Bundle

Almost synonymous terms used in various areas are Topological bundle, Locally trivial fibre bundle, Fibre space, Fibration, Skew product etc. Particular cases are Vector bundle, Tangent bundle, Principal fibre bundle, $\dots$

2020 Mathematics Subject Classification: Primary: 55Rxx Secondary: 14Dxx32Lxx53Cxx55Sxx57Rxx [MSN][ZBL]

A very flexible geometric construction aimed to represent a family of similar objects (fibres or fibers, depending on the preferred spelling) which are parametrized by the index set which itself has an additional topological or geometric structure (topological space, smooth or holomorphic manifold etc.).

The most known examples are the tangent and cotangent bundle of a smooth manifold. The coverings are also a special particular form of a topological bundle (with discrete fibers).

Formal definition of a topological bundle

Let $\pi:E\to B$ be a continuous map between topological spaces, called the total space and the base, and $F$ yet another topological space called the (generic) fiber, such that the preimage $F_b=\pi^{-1}(b)\subset E$ of every point of the base is homeomorphic to $F$. The latter condition means that $E$ is the disjoint union of "fibers", $E=\bigsqcup_{b\in B} F_b$ homeomorphic to each other.

The map $\pi$ is called fibration of $E$ over $B$, if the above representation is locally trivial: any point of the base admits an open neighborhood $U$ such that the restriction of $\pi$ on the preimage $\pi^{-1}(U)$ is topologically equivalent to the Cartesian projection $\pi_2$ of the product $F\times U$ on the second component: $\pi_2(v,b)=b$. Formally this means that there exists a homeomorhism $H_U=H:\pi^{-1}(U)\to F\times U$ such that $\pi=\pi_2\circ H$.

Examples

1. The trivial bundle $E=F\times B$, $\pi=\pi_2: F\times B\to B$, $(v,b)\mapsto b$. In this case all trivializing homeomorphisms are globally defined on the entire total space (as the identity map).
2. Let $E=\R^n\smallsetminus\{0\}$ be the punctured Euclidean space, $B=\mathbb S^{n-1}$ the standard unit sphere and $\pi$ the radial projection $\pi(x)=\|x\|^{-1}\cdot x$. This is a topological bundle with the fiber $F=(0,+\infty)\simeq\R^1$.
3. Let $E=\mathbb S^{n-1}$ as above, $B=\R P^{n-1}$ the real projective space (all lines in $\R^n$ passing through the origin) and $\pi$ the map taking a point $x$ on the sphere into the line $\ell_x$ passing through $x$. The preimage $\pi^{-1}(\ell)$ consists of two antipodal points $x$ and $-x\in\mathbb S^{n-1}$, thus $F$ is a discrete two-point set $\mathbb Z_2=\{-1,1\}$. This is a topological bundle, which cannot be trivial: indeed, if it were, then the total space $\mathbb S^{n-1}$ would consist of two connected components, while it is connected.
4. More generally, let $\pi:M^m\to N^n$ be a differentiable map between two smooth (connected) compact manifolds of dimensions $m\ge n$, which has the maximal rank (equal to $n$) everywhere. One can show then, using the implicit function theorem and partition of unity, that $\pi$ is a topological bundle with a fiber $F$ which itself is a smooth compact manifold.
5. The Hopf fibration $\mathbb S^3\to\mathbb S^2$ with the generic fiber $\mathbb S^1$. It is best realized through the restriction on the sphere $\mathbb S^3=\{|z|^2+|w|^2=1\}\subseteq\C^2$ of the canonical map $(z,w)\mapsto [z:w]\in\C P^1=\C^1\cup\{\infty\}\simeq\mathbb S^2$. The preimage of each point on the projective plane is a line in $\C^2$ which intersects the unit sphere $\mathbb S^3$ by the circle. This fibration can be spectacularly visualized if the sphere $\mathbb S^3$ is punctured (one of its point deleted) to become $\R^3$: fibers are linked between themselves.

