|
|
| Line 1: |
Line 1: |
| 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]]
| |
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.
my website - over the counter std test (visit the next internet site)