Editing Submanifold

{{Short description|Subset of a manifold that is a manifold itself; an injective immersion into a manifold}}
[[Image:immersedsubmanifold selfintersection.jpg|thumb|160px|Immersed manifold straight line with self-intersections]]
In [[mathematics]], a '''submanifold''' of a [[manifold]] <math>M</math> is a [[subset]] <math>S</math> which itself has the structure of a manifold, and for which the [[inclusion map]] <math>S \rightarrow M</math> satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.

==Formal definition==

In the following we assume all manifolds are [[differentiable manifold]]s of [[differentiability class|class]] <math>C^r</math> for a fixed <math>r\geq 1</math>, and all morphisms are differentiable of class <math>C^r</math>.

===Immersed submanifolds===
[[Image:immersedsubmanifold nonselfintersection.jpg|thumb|150px|This image of the open interval (with boundary points identified with the arrow marked ends) is an immersed submanifold.]]

An '''immersed submanifold''' of a manifold <math>M</math> is the image <math>S</math> of an [[immersion (mathematics)|immersion]] map <math>f: N\rightarrow M</math>; in general this image will not be a submanifold as a subset, and an immersion map need not even be [[injective]] (one-to-one) – it can have self-intersections.<ref>{{harvnb|Sharpe|1997|page=26}}.</ref>

More narrowly, one can require that the map <math>f: N\rightarrow M</math> be an injection (one-to-one), in which we call it an [[injective]] [[immersion (mathematics)|immersion]], and define an '''immersed submanifold''' to be the image subset <math>S</math> together with a [[topology (structure)|topology]] and [[differential structure]] such that <math>S</math> is a manifold and the inclusion <math>f</math> is a [[diffeomorphism]]: this is just the topology on <math>N</math>'','' which in general will not agree with the subset topology: in general the subset <math>S</math> is not a submanifold of <math>M</math>'','' in the subset topology.

Given any injective immersion <math>f: N\rightarrow M</math> the [[image (mathematics)|image]] of <math>N</math> in <math>M</math> can be uniquely given the structure of an immersed submanifold so that <math>f: N\rightarrow f(N)</math> is a [[diffeomorphism]]. It follows that immersed submanifolds are precisely the images of injective immersions.

The submanifold topology on an immersed submanifold need not be the [[subspace topology]] inherited from <math>M</math>. In general, it will be [[finer topology|finer]] than the subspace topology (i.e. have more [[open set]]s).

Immersed submanifolds occur in the theory of [[Lie group]]s where [[Lie subgroup]]s are naturally immersed submanifolds. They  also appear in the study of [[foliation|foliations]] where immersed submanifolds provide the right context to prove the [[Frobenius theorem (differential topology)|Frobenius theorem]].

===Embedded submanifolds===

An '''embedded submanifold''' (also called a '''regular submanifold'''), is an immersed submanifold for which the inclusion map is a [[topological embedding]]. That is, the submanifold topology on <math>S</math> is the same as the subspace topology.

Given any [[embedding]] <math>f: N\rightarrow M</math> of a manifold <math>N</math> in <math>M</math> the image <math>f(N)</math> naturally has the structure of an embedded submanifold. That is, embedded submanifolds are precisely the images of embeddings.

There is an intrinsic definition of an embedded submanifold which is often useful. Let <math>M</math> be an <math>n</math>-dimensional manifold, and let <math>k</math> be an integer such that <math>0 \leq k \leq n</math>. A <math>k</math>-dimensional embedded submanifold of <math>M</math> is a subset <math>S \subset M</math> such that for every point <math>p \in S</math> there exists a [[chart (topology)|chart]]  <math>U \subset M, \varphi : U \rightarrow \mathbb{R}^n</math> containing <math>p</math> such that <math>\varphi(S \cap U)</math> is the intersection of a <math>k</math>-dimensional [[plane (mathematics)|plane]] with <math>\varphi(U)</math>. The pairs <math>(S\cap U, \varphi\vert_{S\cap U})</math> form an [[atlas (topology)|atlas]] for the differential structure on <math>S</math>.

