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| [[Image:immersedsubmanifold selfintersection.jpg|thumb|160px|Immersed manifold straight line with selfintersections]]
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| 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.
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| ==Formal definition==
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| In the following we assume all manifolds are [[differentiable manifold]]s of [[differentiability class|class]] ''C''<sup>''r''</sup> for a fixed ''r'' ≥ 1, and all morphisms are differentiable of class ''C''<sup>''r''</sup>.
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| ===Immersed submanifolds===
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| [[Image:immersedsubmanifold nonselfintersection.jpg|thumb|150px|Immersed submanifold open interval with interval ends mapped to arrow marked ends]]
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| An '''immersed manifold''' of a manifold ''M'' is the image ''S'' of an [[immersion (mathematics)|immersion]] map ''f'': ''N'' → ''M''; 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>{{cite book | last = Sharpe | first = R. W. | year=1997| title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program| place=New York| publisher=Springer| page=26}}</ref>
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| More narrowly, one can require that the map ''f'': ''N'' → ''M'' be an inclusion (one-to-one), in which we call it an [[injective]] [[immersion (mathematics)|immersion]], and define an '''immersed submanifold''' to be the image subset ''S'' together with a [[topology (structure)|topology]] and [[differential structure]] such that ''S'' is a manifold and the inclusion ''f'' is a [[diffeomorphism]]: this is just the topology on ''N,'' which in general will not agree with the subset topology: in general the subset ''S'' is not a submanifold of ''M,'' in the subset topology.
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| Given any injective immersion ''f'' : ''N'' → ''M'' the [[image (mathematics)|image]] of ''N'' in ''M'' can be uniquely given the structure of an immersed submanifold so that ''f'' : ''N'' → ''f''(''N'') is a [[diffeomorphism]]. It follows that immersed submanifolds are precisely the images of injective immersions.
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| The submanifold topology on an immersed submanifold need not be the [[relative topology]] inherited from ''M''. In general, it will be [[finer topology|finer]] than the subspace topology (i.e. have more [[open set]]s).
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| Immersed submanifolds occur in the theory of [[Lie group]]s where [[Lie subgroup]]s are naturally immersed submanifolds.
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| ===Embedded submanifolds===
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| 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 ''S'' is the same as the subspace topology.
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| Given any [[embedding]] ''f'' : ''N'' → ''M'' of a manifold ''N'' in ''M'' the image ''f''(''N'') naturally has the structure of an embedded submanifold. That is, embedded submanifolds are precisely the images of embeddings.
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| There is an intrinsic definition of an embedded submanifold which is often useful. Let ''M'' be an ''n''-dimensional manifold, and let ''k'' be an integer such that 0 ≤ ''k'' ≤ ''n''. A ''k''-dimensional embedded submanifold of ''M'' is a subset ''S'' ⊂ ''M'' such that for every point ''p'' ∈ ''S'' there exists a [[chart (topology)|chart]] (''U'' ⊂ ''M'', φ : ''U'' → '''R'''<sup>''n''</sup>) containing ''p'' such that φ(''S'' ∩ ''U'') is the intersection of a ''k''-dimensional [[plane (mathematics)|plane]] with φ(''U''). The pairs (''S'' ∩ ''U'', φ|<sub>''S'' ∩ ''U''</sub>) form an [[atlas (topology)|atlas]] for the differential structure on ''S''.
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| Alexander's theorem and the Jordan-Schoenflies theorem are good examples of smooth embeddings.
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| ===Other variations===
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| 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. Sharpe (1997) defines a type of submanifold which lies somewhere between an embedded submanifold and an immersed submanifold.
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| ==Properties==
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| Given any immersed submanifold ''S'' of ''M'', the [[tangent space]] to a point ''p'' in ''S'' can naturally be thought of as a [[linear subspace]] of the tangent space to ''p'' in ''M''. This follows from the fact that the inclusion map is an immersion and provides an injection
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| : <math>i_{\ast}: T_p S \to T_p M.</math>
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| Suppose ''S'' is an immersed submanifold of ''M''. If the inclusion map ''i'' : ''S'' → ''M'' is [[closed map|closed]] then ''S'' is actually an embedded submanifold of ''M''. Conversely, if ''S'' is an embedded submanifold which is also a [[closed subset]] then the inclusion map is closed. The inclusion map ''i'' : ''S'' → ''M'' is closed if and only if it is a [[proper map]] (i.e. inverse images of [[compact set]]s are compact). If ''i'' is closed then ''S'' is called a '''closed embedded submanifold''' of ''M''. Closed embedded submanifolds form the nicest class of submanifolds.
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| ==Submanifolds of Euclidean space==
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| Manifolds are often ''defined'' as embedded submanifolds of [[Euclidean space]] '''R'''<sup>''n''</sup>, so this forms a very important special case. By the [[Whitney embedding theorem]] any [[second-countable space|second-countable]] smooth ''n''-manifold can be smoothly embedded in '''R'''<sup>2''n''</sup>.
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{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}}
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| *{{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}}
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| *{{cite book | last = Warner | first = Frank W. | title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer | edition = |location = New York | year=1983 | isbn=0-387-90894-3}}
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| [[Category:Differential topology]]
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| [[Category:Manifolds]]
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