Panel analysis - Revision history

129.194.8.73 at 10:55, 16 April 2014

2014-04-16T10:55:36Z

← Older revision		Revision as of 12:55, 16 April 2014
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	~~In [[differential geometry]], a field of [[mathematics]], a '''normal bundle''' is a particular kind of [[vector bundle]], [[complementary angles\|complementary]] to~~ the ~~[[tangent bundle]], and coming from an [[embedding]] (or [[immersion (mathematics)\|immersion]]).~~		Hi there. Allow me start by introducing the author, her name is Myrtle Cleary. To gather coins is a factor that I'm completely addicted to. Her family lives in Minnesota. For many years he's been working as a receptionist.<br><br>my website - over the counter std test ([http://www.siccus.net/blog/15356 visit the next internet site])

	~~==Definition==~~
	~~===Riemannian manifold===~~
	~~Let <math>(M,g)</math> be a [[Riemannian manifold]], and <math>S \subset M</math> a [[Riemannian submanifold]]. Define, for a given <math>p \in S</math>~~, ~~a vector <math>n \in \mathrm{T}_p M</math> to be ''normal'' to <math>S</math> whenever <math>g(n,v)=0</math> for all <math>v\in \mathrm{T}_p S</math> (so that <math>n</math>~~ is ~~[[orthogonal complement\|orthogonal]] to <math>\mathrm{T}_p S</math>)~~. ~~The set <math>\mathrm{N}_p S</math> of all such <math>n</math>~~ is ~~then called the ''normal space'' to <math>S</math> at <math>p</math>.~~

	~~Just as the total space of the [[tangent bundle]] to~~ a ~~manifold is constructed from all [[tangent space]]s to the manifold, the total space of the~~ '~~'normal bundle'' <math>\mathrm{N} S</math> to <math>S</math> is defined as~~
	~~:<math>\mathrm{N}S := \coprod_{p \in S} \mathrm{N}_p S</math>.~~

	~~The '''conormal bundle''' is defined as the [[dual bundle]]~~ to ~~the normal bundle. It can be realised naturally as a sub-bundle of the [[cotangent bundle]]~~.

	~~===General definition===~~
	~~More abstractly, given an [[immersion (mathematics)\|immersion]] <math>i\colon N \to M</math> (for instance an embedding), one can define a normal bundle of ''N''~~ in ~~''M'', by at each point of ''N'', taking the [[quotient space (linear algebra)\|quotient space]] of the tangent space on ''M'' by the tangent space on ''N''~~. For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot (such a choice is equivalent to a [[section (category theory)\|section]] of the projection <math>V \to V/W</math>).

	~~Thus the normal bundle is in general a~~ '~~'quotient'' of the tangent bundle of the ambient space restricted to the subspace.~~

	~~Formally, the normal bundle to ''N'' in ''M'' is a quotient bundle of the tangent bundle on ''M'': one has the [[short exact sequence]] of vector bundles on ''N'':~~
	~~:<math>0 \to TN \to TM\vert_{i(N)} \to T_{M/N} := TM\vert_{i(N)} / TN \to 0</math>~~
	where <math>TM\vert_{i(N)}</math> is the restriction of the tangent bundle on ''M'' to ''N'' (properly, the pullback <math>i^*TM</math> of the tangent bundle on ''M'' to a vector bundle on ''N'' via the map <math>i</math>).

	~~==Stable normal bundle==~~
	~~[[abstraction\|Abstract]] [[manifolds]] have a [[canonical form\|canonical]] tangent bundle, but do not have a normal bundle: only an embedding (or immersion) of a manifold in another yields~~ a ~~normal bundle~~.
	~~However, since every compact manifold can be embedded in <math>\mathbf{R}^N</math>, by the [[Whitney embedding theorem]], every manifold admits a normal bundle, given such an embedding.~~

	~~There is in general no natural choice of embedding, but for a given ''M'', any two embeddings in~~ <~~math~~>~~\mathbf{R}^N~~<~~/math~~> for sufficiently large ''N'' are [[regular homotopy\|regular homotopic]], and hence induce the same normal bundle. The resulting class of normal bundles (it is a class of bundles and not a specific bundle because ''N'' could vary) is called the [[stable normal bundle]].

