties can be dualized.

An embedding can also refer to an embedding functor.

- ↑ Spivak 1999 , p. 49 suggests that "the English" (i.e. the British) use "embedding" instead of "imbedding".
- ↑ "Arrows – Unicode" (PDF). Retrieved 2017-02-07.
- ↑ Hocking & Young 1988 , p. 73. Sharpe 1997 , p. 16.
- ↑ Bishop & Crittenden 1964 , p. 21. Bishop & Goldberg 1968 , p. 40. Crampin & Pirani 1994 , p. 243. do Carmo 1994 , p. 11. Flanders 1989 , p. 53. Gallot, Hulin & Lafontaine 2004 , p. 12. Kobayashi & Nomizu 1963 , p. 9. Kosinski 2007 , p. 27. Lang 1999 , p. 27. Lee 1997 , p. 15. Spivak 1999 , p. 49. Warner 1983 , p. 22.
- ↑ Whitney H.,
*Differentiable manifolds,*Ann. of Math. (2),**37**(1936), pp. 645–680 - ↑ Nash J.,
*The embedding problem for Riemannian manifolds,*Ann. of Math. (2),**63**(1956), 20–63.

In mathematics, a **diffeomorphism** is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

In differential geometry, a subject of mathematics, a **symplectic manifold** is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

In differential geometry, a **Riemannian manifold** or **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with a positive-definite inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p*. A common convention is to take *g* to be smooth, which means that for any smooth coordinate chart *(U,x)* on *M*, the *n*^{2} functions

In the mathematical fields of differential geometry and tensor calculus, **differential forms** are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematical physics, **Minkowski space** is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.

In mathematics, an **isometry** is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In the mathematical field of differential geometry, the **Ricci flow**, sometimes also referred to as **Hamilton's Ricci flow**, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation; however, it exhibits many phenomena not present in the study of the heat equation. Many results for Ricci flow have also been shown for the mean curvature flow of hypersurfaces.

In mathematics, a **submersion** is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

In mathematics, the **homotopy principle** is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as occur in the immersion problem, isometric immersion problem, fluid dynamics, and other areas.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In mathematics, a **submanifold** of a manifold *M* is a subset *S* which itself has the structure of a manifold, and for which the inclusion map *S* → *M* satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.

In mathematics and quantum mechanics, a **Dirac operator** is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors.

In the mathematical field of analysis, the **Nash–Moser theorem**, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized problem is not bounded.

In mathematics, **Floer homology** is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.

In differential geometry, a subfield of mathematics, the **Margulis lemma** is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifolds. Roughly, it states that within a fixed radius, usually called the **Margulis constant**, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup.

In mathematics, an **immersion** is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, *f* : *M* → *N* is an immersion if

In Riemannian geometry and pseudo-Riemannian geometry, the **Gauss-Codazzi equations** are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of a Riemannian or pseudo-Riemannian manifold.

In Riemannian geometry, the **filling radius** of a Riemannian manifold *X* is a metric invariant of *X*. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form.

In the field of differential geometry in mathematics, **mean curvature flow** is an example of a geometric flow of hypersurfaces in a Riemannian manifold. Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly. Except in special cases, the mean curvature flow develops singularities.

In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed

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*Abstract and Concrete Categories (The Joy of Cats)*. - Embedding of manifolds on the Manifold Atlas

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