Cocycle of a bundle

On a nonvoid overlapping $U_{\alpha\beta}=U_\alpha\cap U_\beta$ of two different trivializing charts $U_\alpha$ and $U_\beta$ two homeomorphisms $H_\alpha,H_\beta: \pi^{-1}(U_{\alpha\beta})\to F\times U_{\alpha\beta}$ are defined. Since both $H_\alpha$ and $H_\beta$ conjugate $\pi$ with the Cartesian projection on $U_{\alpha\beta}$, they map each fiber $F_b=\pi^{-1}(b)$ into the same space $F\times\{b\}$. The composition $H_\alpha\circ H_\beta^{-1}$ keeps constant the $b$-component and hence takes the "triangular" form $$ H_\alpha\circ H_\beta^{-1}:(v,b)\mapsto (H_{\alpha\beta}(b,v),b),\qquad H_{\alpha\beta}(\cdot,b)\in\operatorname{Homeo}(F) $$ with the homeomorphisms $H_{\alpha\beta}(\cdot, b)$ continuously depending on $b\in U_{\alpha\beta}$. The collection of these "homeomorphism-valued" functions defined in the intersections $U_{\alpha\beta}$ is called the cocycle associated with a given trivialization of the bundle $\pi$ (or simply the cocycle of the bundle. They homeomorphisms $\{H_{\alpha\beta}\}$ satisfy the following identities, obvious from their construction: $$ H_{\alpha\beta}\circ H_{\beta\alpha}=\operatorname{id},\qquad H_{\alpha\beta}\circ H_{\beta\gamma}\circ H_{\gamma\alpha}=\operatorname{id}, \tag{HC} $$ the second being true on every nonvoid triple intersection $U_{\alpha\beta\gamma}=U_\alpha\cap U_\beta\cap U_\gamma$.

Bundles from cocycles: the abstract "patchwork" construction

Every bundle directly defined by the map $\pi$ implicitly assumes that a trivializing atlas can be produced, thus defining the corresponding cocycle. Conversely, starting from a cocycle (HC) one can explicitly construct an abstract topological space $E$ together with the projection $\pi$. Let $\widetilde E=\bigsqcup F\times U_\alpha$ be the disjoint union of the "cylinders" $F\times U_\alpha$, on which the equivalence relation is defined: $$ (v_\alpha, b_\alpha)\sim(v_\beta,b_\beta) \iff b_\alpha=b_\beta\in U_\alpha\cap U_\beta,\quad v_\alpha=H_{\alpha\beta}(b_\beta)\,v_\beta. $$ The cocycle identities ensure that this is indeed a symmetric and transitive equivalence relation. The quotient space $E=\widetilde E/\sim$ admits the natural projection on the base $B$ which precisely corresponds to the specified cocycle.

Example. One can construct the "product" of any two bundles $\pi_1:E_1\to B$ and $\pi_2:E_2\to B$ over the same base by applying the above construction to the sets $(F_1\times F_2)\times U_\alpha$ and using the Cartesian product of the maps $\{H_{\alpha\beta}^i\}$, $i=1,2$, for the identification, $$ \begin{pmatrix} H^1_{\alpha\beta}&\\&H^2_{\alpha\beta}\end{pmatrix}:(F_1\times F_2)\times U_{\alpha\beta}\to (F_1\times F_2)\times U_{\alpha\beta}. $$

Vector bundles and other additional structures on the fibers

The general construction of bundle easily allows various additional structures, both on the base space and (more importantly) on the fibers. By far the most important special case is that of vector bundles.

To define a vector bundle, one has in addition to the principal definition assume the following:

1. The fiber $F$ is a vector space, and
2. The trivializing homeomorphisms must respect the linear structure of the fibers.

The second assumption means that rather than being arbitrary homeomorphisms, the maps $\{H_{\alpha\beta}\}$ forming the bundle cocycle, must be linear invertible of each "standard fiber" $F\times \{b\}$; if the fiber is identified with the canonical $n$-space $\Bbbk^n$ (over $\Bbbk=\R$ or $\Bbbk=\C$), then the cocycle will consist of invertible continuous matrix-functios $M_{\alpha\beta}:U_{\alpha\beta}\to\operatorname{GL}(n,\Bbbk)$, so that $H_{\alpha\beta}(v,b)=(M_{\alpha\beta}(b)\, v, b)$, $v\in\Bbbk^n$. The cocycle identities become then the identites relating the values of these matrix-valued functions, $$ M_{\alpha\beta}(b)\cdot M_{\beta\alpha}(b)\equiv E,\qquad M_{\alpha\beta}(b)\cdot M_{\beta\gamma}(b)\cdot M_{\gamma\alpha}(b)\equiv E, \tag{MC} $$ where $E$ is the $n\times n$-identical matrix.

For vector bundles all linear constructions become well defined on fibers.

