Editing Vector bundle (section)

==Smooth vector bundles==
[[File:Smooth vs non-smooth vector bundle.png|thumb|300px|The regularity of transition functions describing a vector bundle determines the type of the vector bundle. If the continuous transition functions ''g<sub>UV</sub>'' are used, the resulting vector bundle ''E'' is only continuous but not smooth. If the smooth transition functions ''h<sub>UV</sub>'' are used, then the resulting vector bundle ''F'' is a smooth vector bundle.]]
A vector bundle (''E'', ''p'', ''M'') is '''smooth''', if ''E'' and ''M'' are [[manifold|smooth manifolds]], p: ''E'' → ''M'' is a smooth map, and the local trivializations are [[diffeomorphism]]s. Depending on the required degree of [[smoothness]], there are different corresponding notions of [[continuously differentiable|''C<sup>p</sup>'']] bundles, [[infinitely differentiable]] ''C''<sup>∞</sup>-bundles and [[real analytic]] ''C''<sup>ω</sup>-bundles. In this section we will concentrate on [[C-infinity|''C''<sup>∞</sup>]]-bundles. The most important example of a ''C''<sup>∞</sup>-vector bundle is the [[tangent bundle]] (''TM'', {{pi}}<sub>''TM''</sub>, ''M'') of a ''C''<sup>∞</sup>-manifold ''M''.

A smooth vector bundle can be characterized by the fact that it admits transition functions as described above which are ''smooth'' functions on overlaps of trivializing charts ''U'' and ''V''. That is, a vector bundle ''E'' is smooth if it admits a covering by trivializing open sets such that for any two such sets ''U'' and ''V'', the transition function
:<math>g_{UV}: U\cap V \to \operatorname{GL}(k,\mathbb{R})</math>
is a smooth function into the [[matrix group]] GL(k,'''R'''), which is a [[Lie group]].

Similarly, if the transition functions are:

* ''C<sup>r</sup>'' then the vector bundle is a '''''C<sup>r</sup>'' vector bundle''',
* ''real [[Analytic function|analytic]]'' then the vector bundle is a '''real analytic vector bundle''' (this requires the matrix group to have a real analytic structure),
* ''holomorphic'' then the vector bundle is a '''[[holomorphic vector bundle]]''' (this requires the matrix group to be a [[complex Lie group]]),
* ''algebraic functions'' then the vector bundle is an '''[[algebraic vector bundle]]''' (this requires the matrix group to be an [[algebraic group]]).

The ''C''<sup>∞</sup>-vector bundles (''E'', ''p'', ''M'') have a very important property not shared by more general ''C''<sup>∞</sup>-fibre bundles. Namely, the tangent space ''T<sub>v</sub>''(''E''<sub>''x''</sub>) at any ''v'' ∈ ''E''<sub>''x''</sub> can be naturally identified with the fibre ''E''<sub>''x''</sub> itself. This identification is obtained through the ''vertical lift'' ''vl''<sub>''v''</sub>: ''E<sub>x</sub>'' → ''T''<sub>''v''</sub>(''E''<sub>''x''</sub>), defined as

:<math>\operatorname{vl}_vw[f] := \left.\frac{d}{dt}\right|_{t=0}f(v + tw), \quad f\in C^\infty(E_x).</math>

The vertical lift can also be seen as a natural ''C''<sup>∞</sup>-vector bundle isomorphism ''p*E'' → ''VE'', where (''p*E'', ''p*p'', ''E'') is the pull-back bundle of (''E'', ''p'', ''M'') over ''E'' through ''p'': ''E'' → ''M'', and ''VE'' := Ker(''p''<sub>*</sub>) ⊂ ''TE'' is the ''vertical tangent bundle'', a natural vector subbundle of the tangent bundle (''TE'', {{pi}}<sub>''TE''</sub>, ''E'') of the total space ''E''.

The total space ''E'' of any smooth vector bundle carries a natural vector field ''V''<sub>''v''</sub> := vl<sub>''v''</sub>''v'', known as the ''canonical vector field''. More formally, ''V'' is a smooth section of (''TE'', {{pi}}<sub>''TE''</sub>, ''E''), and it can also be defined as the infinitesimal generator of the [[Lie group action|Lie-group action]] <math>(t,v) \mapsto e^{tv}</math> given by the fibrewise scalar multiplication. The canonical vector field ''V'' characterizes completely the smooth vector bundle structure in the following manner. As a preparation, note that when ''X'' is a smooth vector field on a smooth manifold ''M'' and ''x'' ∈ ''M'' such that ''X''<sub>''x''</sub> = 0, the linear mapping
:<math>C_x(X): T_x M \to T_x M; \quad C_x(X) Y = (\nabla_Y X)_x</math>

does not depend on the choice of the linear [[covariant derivative]] ∇ on ''M''. The canonical vector field ''V'' on ''E'' satisfies the axioms

# The flow (''t'', ''v'') → Φ<sup>''t''</sup><sub>''V''</sub>(''v'') of ''V'' is globally defined.
# For each ''v'' ∈ ''V'' there is a unique lim<sub>t→∞</sub> Φ<sup>''t''</sup><sub>''V''</sub>(''v'') ∈ ''V''.
# ''C''<sub>v</sub>(''V'')∘''C''<sub>v</sub>(''V'') = ''C''<sub>v</sub>(''V'') whenever ''V''<sub>''v''</sub> = 0.
# The [[zero set]] of ''V'' is a smooth [[submanifold]] of ''E'' whose [[codimension]] is equal to the rank of ''C''<sub>v</sub>(''V'').

Conversely, if ''E'' is any smooth manifold and ''V'' is a smooth vector field on ''E'' satisfying 1–4, then there is a unique vector bundle structure on ''E'' whose canonical vector field is ''V''.

For any smooth vector bundle (''E'', ''p'', ''M'') the total space ''TE'' of its tangent bundle (''TE'', {{pi}}<sub>''TE''</sub>, ''E'') has a natural [[secondary vector bundle structure]] (''TE'', ''p''<sub>*</sub>, ''TM''), where ''p''<sub>*</sub> is the [[Pushforward (differential)|push-forward]] of the canonical projection ''p'': ''E'' → ''M''. The vector bundle operations in this secondary vector bundle structure are the push-forwards +<sub>*</sub>: ''T''(''E'' × ''E'') → ''TE'' and λ<sub>*</sub>: ''TE'' → ''TE'' of the original addition +: ''E'' × ''E'' → ''E'' and scalar multiplication λ: ''E'' → ''E''.