	~~==Dual to tangent bundle==~~
	~~The normal bundle is dual to the tangent bundle in the sense of [[K~~-~~theory]]:~~
	~~by the above short exact sequence,~~
	~~:<math>[TN] + [T_{M/N}] = [TM]</math>~~
	in the ~~[[Grothendieck group]].~~
	~~In case of an immersion in <math>\mathbf{R}^N</math>, the tangent bundle of the ambient space is trivial~~ (~~since <math>\mathbf{R}^N</math> is contractible, hence~~ [~~[parallelizable]]), so <math>[TN] + [T_{M/N}] = 0</math>, and thus <math>[T_{M/N}] = -[TN]</math>.~~

	~~This is useful in the computation of [[characteristic classes]], and allows one to prove lower bounds on immersibility and embeddability of manifolds in [[Euclidean space]].~~

	~~==For symplectic manifolds==~~
	Suppose a manifold <math>X</math> is embedded in to a [[symplectic manifold]] <math>(M,\omega)</math>, such that the pullback of the symplectic form has constant rank on <math>X</math>. Then one can define the symplectic normal bundle to X as the vector bundle over X with fibres
	:~~<math> (T_{i(x)}X)^\omega/(T_{i(x)}X\cap (T_{i(x)}X)^\omega), \quad x\in X,<~~/~~math>~~
	~~where <math>i:X\rightarrow M<~~/math> denotes the embedding. Notice that the constant rank condition ensures that these normal spaces fit together to form a bundle. Furthermore, any fibre inherits the structure of a symplectic vector space.

	~~By [[Darboux's theorem]], the constant rank embedding is locally determined by <math>i*(TM)</math>~~. ~~The isomorphism~~
	~~:<math> i^(TM)\cong TX~~/~~\nu \oplus (TX)^\omega~~/~~\nu \oplus(\nu\oplus \nu^), \quad \nu=TX\cap (TX)^\omega,</math>~~
	~~of symplectic vector bundles over <math>X</math> implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.~~

	~~==Algebraic geometry==~~
	~~In [[algebraic geometry]],~~ the ~~normal bundle ''N''<sub>''X''</sub>''Y'' of a [[regular embedding]~~] i: ''X'' → ''Y'', defined by some sheaf of ideals ''I'' is the vector bundle on ''X'' corresponding to the dual of the sheaf ''I''/''I''<sup>2</sup>. The regularity of the embedding ensures that this sheaf is locally free and agrees with the ''normal cone'' C<sub>''X''</sub>''Y'', which is defined as <math>Spec \oplus_{n \geq 0} I^n / I^{n+1}</math>.<ref>{{Citation \| last1=Fulton \| first1=William \| author1-link=William Fulton (mathematician) \| title=Intersection theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] \| isbn=978-3-540-62046-4; 978-0-387-98549-7 \| id={{MathSciNet \| id = 1644323}} \| year=1998 \| volume=2}}, section B.7</ref>

	~~==References==~~
	~~<references />~~

	~~{{DEFAULTSORT:Normal Bundle}}~~
	~~[[Category:Algebraic geometry]]~~
	~~[[Category:Differential geometry]]~~
	~~[[Category:Differential topology]]~~
	~~[[Category:Vector bundles]]~~

