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| [[Image:Moebiusstrip.png|thumb|250px|right|The [[Möbius strip]] is a [[line bundle]] over the [[N-sphere|1-sphere]] '''S'''<sup>1</sup>. Locally around every point in '''S'''<sup>1</sup>, it [[homeomorphism|looks like]] ''U'' × '''R''' (where ''U'' is an open arc including the point), but the total bundle is different from '''S'''<sup>1</sup> × '''R''' (which is a [[Cartesian product|cylinder]] instead). ]]
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| In [[mathematics]], a '''vector bundle''' is a [[topology|topological]] construction that makes precise the idea of a family of [[vector space]]s parameterized by another space ''X'' (for example ''X'' could be a [[topological space]], a [[manifold]], or an [[algebraic variety]]): to every point ''x'' of the space ''X'' we associate (or "attach") a vector space ''V''(''x'') in such a way that these vector spaces fit together to form another space of the same kind as ''X'' (e.g. a topological space, manifold, or algebraic variety), which is then called a '''vector bundle over ''X'''''.
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| The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space ''V'' such that ''V''(''x'') = ''V'' for all ''x'' in ''X'': in this case there is a copy of ''V'' for each ''x'' in ''X'' and these copies fit together to form the vector bundle ''X'' × ''V'' over ''X''. Such vector bundles are said to be [[Fiber bundle#Trivial bundle|''trivial'']]. A more complicated (and prototypical) class of examples are the [[tangent bundle]]s of [[manifold|smooth (or differentiable) manifolds]]: to every point of such a manifold we attach the [[tangent space]] to the manifold at that point. Tangent bundles are not, in general, trivial bundles: for example, the tangent bundle of the sphere is non-trivial by the [[hairy ball theorem]]. In general, a manifold is said to be [[Parallelizable manifold|parallelizable]] if and only if its tangent bundle is trivial.
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| Vector bundles are almost always required to be ''locally trivial'', however, which means they are examples of [[fiber bundle]]s. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the [[category of topological spaces]].
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| ==Definition and first consequences==
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| A '''real vector bundle''' consists of:
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| # topological spaces ''X'' (''base space'') and ''E'' (''total space'')
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| # a [[continuous function|continuous]] [[surjection]] π : ''E'' → ''X'' (''bundle projection'')
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| # for every ''x'' in ''X'', the structure of a [[Hamel dimension|finite-dimensional]] [[real number|real]] [[vector space]] on the [[Fiber bundle|fiber]] π<sup>−1</sup>({''x''})
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| where the following compatibility condition is satisfied: for every point in ''X'', there is an open neighborhood ''U'', a [[natural number]] ''k'', and a [[homeomorphism]] | |
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| :<math>\varphi: U \times \mathbf{R}^{k} \to \pi^{-1}(U) </math>
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| such that for all ''x'' ∈ ''U'',
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| *<math> (\pi \circ \varphi)(x,v) = x </math> for all vectors ''v'' in '''R'''<sup>''k''</sup>, and
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| * the map <math> v {{\mapsto}} \varphi (x, v)</math> is an isomorphism between the vector spaces '''R'''<sup>''k''</sup> and π<sup>−1</sup>({''x''}).
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| The open neighborhood ''U'' together with the homeomorphism φ is called a '''local trivialization''' of the vector bundle. The local trivialization shows that ''locally'' the map π "looks like" the projection of ''U'' × '''R'''<sup>''k''</sup> on ''U''.
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| Every fiber π<sup>−1</sup>({''x''}) is a finite-dimensional real vector space and hence has a dimension ''k''<sub>''x''</sub>. The local trivializations show that the [[Function (mathematics)|function]] ''x'' {{mapsto}} ''k<sub>x</sub>'' is [[locally constant]], and is therefore constant on each [[Locally connected space|connected component]] of ''X''. If ''k<sub>x</sub>'' is equal to a constant ''k'' on all of ''X'', then ''k'' is called the '''rank''' of the vector bundle, and ''E'' is said to be a '''vector bundle of rank ''k'''''. Often the definition of a vector bundle includes that the rank is well defined, so that ''k<sub>x</sub>'' is constant. Vector bundles of rank 1 are called [[line bundle]]s, while those of rank 2 are less commonly called plane bundles.
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| The [[Cartesian product]] ''X'' × '''R'''<sup>''k''</sup> , equipped with the projection ''X'' × '''R'''<sup>''k''</sup> → ''X'', is called the '''trivial bundle''' of rank ''k'' over ''X''.
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| ===Transition functions===
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| Given a vector bundle ''E'' → ''X'' of rank ''k'', and a pair of neighborhoods ''U'' and ''V'' over which the bundle trivializes via
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| :<math>\begin{align}
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| \varphi_U : U\times \mathbf{R}^k &\xrightarrow{\cong} \pi^{-1}(U), \\
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| \varphi_V : V\times \mathbf{R}^k &\xrightarrow{\cong} \pi^{-1}(V)
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| \end{align}</math>
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| the composite function
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| :<math>\varphi_V^{-1}\circ\varphi_U : (U\cap V)\times\mathbf{R}^k\to (U\cap V)\times\mathbf{R}^k</math>
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| is well-defined on the overlap, and satisfies
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| :<math>\varphi_V^{-1}\circ\varphi_U (x,v) = \left (x,g_{UV}(x)v \right)</math>
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| for some GL(''k'')-valued function
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| :<math>g_{UV}:U\cap V\to \operatorname{GL}(k).</math>
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| These are called the '''transition functions''' (or the '''coordinate transformations''') of the vector bundle.
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| The set of transition functions forms a [[Čech cocycle]] in the sense that
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| :<math>g_{UU}(x) = I, \quad g_{UV}(x)g_{VW}(x)g_{WU}(x) = I</math>
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| for all ''U'', ''V'', ''W'' over which the bundle trivializes. Thus the data (''E'', ''X'', π, '''R'''<sup>''k''</sup>) defines a [[fiber bundle]]; the additional data of the ''g''<sub>''UV''</sub> specifies a GL(''k'') structure group in which the action on the fiber is the standard action of GL(''k'').
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| Conversely, given a fiber bundle (''E'', ''X'', π, '''R'''<sup>''k''</sup>) with a GL(''k'') cocycle acting in the standard way on the fiber '''R'''<sup>''k''</sup>, there is associated a vector bundle. This is sometimes taken as the definition of a vector bundle.
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| ==Vector bundle morphisms==
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| A '''[[morphism]]''' from the vector bundle π<sub>1</sub> : ''E''<sub>1</sub> → ''X''<sub>1</sub> to the vector bundle π<sub>2</sub> : ''E''<sub>2</sub> → ''X''<sub>2</sub> is given by a pair of continuous maps ''f'' : ''E''<sub>1</sub> → ''E''<sub>2</sub> and ''g'' : ''X''<sub>1</sub> → ''X''<sub>2</sub> such that
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| * ''g'' ∘ π<sub>1</sub> = π<sub>2</sub> ∘ ''f''
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| <div style="text-align: center;">[[Image:BundleMorphism-01.png]]</div>
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| * for every ''x'' in ''X''<sub>1</sub>, the map π<sub>1</sub><sup>−1</sup>({''x''}) → π<sub>2</sub><sup>−1</sup>({''g''(''x'')}) induced by ''f'' is a [[linear map]] between vector spaces.
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| Note that ''g'' is determined by ''f'' (because π<sub>1</sub> is surjective), and ''f'' is then said to '''cover ''g'''''.
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| The class of all vector bundles together with bundle morphisms forms a [[category (mathematics)|category]]. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a [[bundle map]] between [[fiber bundle]]s, and are also often called '''(vector) bundle homomorphisms'''.
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| A bundle homomorphism from ''E''<sub>1</sub> to ''E''<sub>2</sub> with an inverse which is also a bundle homomorphism (from ''E''<sub>2</sub> to ''E''<sub>1</sub>) is called a '''(vector) bundle isomorphism''', and then ''E''<sub>1</sub> and ''E''<sub>2</sub> are said to be '''isomorphic''' vector bundles. An isomorphism of a (rank ''k'') vector bundle ''E'' over ''X'' with the trivial bundle (of rank ''k'' over ''X'') is called a '''trivialization''' of ''E'', and ''E'' is then said to be '''trivial''' (or '''trivializable'''). The definition of a vector bundle shows that any vector bundle is '''locally trivial'''.
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| We can also consider the category of all vector bundles over a fixed base space ''X''. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the [[identity function|identity map]] on ''X''. That is, bundle morphisms for which the following diagram [[commutative diagram|commutes]]:
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| <div style="text-align: center;">[[Image:BundleMorphism-02.png]]</div>
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| (Note that this category is ''not'' [[abelian category|abelian]]; the [[kernel (category theory)|kernel]] of a morphism of vector bundles is in general not a vector bundle in any natural way.)
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| A vector bundle morphism between vector bundles π<sub>1</sub> : ''E''<sub>1</sub> → ''X''<sub>1</sub> and π<sub>2</sub> : ''E''<sub>2</sub> → ''X''<sub>2</sub> covering a map ''g'' from ''X''<sub>1</sub> to ''X''<sub>2</sub> can also be viewed as a vector bundle morphism over ''X''<sub>1</sub> from ''E''<sub>1</sub> to the [[pullback bundle]] ''g*E''<sub>2</sub>.
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| ==Sections and locally free sheaves==
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| [[Image:Surface normal.png|right|thumb|300px|The map associating a [[Normal vector|normal]] to each point on a surface can be thought of as a section. The surface is the space ''X'', and at each point ''x'' there is a vector in the vector space attached at ''x''.]]
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| Given a vector bundle π : ''E'' → ''X'' and an open subset ''U'' of ''X'', we can consider '''sections''' of π on ''U'', i.e. continuous functions ''s'' : ''U'' → ''E'' where the composite π∘''s'' is such that {{nowrap|1=(π∘''s'')(''u'') = ''u''}} for all ''u'' in ''U''. Essentially, a section assigns to every point of ''U'' a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but [[vector field]]s on that manifold.
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| Let ''F''(''U'') be the set of all sections on ''U''. ''F''(''U'') always contains at least one element, namely the '''zero section''': the function ''s'' that maps every element ''x'' of ''U'' to the zero element of the vector space π<sup>−1</sup>({''x''}). With the pointwise addition and scalar multiplication of sections, ''F''(''U'') becomes itself a real vector space. The collection of these vector spaces is a [[sheaf (mathematics)|sheaf]] of vector spaces on ''X''.
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| If ''s'' is an element of ''F''(''U'') and α : ''U'' → '''R''' is a continuous map, then α''s'' (pointwise scalar multiplication) is in ''F''(''U''). We see that ''F''(''U'') is a [[module (mathematics)|module]] over the ring of continuous real-valued functions on ''U''. Furthermore, if O<sub>''X''</sub> denotes the structure sheaf of continuous real-valued functions on ''X'', then ''F'' becomes a sheaf of O<sub>''X''</sub>-modules.
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| Not every sheaf of O<sub>''X''</sub>-modules arises in this fashion from a vector bundle: only the [[locally free sheaf|locally free]] ones do. (The reason: locally we are looking for sections of a projection ''U'' × '''R'''<sup>''k''</sup> → ''U''; these are precisely the continuous functions ''U'' → '''R'''<sup>''k''</sup>, and such a function is an ''k''-tuple of continuous functions ''U'' → '''R'''.)
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| Even more: the category of real vector bundles on ''X'' is [[category theory|equivalent]] to the category of locally free and finitely generated sheaves of O<sub>''X''</sub>-modules.
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| So we can think of the category of real vector bundles on ''X'' as sitting inside the category of sheaves of O<sub>''X''</sub>-modules; this latter category is abelian, so this is where we can compute kernels and cokernels of morphisms of vector bundles.
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| Note that a rank ''n'' vector bundle is trivial if and only if it has ''n'' linearly independent global sections.
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| ==<span id="directsum"></span> Operations on vector bundles ==
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| Most operations on vector spaces can be extended to vector bundles by performing the vector space operation ''fiberwise''.
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| For example, if ''E'' is a vector bundle over ''X'', then the there is a bundle ''E*'' over ''X'', called the '''[[dual bundle]]''', whose fiber at ''x''∈''X'' is the [[dual vector space]] (''E<sub>x</sub>'')*. Formally ''E''* can be defined as the set of pairs (''x'', φ), where ''x'' ∈ ''X'' and φ ∈ (''E''<sub>''x''</sub>)*. The dual bundle is locally trivial because the [[transpose|dual space]] of the inverse of a local trivialization of ''E'' is a local trivialization of ''E*'': the key point here is that the operation of taking the dual vector space is [[functorial]].
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| There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles ''E'', ''F'' on ''X'' (over the given field). A few examples follow.
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| * The '''Whitney sum''' (named for [[Hassler Whitney]]) or '''direct sum bundle''' of ''E'' and ''F'' is a vector bundle ''E'' ⊕ ''F'' over ''X'' whose fiber over ''x'' is the [[Direct sum of modules|direct sum]] ''E<sub>x</sub>'' ⊕ ''F<sub>x</sub>'' of the vector spaces ''E<sub>x</sub>'' and ''F<sub>x</sub>''.
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| * The '''[[tensor product bundle]]''' ''E'' ⊗ ''F'' is defined in a similar way, using fiberwise [[tensor product]] of vector spaces.
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| * The '''Hom-bundle''' Hom(''E'', ''F'') is a vector bundle whose fiber at ''x'' is the space of linear maps from ''E<sub>x</sub>'' to ''F<sub>x</sub>'' (which is often denoted Hom(''E''<sub>''x''</sub>, ''F<sub>x</sub>'') or ''L''(''E''<sub>''x''</sub>, ''F''<sub>''x''</sub>)). The Hom-bundle is so-called (and useful) because there is a bijection between vector bundle homomorphisms from ''E'' to ''F'' over ''X'' and sections of Hom(''E'', ''F'') over ''X''.
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| * The [[dual bundle|dual vector bundle]] ''E*'' is the Hom bundle Hom(''E'', '''R''' × ''X'') of bundle homomorphisms of ''E'' and the trivial bundle '''R''' × ''X''. There is a canonical vector bundle isomorphism Hom(''E'', ''F'') = ''E*'' ⊗ ''F''.
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| Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the category of vector spaces can also be performed on the category of vector bundles in a [[functor]]ial manner. This is made precise in the language of [[smooth functor]]s. An operation of a different nature is the '''[[pullback bundle]]''' construction. Given a vector bundle ''E'' → ''Y'' and a continuous map ''f'' : ''X'' → ''Y'' one can "pull back" ''E'' to a vector bundle ''f*E'' over ''X''. The fiber over a point ''x'' ∈ ''X'' is essentially just the fiber over ''f''(''x'') ∈ ''Y''. Hence, Whitney summing ''E'' ⊕ ''F'' can be defined as the pullback bundle of the diagonal map from ''X'' to ''X'' x ''X'' where the bundle over ''X'' x ''X'' is ''E'' x ''F''.
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| ==Additional structures and generalizations==
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| Vector bundles are often given more structure. For instance, vector bundles may be equipped with a [[metric (vector bundle)|vector bundle metric]]. Usually this metric is required to be [[definite bilinear form|positive definite]], in which case each fibre of ''E'' becomes a Euclidean space. A vector bundle with a [[Linear complex structure|complex structure]] corresponds to a complex vector bundle, which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings be complex-linear in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resulting [[reduction of the structure group of a bundle]]. Vector bundles over more general [[topological field]]s may also be used.
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| If instead of a finite-dimensional vector space, if the fiber ''F'' is taken to be a [[Banach space]] then a '''[[Banach bundle]]''' is obtained.<ref>{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Differential and Riemannian manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94338-1 | year=1995}}</ref> Specifically, one must require that the local trivializations are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions
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| :<math>g_{UV} : U\cap V \to \operatorname{GL}(F)</math>
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| are continuous mappings of [[Banach manifold]]s. In the corresponding theory for C<sup>''p''</sup> bundles, all mappings are required to be C<sup>''p''</sup>.
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| Vector bundles are special [[fiber bundle]]s, those whose fibers are vector spaces and whose cocycle respects the vector space structure. More general fiber bundles can be constructed in which the fiber may have other structures; for example [[sphere bundle]]s are fibered by spheres.
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| ==Smooth vector bundles==
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| 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''<sup>∞</sup>-bundles. The most important example of a ''C''<sup>∞</sup>-vector bundle is the [[tangent bundle]] (''TM'',π<sub>''TM''</sub>,''M'') of a ''C''<sup>∞</sup>-manifold ''M''.
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| 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
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| :<math>\operatorname{vl}_vw[f] := \frac{d}{dt}\Big|_{t=0}f(v+tw), \quad f\in C^\infty(E_x).</math>
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| 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'', π<sub>''TE''</sub>, ''E'') of the total space ''E''.
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| The ''slit vector bundle'' ''E''/0, obtained from (''E'', ''p'', ''M'') by removing the zero section 0 ⊂ ''E'', 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'', π<sub>''TE''</sub>, ''E''), and it can also be defined as the infinitesimal generator of the Lie-group action
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| :<math>\begin{cases}
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| \Phi_V:\mathbf{R} \times (E\setminus 0) \to (E\setminus 0) \\
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| (t,v)\mapsto \Phi_V^t(v) := e^tv.
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| \end{cases}</math>
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| For any smooth vector bundle (''E'', ''p'', ''M'') the total space ''TE'' of its tangent bundle (''TE'', π<sub>''TE''</sub>, ''E'') has a natural [[secondary vector bundle structure]] (''TE'', ''p''<sub>*</sub>,''TM''), where ''p''<sub>*</sub> is the 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''.
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| ==K-theory==
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| The K-theory group, ''K''(''X''), of a manifold is defined as the abelian group generated by isomorphism classes [E] of (complex) vector bundles modulo the relation that whenever we have an [[exact sequence]]
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| :0 → ''A'' → ''B'' → ''C'' → 0
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| then
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| :[''B''] = [''A''] + [''C'']
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| in [[topological K-theory]]. [[KO-theory]] is a version of this construction which considers real vector bundles. K-theory with [[compact support]]s can also be defined, as well as higher K-theory groups.
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| The famous [[Bott periodicity|periodicity theorem]] of [[Raoul Bott]] asserts that the K-theory of any space X is isomorphic to that of the [[Cartesian product]] ''X'' × [[2-sphere|'''S'''<sup>2</sup>]].
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| In [[algebraic geometry]], one considers the K-theory groups consisting of [[coherent sheaf|coherent sheaves]] on a [[scheme (mathematics)|scheme]] ''X'', as well as the K-theory groups of vector bundles on the scheme with the above equivalence relation. The two constructs are the same provided that the underlying scheme is [[smooth morphism|smooth]].
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| ==See also==
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| ===General notions===
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| * [[Grassmannian]]: classifying spaces for vector bundle, among which [[Projective space]]s for [[line bundle]]s
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| * [[Characteristic class]]
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| * [[Splitting principle]]
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| ===Topology and differential geometry===
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| * [[Fiber bundle]]: the general topological notion, among which [[Covering space]]s
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| * [[Connection (vector bundle)]]: the notion needed to differentiate sections of vector bundles.
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| * [[Sheaf (mathematics)]]
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| * [[Topological K-theory]]
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| ===Algebraic and analytic geometry===
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| * [[Algebraic vector bundle]]
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| * [[Coherent sheaf]], in particular [[Picard group]]
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| * [[Holomorphic vector bundle]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{Citation | last=Hatcher | first=Allen | author-link=Allen Hatcher | title=Vector Bundles & K-Theory | url=http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html | edition=2.0 | year=2003}}
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| * {{citation|first=Jeffrey M.|last=Lee|title=Manifolds and Differential Geometry| url=http://www.ams.org/bookstore-getitem/item=gsm-107| series=Graduate Studies in Mathematics|volume=Vol. 107 |publisher=American Mathematical Society|publication-place=Providence|year=2009}} . ISBN 978-0-8218-4815-9
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| *{{Citation | author=Lee, John M. | title=Introduction to Smooth Manifolds | url=http://www.math.washington.edu/~lee/Books/smooth.html | location=New York | publisher=Springer | year=2003 | isbn=0-387-95448-1}} see Ch.5
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| * {{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-3-540-42627-1 | year=2002}}, see section 1.5.
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| * {{Citation | last1=Abraham | first1=Ralph H. | author1-link=Ralph Abraham | last2=Marsden | first2=Jerrold E. | author2-link=Jerrold E. Marsden | title=Foundations of mechanics | publisher=Benjamin-Cummings | location=London | isbn=978-0-8053-0102-1 | year=1978}}, see section 1.5
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| ==External links==
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| * {{springer|title=Vector bundle|id=p/v096380}}
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| * [http://mathoverflow.net/questions/7836/why-is-it-useful-to-study-vector-bundles Why is it useful to study vector bundles ?] on [[MathOverflow]]
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| * [http://mathoverflow.net/questions/16240/why-is-it-useful-to-classify-the-vector-bundles-of-a-space Why is it useful to classify the vector bundles of a space ?]
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| [[Category:Differential topology]]
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| [[Category:Algebraic topology]]
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| [[Category:Vector bundles| ]]
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| [[Category:Vectors]]
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