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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a vector bundle is a
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
construction that makes precise the idea of a family of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s parameterized by another
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
X (for example X could be a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
): 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. 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\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
s of smooth (or differentiable) manifolds: to every point of such a manifold we attach the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
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 The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, if ...
. In general, a manifold is said to be
parallelizable In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a basis of the tangent space at p. Equi ...
if, and only if, its tangent bundle is trivial. Vector bundles are almost always required to be ''locally trivial'', however, which means they are examples of
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
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 bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...
s 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 In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...
.


Definition and first consequences

A real vector bundle consists of: # topological spaces X (''base space'') and E (''total space'') # a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
\pi:E\to X (''bundle projection'') # for every x in X, the structure of a
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
on the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
\pi^(\) where the following compatibility condition is satisfied: for every point p in X, there is an open neighborhood U\subseteq X of p, a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
k, and a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
:\varphi\colon U \times \R^k \to \pi^(U) such that for all x in U, * (\pi \circ \varphi)(x,v) = x for all vectors v in \R^k, and * the map v \mapsto \varphi (x, v) is a linear isomorphism between the vector spaces \R^k and \pi^(\). The open neighborhood U together with the homeomorphism \varphi is called a local trivialization of the vector bundle. The local trivialization shows that ''locally'' the map \pi "looks like" the projection of U\times\R^k on U. Every fiber \pi^(\) is a finite-dimensional real vector space and hence has a dimension k_x. The local trivializations show that the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
x\to k_x is
locally constant In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. ...
, and is therefore constant on each connected component of X. If k_x 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_x is constant. Vector bundles of rank 1 are called
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
s, while those of rank 2 are less commonly called plane bundles. The
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
X\times\R^k, equipped with the projection X\times\R^k\to X, is called the trivial bundle of rank k over X.


Transition functions

Given a vector bundle E\to X of rank k, and a pair of neighborhoods U and V over which the bundle trivializes via :\begin \varphi_U\colon U\times \R^k &\mathrel \pi^(U), \\ \varphi_V\colon V\times \R^k &\mathrel \pi^(V) \end the composite function :\varphi_U^\circ\varphi_V \colon (U\cap V)\times\R^k\to (U\cap V)\times\R^k is well-defined on the overlap, and satisfies :\varphi_U^\circ\varphi_V (x,v) = \left (x,g_(x)v \right) for some \text(k)-valued function :g_\colon U\cap V\to \operatorname(k). These are called the transition functions (or the coordinate transformations) of the vector bundle. The set of transition functions forms a
Čech cocycle Čech (feminine Čechová) is a Czech surname meaning Czech. It was used to distinguish an inhabitant of Bohemia from Slovaks, Moravians and other ethnic groups. Notable people with the surname include: * Dana Čechová (born 1983), Czech tabl ...
in the sense that :g_(x) = I, \quad g_(x)g_(x)g_(x) = I for all U,V,W over which the bundle trivializes satisfying U\cap V\cap W\neq \emptyset. Thus the data (E,X,\pi,\R^k) defines a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
; the additional data of the g_ specifies a \text(k) structure group in which the action on the fiber is the standard action of \text(k). Conversely, given a fiber bundle (E,X,\pi,\R^k) with a \text(k) cocycle acting in the standard way on the fiber \R^k, there is associated a vector bundle. This is an example of the
fibre bundle construction theorem In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of Transition map, transition functions. The theorem also gives conditions under which two such bu ...
for vector bundles, and can be taken as an alternative definition of a vector bundle.


Subbundles

One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle \pi: E\to X over a topological space, a subbundle is simply a subspace F\subset E for which the restriction \left.\pi\_F of \pi to F gives \left.\pi\_F: F \to X the structure of a vector bundle also. In this case the fibre F_x\subset E_x is a vector subspace for every x\in X. A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example the
Möbius band Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...
, a non-trivial
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle.


Vector bundle morphisms

A
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
from the vector bundle 1: ''E''1 → ''X''1 to the vector bundle 2: ''E''2 → ''X''2 is given by a pair of continuous maps ''f'': ''E''1 → ''E''2 and ''g'': ''X''1 → ''X''2 such that : ''g'' ∘ 1 = 2 ∘ ''f'' :: : for every ''x'' in ''X''1, the map 1−1() → 2−1() induced by ''f'' is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
between vector spaces. Note that ''g'' is determined by ''f'' (because 1 is surjective), and ''f'' is then said to cover ''g''. The class of all vector bundles together with bundle morphisms forms a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. 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 In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. Ther ...
between
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s, and are sometimes called (vector) bundle homomorphisms. A bundle homomorphism from ''E''1 to ''E''2 with an inverse which is also a bundle homomorphism (from ''E''2 to ''E''1) is called a (vector) bundle isomorphism, and then ''E''1 and ''E''2 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. 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 map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on ''X''. That is, bundle morphisms for which the following diagram commutes: : (Note that this category is ''not'' abelian; the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of a morphism of vector bundles is in general not a vector bundle in any natural way.) A vector bundle morphism between vector bundles 1: ''E''1 → ''X''1 and 2: ''E''2 → ''X''2 covering a map ''g'' from ''X''1 to ''X''2 can also be viewed as a vector bundle morphism over ''X''1 from ''E''1 to the
pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
''g''*''E''2.


