Editing Vector bundle (section)

==Definition and first consequences==
[[File:Vector bundle.png|thumb|300px|A vector bundle <math>E</math> over a base <math>M</math>. A point <math>m_1</math> in <math>M(=X)</math> corresponds to the [[Origin (mathematics)|origin]] in a fibre <math>E_{m_1}</math> of the vector bundle <math>E</math>, and this fibre is mapped down to the point <math>m_1</math> by the [[projection (mathematics)|projection]] <math>\pi: E \to M</math>.]]
A '''real vector bundle''' consists of:
# topological spaces <math>X</math> (''base space'') and <math>E</math> (''total space'')
# a [[continuous function|continuous]] [[surjection]] <math>\pi:E\to X</math> (''bundle projection'')
# for every <math>x</math>  in <math>X</math>, the structure of a [[Hamel dimension|finite-dimensional]] [[real number|real]] [[vector space]] on the [[Fiber bundle|fiber]] <math>\pi^{-1}(\{x\})</math>

where the following compatibility condition is satisfied: for every point <math>p</math> in <math>X</math>, there is an [[open neighborhood]] <math>U\subseteq X</math> of <math>p</math>, a [[natural number]] <math>k</math>, and a [[homeomorphism]]

:<math>\varphi\colon  U \times \R^k \to \pi^{-1}(U) </math>

such that for all <math>x</math> in <math>U</math>,

* <math> (\pi \circ \varphi)(x,v) = x  </math> for all [[Euclidean vector|vectors]] <math>v</math> in <math>\R^k</math>, and
* the map <math> v \mapsto \varphi (x, v)</math>  is a [[linear map|linear]] [[isomorphism]] between the vector spaces <math>\R^k</math> and <math>\pi^{-1}(\{x\})</math>.

The open neighborhood <math>U</math> together with the homeomorphism <math>\varphi</math> is called a '''[[local trivialization]]''' of the vector bundle. The local trivialization shows that ''locally'' the map <math>\pi</math> "looks like" the projection of <math>U\times\R^k</math> on <math>U</math>.

Every fiber <math>\pi^{-1}(\{x\})</math> is a finite-dimensional real vector space and hence has a [[dimension]] <math>k_x</math>. The local trivializations show that the [[Function (mathematics)|function]] <math>x\to k_x</math> is [[locally constant]], and is therefore constant on each [[Locally connected space|connected component]] of <math>X</math>. If <math>k_x</math> is equal to a constant <math>k</math> on all of <math>X</math>, then <math>k</math> is called the '''rank''' of the vector bundle, and <math>E</math> is said to be a '''vector bundle of rank <math>k</math>'''.  Often the definition of a vector bundle includes that the rank is well defined, so that <math>k_x</math> is constant. Vector bundles of rank 1 are called [[line bundle]]s, while those of rank 2 are less commonly called plane bundles.

The [[Cartesian product]] <math>X\times\R^k</math>, equipped with the projection <math>X\times\R^k\to X</math>, is called the '''trivial bundle''' of rank <math>k</math> over <math>X</math>.

===Transition functions===
[[File:Transition functions.png|thumb|300px|Two trivial vector bundles over [[open set]]s <math>U_\alpha</math> and <math>U_\beta</math> may be [[Gluing (topology)|glued]] over the intersection <math>U_{\alpha\beta}</math> by transition functions <math>g_{\alpha \beta}</math> which serve to stick the shaded grey regions together after applying a [[linear transformation]] to the fibres (note the transformation of the blue [[quadrilateral]] under the effect of <math>g_{\alpha\beta}</math>). Different choices of transition functions may result in different vector bundles which are non-trivial after gluing is complete.]]
[[File:Mobius transition functions.png|thumb|300px|The [[Möbius strip]] can be constructed by a non-trivial gluing of two trivial bundles on open [[subset]]s ''U'' and ''V'' of the [[1-sphere|circle ''S<sup>1</sup>'']]. When glued trivially (with ''g<sub>UV</sub>=1'') one obtains the trivial bundle, but with the non-trivial gluing of ''g<sub>UV</sub>=1'' on one overlap and ''g<sub>UV</sub>=-1'' on the second overlap, one obtains the non-trivial bundle ''E'', the Möbius strip. This can be visualised as a "twisting" of one of the local [[Chart (topology)|charts]].]]
Given a vector bundle <math>E\to X</math> of rank <math>k</math>, and a pair of neighborhoods <math>U</math> and <math>V</math> over which the bundle trivializes via

:<math>\begin{align}
\varphi_U\colon U\times \R^k &\mathrel{\xrightarrow{\cong}} \pi^{-1}(U), \\
\varphi_V\colon V\times \R^k &\mathrel{\xrightarrow{\cong}} \pi^{-1}(V)
\end{align}</math>

the [[Composite Function|composite function]]
:<math>\varphi_U^{-1}\circ\varphi_V \colon (U\cap V)\times\R^k\to (U\cap V)\times\R^k</math>

is well-defined on the overlap, and satisfies
:<math>\varphi_U^{-1}\circ\varphi_V (x,v) = \left (x,g_{UV}(x)v \right)</math>

for some <math>\text{GL}(k)</math>-valued function
:<math>g_{UV}\colon U\cap V\to \operatorname{GL}(k).</math>

These are called the '''[[Transition map|transition functions]]''' (or the '''coordinate transformations''') of the vector bundle.

The [[Set (mathematics)|set]] of transition functions forms a [[Čech cocycle]] in the sense that
:<math>g_{UU}(x) = I, \quad g_{UV}(x)g_{VW}(x)g_{WU}(x) = I</math>

for all <math>U,V,W</math> over which the bundle trivializes satisfying <math> U\cap V\cap W\neq \emptyset</math>.  Thus the data <math>(E,X,\pi,\R^k)</math> defines a [[fiber bundle]]; the additional data of the <math>g_{UV}</math> specifies a <math>\text{GL}(k)</math> structure group in which the [[Group action|action]] on the fiber is the standard action of <math>\text{GL}(k)</math>.

Conversely, given a fiber bundle <math>(E,X,\pi,\R^k)</math> with a <math>\text{GL}(k)</math> cocycle acting in the standard way on the fiber <math>\R^k</math>, there is [[associated bundle|associated]] a vector bundle.  This is an example of the [[fibre bundle construction theorem]] for vector bundles, and can be taken as an alternative definition of a vector bundle.

===Subbundles===
{{Main article|Subbundle}}
[[File:Subbundle.png|thumb|300px|A line subbundle <math>L</math> of a trivial rank 2 vector bundle <math>E</math> over a one-dimensional manifold <math>M</math>.]]
One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle <math>\pi: E\to X</math> over a topological space, a subbundle is simply a [[Linear subspace|subspace]] <math>F\subset E</math> for which the [[Restriction of a map|restriction]] <math>\left.\pi\right|_F</math> of <math>\pi</math> to <math>F</math> gives <math>\left.\pi\right|_F: F \to X</math> the structure of a vector bundle also. In this case the fibre <math>F_x\subset E_x</math> is a vector subspace for every <math>x\in X</math>.

A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle over a compact space can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example, the [[Möbius band]], a non-trivial [[line bundle]] over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle.