Editing Embedding (section)

===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>.