Sections and locally free sheaves

Given a vector bundle : ''E'' → ''X'' and an open subset ''U'' of ''X'', we can consider sections of on ''U'', i.e. continuous functions ''s'': ''U'' → ''E'' where the composite  ∘ ''s'' is such that 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 fields on that manifold. 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 −1(). 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 Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper se ...
of vector spaces on ''X''. 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 Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
over the ring of continuous real-valued functions on ''U''. Furthermore, if O''X'' denotes the structure sheaf of continuous real-valued functions on ''X'', then ''F'' becomes a sheaf of O''X''-modules. Not every sheaf of O''X''-modules arises in this fashion from a vector bundle: only the locally free ones do. (The reason: locally we are looking for sections of a projection ''U'' × R''k'' → ''U''; these are precisely the continuous functions ''U'' → R''k'', and such a function is a ''k''-tuple of continuous functions ''U'' → R.) Even more: the category of real vector bundles on ''X'' is
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
to the category of locally free and finitely generated sheaves of O''X''-modules. So we can think of the category of real vector bundles on ''X'' as sitting inside the category of sheaves of O''X''-modules; this latter category is abelian, so this is where we can compute kernels and cokernels of morphisms of vector bundles. A rank ''n'' vector bundle is trivial if and only if it has ''n'' linearly independent global sections.


Operations on vector bundles

Most operations on vector spaces can be extended to vector bundles by performing the vector space operation ''fiberwise''. For example, if ''E'' is a vector bundle over ''X'', then there is a bundle ''E*'' over ''X'', called the
dual bundle In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. Definition The dual bundle of a vector bundle \pi: E \to X is the vector bundle \pi^*: E^* \to X whose fibers are the dual sp ...
, whose fiber at ''x'' ∈ ''X'' is the
dual vector space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
(''Ex'')*. Formally ''E*'' can be defined as the set of pairs (''x'', φ), where ''x'' ∈ ''X'' and φ ∈ (''E''''x'')*. The dual bundle is locally trivial because the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
. 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. * The Whitney sum (named for
Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration t ...
) or direct sum bundle of ''E'' and ''F'' is a vector bundle ''E'' ⊕ ''F'' over ''X'' whose fiber over ''x'' is the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
''Ex'' ⊕ ''Fx'' of the vector spaces ''Ex'' and ''Fx''. * The
tensor product bundle In differential geometry, the tensor product of vector bundles ''E'', ''F'' (over same space X) is a vector bundle, denoted by ''E'' ⊗ ''F'', whose fiber over a point x \in X is the tensor product of vector spaces ''E'x'' ⊗ ''F'x''.To co ...
''E'' ⊗ ''F'' is defined in a similar way, using fiberwise
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of vector spaces. * The Hom-bundle Hom(''E'', ''F'') is a vector bundle whose fiber at ''x'' is the space of linear maps from ''Ex'' to ''Fx'' (which is often denoted Hom(''E''''x'', ''Fx'') or ''L''(''E''''x'', ''F''''x'')). 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''. * Building on the previous example, given a section ''s'' of an endomorphism bundle Hom(''E'', ''E'') and a function ''f'': ''X'' → R, one can construct an eigenbundle by taking the fiber over a point ''x'' ∈ ''X'' to be the ''f''(''x'')-
eigenspace In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of the linear map ''s''(''x''): ''E''''x'' → ''E''''x''. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case of ''s'' being the zero section and ''f'' having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in ''E'', while everywhere else the fiber is the trivial 0-dimensional vector space. * The
dual vector bundle In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. Definition The dual bundle of a vector bundle \pi: E \to X is the vector bundle \pi^*: E^* \to X whose fibers are the dual s ...
''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''. 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 In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
ial manner. This is made precise in the language of
smooth functor In differential topology, a branch of mathematics, a smooth functor is a type of functor defined on finite-dimensional real number, real vector spaces. Intuitively, a smooth functor is infinitely differentiable, smooth in the sense that it sends sm ...
s. An operation of a different nature is the
pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
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'' where the bundle over ''X'' × ''X'' is ''E'' × ''F''. Remark: Let ''X'' be a compact space. Any vector bundle ''E'' over ''X'' is a direct summand of a trivial bundle; i.e., there exists a bundle ''E'' such that ''E'' ⊕ ''E'' is trivial. This fails if ''X'' is not compact: for example, the
tautological line bundle In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector ...
over the infinite real projective space does not have this property.


