Editing Embedding (section)

==Topology and geometry==
===General topology===

In [[general topology]], an embedding is a [[homeomorphism]] onto its image.<ref>{{harvnb|Hocking|Young|1988|page=73}}. {{harvnb|Sharpe|1997|page=16}}.</ref> More explicitly, an injective [[continuous function (topology)|continuous]] map <math>f : X \to Y</math> between [[topological space]]s <math>X</math> and <math>Y</math> is a '''topological embedding''' if <math>f</math> yields a homeomorphism between <math>X</math> and <math>f(X)</math> (where <math>f(X)</math> carries the [[subspace topology]] inherited from <math>Y</math>). Intuitively then, the embedding <math>f : X \to Y</math> lets us treat <math>X</math> as a [[subspace topology|subspace]] of <math>Y</math>. Every embedding is injective and [[continuous function (topology)|continuous]]. Every map that is injective, continuous and either [[open map|open]] or [[closed map|closed]] is an embedding; however there are also embeddings that are neither open nor closed. The latter happens if the image <math>f(X)</math> is neither an [[open set]] nor a [[closed set]] in <math>Y</math>.

For a given space <math>Y</math>, the existence of an embedding <math>X \to Y</math> is a [[topological invariant]] of <math>X</math>. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.

====Related definitions====

If the domain of a function <math>f : X \to Y</math> is a [[topological space]] then the function is said to be ''{{visible anchor|locally injective at a point}}'' if there exists some [[Neighbourhood (mathematics)|neighborhood]] <math>U</math> of this point such that the restriction <math>f\big\vert_U : U \to Y</math> is injective. It is called ''{{visible anchor|locally injective}}'' if it is locally injective around every point of its domain. Similarly, a ''{{visible anchor|local topological embedding|text=local (topological, resp. smooth) embedding}}'' is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp. smooth) embedding. 

Every injective function is locally injective but not conversely. [[Local diffeomorphism]]s, [[local homeomorphism]]s, and smooth [[Immersion (mathematics)|immersion]]s are all locally injective functions that are not necessarily injective. The [[inverse function theorem]] gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every [[Fiber (mathematics)|fiber]] of a locally injective function <math>f : X \to Y</math> is necessarily a [[Discrete space|discrete subspace]] of its [[Domain of a function|domain]] <math>X.</math>

===Differential topology===

In [[differential topology]]:
Let <math>M</math> and <math>N</math> be smooth [[manifold]]s and <math>f:M\to N</math> be a smooth map. Then <math>f</math> is called an [[immersion (mathematics)|immersion]] if its [[pushforward (differential)|derivative]] is everywhere injective. An '''embedding''', or a '''smooth embedding''', is defined to be an immersion that is an embedding in the topological sense mentioned above (i.e. [[homeomorphism]] onto its image).<ref>{{harvnb|Bishop|Crittenden|1964|page=21}}. {{harvnb|Bishop|Goldberg|1968|page=40}}. {{harvnb|Crampin|Pirani|1994|page=243}}. {{harvnb|do Carmo|1994|page=11}}. {{harvnb|Flanders|1989|page=53}}. {{harvnb|Gallot|Hulin|Lafontaine|2004|page=12}}. {{harvnb|Kobayashi|Nomizu|1963|page=9}}. {{harvnb|Kosinski|2007|page=27}}. {{harvnb|Lang|1999|page=27}}. {{harvnb|Lee|1997|page=15}}. {{harvnb|Spivak|1999|page=49}}. {{harvnb|Warner|1983|page=22}}.</ref> 
 
In other words, the domain of an embedding is [[diffeomorphism|diffeomorphic]] to its image, and in particular the image of an embedding must be a [[submanifold]]. An immersion is precisely a '''local embedding''', i.e. for any point <math>x\in M</math> there is a neighborhood <math>x\in U\subset M</math> such that <math>f:U\to N</math> 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 <math>N = \mathbb{R}^n</math>. The interest here is in how large <math>n</math> must be for an embedding, in terms of the dimension <math>m</math> of <math>M</math>. The [[Whitney embedding theorem]]<ref>Whitney H., ''Differentiable manifolds,'' Ann. of Math. (2), '''37''' (1936), pp. 645–680</ref> states that <math>n = 2m</math> is enough, and is the best possible linear bound. For example, the [[real projective space]] <math>\mathbb{R}\mathrm{P}^m</math> of dimension <math>m</math>, where <math>m</math> is a power of two, requires <math>n = 2m</math> for an embedding. However, this does not apply to immersions; for instance, <math>\mathbb{R}\mathrm{P}^2</math> can be immersed in <math>\mathbb{R}^3</math> as is explicitly shown by [[Boy's surface]]&mdash;which has self-intersections. The [[Roman surface]] fails to be an immersion as it contains [[cross-cap]]s.

{{Anchor|ProperEmbedding}}An embedding is '''proper''' if it behaves well with respect to [[Topological manifold#Manifolds with boundary|boundaries]]: one requires the map <math>f: X \rightarrow Y</math> to be such that

*<math>f(\partial X) = f(X) \cap \partial Y</math>, and
*<math>f(X)</math> is [[Transversality (mathematics)|transverse]] to <math>\partial Y</math> in any point of <math>f(\partial X)</math>.

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

===Riemannian and pseudo-Riemannian geometry===

In [[Riemannian geometry]] and pseudo-Riemannian geometry:
Let <math>(M,g)</math> and <math>(N,h)</math> be [[Riemannian manifold]]s or more generally [[pseudo-Riemannian manifold]]s.
An '''isometric embedding''' is a smooth embedding <math>f:M\rightarrow N</math> that preserves the  (pseudo-)[[Riemannian metric|metric]] in the sense that <math>g</math> is equal to the [[pullback (differential geometry)|pullback]] of <math>h</math> by <math>f</math>, i.e. <math>g=f^{*}h</math>. Explicitly, for any two tangent vectors <math>v,w\in T_x(M)</math> we have

:<math>g(v,w)=h(df(v),df(w)).</math>

Analogously, '''isometric immersion''' is an immersion between (pseudo)-Riemannian manifolds that preserves the (pseudo)-Riemannian metrics.

Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) that preserves length of [[curve]]s (cf. [[Nash embedding theorem]]).<ref>Nash J.,  ''The embedding problem for Riemannian manifolds,'' Ann. of Math. (2), '''63''' (1956), 20–63.</ref>