Following the way, one may define vector bundles with extra algebraic structures on the fibers. For instance, if the cocycle defining the bundle, consists of orthogonal matrices, $M_{\alpha\beta}:U_{\alpha\beta}\to\operatorname{SO}(n,\R)$, then the fibers of the bundle naturally acquire the structure of Euclidean spaces. Other natural examples are bundles whose fibers have the Hermitian structure (the cocycle should consist of unitary matrix functions then) or symplectic spaces (with canonical cocycle matrices preserving the symplectic structure).

Tangent and cotangent bundle of a smooth manifold

If $M$ is a smooth manifold with the atlas of coordinate charts $\{U_\alpha\}$ and the maps $h_\alpha:U_\alpha\to\R^m$, then the differentials of these maps $\rd h_\alpha$ allow to identify the tangent space $T_a M$ at $a\in U_{\alpha}$ with $\R^m$ and the union $\bigsqcup_{a\in U_\alpha}T_a M$ with $\R^m\times U_\alpha$ (we write the tangent vector first). For a point $a\in U_{\alpha\beta}$ there are two identifications which differ by the Jacobian matrix of the transition map $h_{\alpha\beta}=h_\alpha\circ h_\beta^{-1}$. This shows that the tangent bundle $TM$ is indeed a vector bundle in the sense of the above definition.

The cotangent bundle is also trivialized by every atlas $\{h_\alpha:U_\alpha\to\R^m\}$ on $M$, yet in this case the direction of arrows should be reverted: the cotangent space $T_a^*M$ is identified with $\R^n$ by the linear map $(\rd h_\alpha^*)$, thus the corresponding cocycle will consist of the transposed inverse Jacobian matrices.

Equivalence of cocycles

The trivializing maps defining the structure of a bundle (vector or topological) are by no means unique, even if the covering domains $U_\alpha$ remain the same. E.g., one can replace the collection of maps $\{H_{\alpha}\}$ trivializing a vector bundle, by another collection $\{H'_{\alpha}\}$, post-composing them with the maps $F\times U_\alpha\to F\times U_\alpha$, $(v,b)\mapsto (C_\alpha(b)\,v, b)$ with invertible continuous matrix functions $C_\alpha:U_\alpha\to\operatorname{GL}(n,\Bbbk)$. The corresponding matrix cocycle $\{M_{\alpha\beta}\}$ will be replaced then by the new matrix cocycle $\{M'_{\alpha\beta}(b)\}$, $$ M'_{\alpha\beta}(b)=C_\alpha(b)M_{\alpha\beta}(b)C_\beta^{-1}(b),\qquad b\in U_{\alpha\beta}. \tag{CE} $$ Two matrix cocycles related by these identities, are called equivalent and clearly define the same bundle.

Example. The trivial cocycle $\{M_{\alpha\beta}(b)\}=\{E\}$ which consists of identity matrices, corresponds to the trival bundle $F\times B$: the trivializing maps agree with each other on the intersections and hence define the global trivializing map $H:E\to F\times B$. A cocycle equal to the trivial one in the sense of (CE) is called solvable: its solution is a collection of invertible matrix functions $C_\alpha:U_\alpha\to\operatorname{GL}(n,\Bbbk)$ such that on the overlapping of the domains $U_{\alpha\beta}=U_\alpha\cap U_\beta$ the identities $$ M_{\alpha\beta}(b)=C_{\alpha}^{-1}(b)C_\beta(b),\qquad \forall\alpha,\beta,\ b\in U_{\alpha}\cap U_\beta. $$ Thus solvability of cocycle is an analytic equivalent of the topological triviality of the bundle.

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Special classes of bundles

Together with vector bundles, there are other special classes of bundles.

Fibrations

If all three spaces occurring in the definition of the topological bundle (the total space $E$, the base $B$ and the generic fiber $F$) are smooth manifolds and all the maps (the projection $\pi$ and all the trivalizing maps $H_\alpha$) are differentiable, then the bundle is often called a fibration, or locally trivial fibre bundle.

For a fibration every tangent space $T_x E$ is mapped by the differential $\rd \pi:T_x E\to T_a B$, $a=\pi(x)$ surjectively, with the kernel being the tangent space to the fiber $F_a$ at $x$: $\operatorname{Ker}\rd \pi(x)=T_x F_a$. The direction tangent to the fibers is often referred to as vertical, with the idea that the base is "horizontal". However, the accurate definition of the horizontal direction can be made only in terms of an appropriate connection on the bundle.