Additional structures and generalizations

Vector bundles are often given more structure. For instance, vector bundles may be equipped with a vector bundle metric. Usually this metric is required to be
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
, in which case each fibre of ''E'' becomes a Euclidean space. A vector bundle with a complex structure corresponds to a
complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...
, 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 In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''-subbundle of the frame bundle#Tangent frame bundle, tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion o ...
. Vector bundles over more general
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is w ...
s may also be used. If instead of a finite-dimensional vector space, if the fiber ''F'' is taken to be a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
then a
Banach bundle In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete metric space, complete normed vector space, possibly of infinite dimension. Definition of a Banach bundle Let ''M'' be a Banach manifold of ...
is obtained. 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 :g_ \colon U\cap V \to \operatorname(F) are continuous mappings of
Banach manifold In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). B ...
s. In the corresponding theory for C''p'' bundles, all mappings are required to be C''p''. Vector bundles are special
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
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 In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S^n of some dimension ''n''. Similarly, in a disk bundle, the fibers are disks D^n. From a topological perspective, there is no difference betwe ...
s are fibered by spheres.


Smooth vector bundles

A vector bundle (''E'', ''p'', ''M'') is smooth, if ''E'' and ''M'' are
smooth manifolds In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, p: ''E'' → ''M'' is a smooth map, and the local trivializations are
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
s. Depending on the required degree of smoothness, there are different corresponding notions of ''Cp'' bundles,
infinitely differentiable In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
''C''-bundles and
real analytic In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...
''C''ω-bundles. In this section we will concentrate on ''C''-bundles. The most important example of a ''C''-vector bundle is the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
(''TM'', ''TM'', ''M'') of a ''C''-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 :g_: U\cap V \to \operatorname(k,\mathbb) is a smooth function into the matrix group GL(k,R), which is a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
. Similarly, if the transition functions are: * ''Cr'' then the vector bundle is a ''Cr'' vector bundle, * ''real 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 In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
(this requires the matrix group to be a
complex Lie group In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
), * ''algebraic functions'' then the vector bundle is an
algebraic vector bundle In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
(this requires the matrix group to be an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
). The ''C''-vector bundles (''E'', ''p'', ''M'') have a very important property not shared by more general ''C''-fibre bundles. Namely, the tangent space ''Tv''(''E''''x'') at any ''v'' ∈ ''E''''x'' can be naturally identified with the fibre ''E''''x'' itself. This identification is obtained through the ''vertical lift'' ''vl''''v'': ''Ex'' → ''T''''v''(''E''''x''), defined as :\operatorname_vw := \left.\frac\_f(v + tw), \quad f\in C^\infty(E_x). The vertical lift can also be seen as a natural ''C''-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''*) ⊂ ''TE'' is the ''vertical tangent bundle'', a natural vector subbundle of the tangent bundle (''TE'', ''TE'', ''E'') of the total space ''E''. The total space ''E'' of any smooth vector bundle carries a natural vector field ''V''''v'' := vl''v''''v'', known as the ''canonical vector field''. More formally, ''V'' is a smooth section of (''TE'', ''TE'', ''E''), and it can also be defined as the infinitesimal generator of the Lie-group action (t,v) \mapsto e^ 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''''x'' = 0, the linear mapping :C_x(X): T_x M \to T_x M; \quad C_x(X) Y = (\nabla_Y X)_x 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''t''''V''(''v'') of ''V'' is globally defined. # For each ''v'' ∈ ''V'' there is a unique limt→∞ Φ''t''''V''(''v'') ∈ ''V''. # ''C''v(''V'')∘''C''v(''V'') = ''C''v(''V'') whenever ''V''''v'' = 0. # The zero set of ''V'' is a smooth submanifold of ''E'' whose codimension is equal to the rank of ''C''v(''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'', ''TE'', ''E'') has a natural
secondary vector bundle structure In mathematics, particularly differential topology, the secondary vector bundle structure refers to the natural vector bundle structure on the total space ''TE'' of the tangent bundle of a smooth vector bundle , induced by the push-forward of t ...
(''TE'', ''p''*, ''TM''), where ''p''* 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 +*: ''T''(''E'' × ''E'') → ''TE'' and λ*: ''TE'' → ''TE'' of the original addition +: ''E'' × ''E'' → ''E'' and scalar multiplication λ: ''E'' → ''E''.


K-theory

The K-theory group, , of a compact Hausdorff topological space is defined as the abelian group generated by isomorphism classes of
complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...
s modulo the relation that whenever we have an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...
: 0 \to A \to B \to C \to 0 then : = + in
topological K-theory In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...
. KO-theory is a version of this construction which considers real vector bundles. K-theory with
compact support In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
s can also be defined, as well as higher K-theory groups. The famous periodicity theorem of
Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...
asserts that the K-theory of any space is isomorphic to that of the , the double suspension of . In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, one considers the K-theory groups consisting of
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
, 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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
.


See also


General notions

*
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
:
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free acti ...
s for vector bundle, among which
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s for
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
s *
Characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
*
Splitting principle In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computation ...
* Stable bundle


Topology and differential geometry

*
Gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
: the general study of connections on vector bundles and principal bundles and their relations to physics. * Connection: the notion needed to differentiate sections of vector bundles.


Algebraic and analytic geometry

*
Algebraic vector bundle In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
*
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
*
Holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...


Notes


Sources

*. *. *, see section 1.5. *. *. * see Ch.5 *.


External links

*
Why is it useful to study vector bundles ?
on
MathOverflow MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a ...

Why is it useful to classify the vector bundles of a space ?
{{Manifolds