145.53.248.60: /* Fixed effect models */ typo fix

2013-12-26T22:50:11Z

Fixed effect models: typo fix

← Older revision		Revision as of 00:50, 27 December 2013
Line 1:		Line 1:
	~~Marvella~~ is ~~what you can contact her but it's not~~ the ~~most~~ [~~http:~~//~~www~~.~~picsearch~~.~~com~~/~~Gonorrhea-pictures~~.~~html feminine~~] ~~name out there. His family members lives in~~ [~~http~~:/~~/www~~.~~livestrong~~.~~com/article/79944~~-~~anal-herpes-symptoms~~/ ~~South Dakota~~] but ~~his spouse desires~~ [~~http:~~//~~www~~.~~gaysphere~~.~~net~~/~~blog~~/~~167593 www~~.~~gaysphere.net~~] ~~std home test~~ [~~http~~://~~www~~.~~youporn~~-~~nederlandse.com~~/~~user~~/~~CMidgett click~~ the ~~next internet page~~] ~~them to move~~. ~~He utilized to be unemployed but now he~~ is ~~a meter reader~~. ~~To collect badges~~ is ~~what her family~~ and ~~her enjoy~~.<br><br>~~Here~~ is ~~my at home std test page -~~ [~~http~~://~~essexsinglesnights~~.co.uk/~~groups~~/~~cures-for-any-candida-tips-~~to~~-use-now~~/ ~~http~~://~~essexsinglesnights~~.~~co.uk~~/~~groups~~/~~cures~~-~~for~~-~~any~~-~~candida~~-~~tips~~-to-~~use~~-~~now~~/]		In [[differential geometry]], a field of [[mathematics]], a '''normal bundle''' is a particular kind of [[vector bundle]], [[complementary angles\|complementary]] to the [[tangent bundle]], and coming from an [[embedding]] (or [[immersion (mathematics)\|immersion]]).

			==Definition==
			===Riemannian manifold===
			Let <math>(M,g)</math> be a [[Riemannian manifold]], and <math>S \subset M</math> a [[Riemannian submanifold]]. Define, for a given <math>p \in S</math>, a vector <math>n \in \mathrm{T}_p M</math> to be ''normal'' to <math>S</math> whenever <math>g(n,v)=0</math> for all <math>v\in \mathrm{T}_p S</math> (so that <math>n</math> is [[orthogonal complement\|orthogonal]] to <math>\mathrm{T}_p S</math>). The set <math>\mathrm{N}_p S</math> of all such <math>n</math> is then called the ''normal space'' to <math>S</math> at <math>p</math>.

			Just as the total space of the [[tangent bundle]] to a manifold is constructed from all [[tangent space]]s to the manifold, the total space of the ''normal bundle'' <math>\mathrm{N} S</math> to <math>S</math> is defined as
			:<math>\mathrm{N}S := \coprod_{p \in S} \mathrm{N}_p S</math>.

			The '''conormal bundle''' is defined as the [[dual bundle]] to the normal bundle. It can be realised naturally as a sub-bundle of the [[cotangent bundle]].

			===General definition===
			More abstractly, given an [[immersion (mathematics)\|immersion]] <math>i\colon N \to M</math> (for instance an embedding), one can define a normal bundle of ''N'' in ''M'', by at each point of ''N'', taking the [[quotient space (linear algebra)\|quotient space]] of the tangent space on ''M'' by the tangent space on ''N''. For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot (such a choice is equivalent to a [[section (category theory)\|section]] of the projection <math>V \to V/W</math>).

			Thus the normal bundle is in general a ''quotient'' of the tangent bundle of the ambient space restricted to the subspace.

			Formally, the normal bundle to ''N'' in ''M'' is a quotient bundle of the tangent bundle on ''M'': one has the [[short exact sequence]] of vector bundles on ''N'':
			:<math>0 \to TN \to TM\vert_{i(N)} \to T_{M/N} := TM\vert_{i(N)} / TN \to 0</math>
			where <math>TM\vert_{i(N)}</math> is the restriction of the tangent bundle on ''M'' to ''N'' (properly, the pullback <math>i^*TM</math> of the tangent bundle on ''M'' to a vector bundle on ''N'' via the map <math>i</math>).

