Editing Vector bundle (section)

===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.