[[Alexander's theorem]] and the [[Schoenflies theorem|Jordan–Schoenflies theorem]] are good examples of smooth embeddings.

===Other variations===

There are some other variations of submanifolds used in the literature. A [[neat submanifold]] is a manifold whose boundary agrees with the boundary of the entire manifold.<ref>{{harvnb|Kosinski|2007|page=27}}.</ref> Sharpe (1997) defines a type of submanifold which lies somewhere between an embedded submanifold and an immersed submanifold.

Many authors define topological submanifolds also. These are the same as <math>C^r</math> submanifolds with <math>r = 0</math>.<ref>{{harvnb|Lang|1999|pages=25–26}}. {{harvnb|Choquet-Bruhat|1968|page=11}}</ref> An embedded topological submanifold is not necessarily regular in the sense of the existence of a local chart at each point extending the embedding. Counterexamples include [[wild arc]]s and [[wild knot]]s.

==Properties==

Given any immersed submanifold <math>S</math> of <math>M</math>, the [[tangent space]] to a point <math>p</math> in <math>S</math> can naturally be thought of as a [[linear subspace]] of the tangent space to <math>p</math> in <math>M</math>. This follows from the fact that the inclusion map is an immersion and provides an injection
: <math>i_{\ast}: T_p S \to T_p M.</math>

Suppose ''S'' is an immersed submanifold of <math>M</math>. If the inclusion map <math>i: S\to M</math> is [[closed map|closed]] then <math>S</math> is actually an embedded submanifold of <math>M</math>. Conversely, if <math>S</math> is an embedded submanifold which is also a [[closed subset]] then the inclusion map is closed. The inclusion map <math>i: S\to M</math> is closed if and only if it is a [[proper map]] (i.e. inverse images of [[compact set]]s are compact). If <math>i</math> is closed then <math>S</math> is called a '''closed embedded submanifold''' of <math>M</math>. Closed embedded submanifolds form the nicest class of submanifolds.

==Submanifolds of real coordinate space==

Smooth manifolds are sometimes ''defined'' as embedded submanifolds of [[real coordinate space]] <math>\mathbb{R}^n</math>, for some <math>n</math>. This point of view is equivalent to the usual, abstract approach, because, by the [[Whitney embedding theorem]], any [[second-countable space|second-countable]] smooth (abstract) <math>m</math>-manifold can be smoothly embedded in <math>\mathbb{R}^{2m}</math>.

==Notes==
{{Reflist}}

==References==
* {{cite book|last=Choquet-Bruhat|first=Yvonne|author-link=Yvonne Choquet-Bruhat|title=Géométrie différentielle et systèmes extérieurs|publisher=Dunod|location=Paris|year=1968}}
*{{cite book|last=Kosinski|first=Antoni Albert|year=2007|orig-year=1993|title=Differential manifolds|location=Mineola, New York|publisher=Dover Publications|isbn=978-0-486-46244-8}}
*{{Cite book| isbn = 978-0-387-98593-0 | title = Fundamentals of Differential Geometry | last1 = Lang | first1 = Serge |author-link1=Serge Lang| year = 1999 |publisher=Springer|location=New York| series = [[Graduate Texts in Mathematics]]}}
*{{cite book | first = John | last = Lee | year = 2003 | title = Introduction to Smooth Manifolds | series = Graduate Texts in Mathematics '''218''' | location = New York | publisher = Springer | isbn = 0-387-95495-3}}
*{{cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer | location = New York | year=1997 | isbn=0-387-94732-9}}
*{{cite book | last = Warner | first = Frank W. | title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer |location = New York | year=1983 | isbn=0-387-90894-3}}

{{Manifolds}}
{{Authority control}}

[[Category:Differential topology]]
[[Category:Manifolds]]