			==Stable normal bundle==
			[[abstraction\|Abstract]] [[manifolds]] have a [[canonical form\|canonical]] tangent bundle, but do not have a normal bundle: only an embedding (or immersion) of a manifold in another yields a normal bundle.
			However, since every compact manifold can be embedded in <math>\mathbf{R}^N</math>, by the [[Whitney embedding theorem]], every manifold admits a normal bundle, given such an embedding.

			There is in general no natural choice of embedding, but for a given ''M'', any two embeddings in <math>\mathbf{R}^N</math> for sufficiently large ''N'' are [[regular homotopy\|regular homotopic]], and hence induce the same normal bundle. The resulting class of normal bundles (it is a class of bundles and not a specific bundle because ''N'' could vary) is called the [[stable normal bundle]].

			==Dual to tangent bundle==
			The normal bundle is dual to the tangent bundle in the sense of [[K-theory]]:
			by the above short exact sequence,
			:<math>[TN] + [T_{M/N}] = [TM]</math>
			in the [[Grothendieck group]].
			In case of an immersion in <math>\mathbf{R}^N</math>, the tangent bundle of the ambient space is trivial (since <math>\mathbf{R}^N</math> is contractible, hence [[parallelizable]]), so <math>[TN] + [T_{M/N}] = 0</math>, and thus <math>[T_{M/N}] = -[TN]</math>.

			This is useful in the computation of [[characteristic classes]], and allows one to prove lower bounds on immersibility and embeddability of manifolds in [[Euclidean space]].

			==For symplectic manifolds==
			Suppose a manifold <math>X</math> is embedded in to a [[symplectic manifold]] <math>(M,\omega)</math>, such that the pullback of the symplectic form has constant rank on <math>X</math>. Then one can define the symplectic normal bundle to X as the vector bundle over X with fibres
			:<math> (T_{i(x)}X)^\omega/(T_{i(x)}X\cap (T_{i(x)}X)^\omega), \quad x\in X,</math>
			where <math>i:X\rightarrow M</math> denotes the embedding. Notice that the constant rank condition ensures that these normal spaces fit together to form a bundle. Furthermore, any fibre inherits the structure of a symplectic vector space.

			By [[Darboux's theorem]], the constant rank embedding is locally determined by <math>i*(TM)</math>. The isomorphism
			:<math> i^(TM)\cong TX/\nu \oplus (TX)^\omega/\nu \oplus(\nu\oplus \nu^), \quad \nu=TX\cap (TX)^\omega,</math>
			of symplectic vector bundles over <math>X</math> implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.

			==Algebraic geometry==
			In [[algebraic geometry]], the normal bundle ''N''<sub>''X''</sub>''Y'' of a [[regular embedding]] i: ''X'' → ''Y'', defined by some sheaf of ideals ''I'' is the vector bundle on ''X'' corresponding to the dual of the sheaf ''I''/''I''<sup>2</sup>. The regularity of the embedding ensures that this sheaf is locally free and agrees with the ''normal cone'' C<sub>''X''</sub>''Y'', which is defined as <math>Spec \oplus_{n \geq 0} I^n / I^{n+1}</math>.<ref>{{Citation \| last1=Fulton \| first1=William \| author1-link=William Fulton (mathematician) \| title=Intersection theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] \| isbn=978-3-540-62046-4; 978-0-387-98549-7 \| id={{MathSciNet \| id = 1644323}} \| year=1998 \| volume=2}}, section B.7</ref>

			==References==
			<references />

			{{DEFAULTSORT:Normal Bundle}}
			[[Category:Algebraic geometry]]
			[[Category:Differential geometry]]
			[[Category:Differential topology]]
			[[Category:Vector bundles]]

204.64.50.216: Disambiguated: index (mathematics) → Indexed family

2012-06-20T15:32:52Z

Disambiguated: index (mathematics) → Indexed family

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