$G$-bundles and principal bundles

Assume that $\pi:E\to B$ is a fibration as above and the fiber $F$ has a structure of a homogeneous space on which a Lie group $G$ acts freely and transitively (say, by the right multiplication), generating thus the action of $G$ on the total space $E$ which is continuous. Then this action should be consistent with the local trivializations $H_\alpha:\pi^{-1}(U_\alpha)\to G\times U_\alpha$: the corresponding transition maps $H_{\alpha\beta}(\cdot,b):G\to G$ must commute with the right action of $G$. This means that $$ \forall g\in G, \ b\in B,\qquad H_{\alpha\beta}(g,b)=H_{\alpha\beta}(e\cdot g,b)=H_{\alpha\beta}(e,b)\cdot g =g_{\alpha\beta}(b)\cdot g, \tag{T} $$ where $g_{\alpha\beta}=H_{\alpha\beta}(e)\in G$ is the uniquely defined group element (depending continuously on $b\in B$), and $e\in G$ is the unit of the group. Thus the $G$-bundle is completely determined by the cocycle $\{g_{\alpha\beta}:U_{\alpha\beta}\to G\}$ satisfying the cocycle identites, $$ g_{\alpha\beta}(\cdot)g_{\beta\alpha}(\cdot)\equiv e,\qquad g_{\alpha\beta}(\cdot)g_{\beta\gamma}(\cdot)g_{\gamma\alpha}(\cdot)\equiv e. \tag{GC} $$ Such a bundle (defined by the left $G$-action of multiplication by $g_{\alpha\beta}$ in the transition maps (T)) is called a principal $G$-bundle with the structural group $G$.

This construction allows to associate (tautologically) with each vector bundle $\pi:E\to B$ with a fiber $\Bbbk^n$ a principal $G$-bundle $\varPi:\mathbf E\to B$ with the same base, where $G=\operatorname{GL}(n,\Bbbk)$. Analytically this is achieved by considering the same matrix cocycle (MC) and re-interpreting it as the $G$-valued cocycle (GC), $g_{\alpha\beta}=M_{\alpha\beta}$. This bundle is (not surprisingly) called the associated principal bundle. If the matrix cocycle $\{M_{\alpha\beta}\}$ takes values in a subgroup $G\subsetneq\operatorname{GL}(n,\Bbbk)$, then the associated principal bundle may have a "smaller" fiber (say, the orthogonal group).

Example. The principal bundle associated with the tangent vector bundle $TM$ is the bundle whose fibers are frames (linear independent ordered tuples of tangent vectors spanning the tangent space $T_aM$ at each point $a\in M$).

The importance of vector bundles and principal bundles roots in the fact that their fibers have a natural structure of a parallelizable manifolds (homogeneous spaces), looking the same near each point.

Line bundles and the "genuine" cohomology

The case of vector bundles of rank $1$ is especially important: first, because in this case the vector bundle is indistinguishable from the associated principal bundle, but mainly because the corresponding group $G=\operatorname{GL}(1,\Bbbk)\simeq\Bbbk^*$ is commutative. This allows to activate the powerful machinery of the sheaf theory and the respective (Cech) cohomology theory.

Algebraic and analytic vector bundles

In the algebraic category it is natural to consider bundles over schemes (ringed spaces) rather than over smooth manifolds. This requires quite a number of changes in the construction, see Vector bundle, algebraic.

Similar difficulties arise in an attempt to define vector bundles over analytic spaces, see Vector bundle, analytic. In both versions one has to use the language and constructions from the sheaf theory.

Bundles with a discrete fiber and topological coverings

If the fiber $F$ is a topological space with the discrete topology, the corresponding bundle is generally referred to as a covering. Indeed, since $F$ is completely disjoint (each point $v$ is both open and closed), the preimage $\pi^{-1}(U)=\bigsqcup_{v\in F} U_v$ is a disjoint union of the sets $U_v$ homeomorphic to $U$.

More features

In several areas of applications other types of fibers may be important, among them:

• Sphere bundles $F\simeq\mathbb S^k$,
• Projective bundles $F\simeq \mathbb P^k$ (real or complex projective spaces),
• Quaternionic bundles.

The construction of the bundle is so flexible that almost any specific flavor can be incorporated into it. In particular, one can consider holomorphic fibrations with the total space, base and the generic fiber are complex analytic manifolds and the projection and the trivializations are holomorphic maps.

Finally, one can allow for singulatities, assuming that the local structure of the Cartesian product holds only outside of a "small" subset $\varSigma$ of $B$, on which the "bundle" is singular. While formally one can simply omit the exceptional locus and consider the "genuine" bundle $\pi':E'\to B'$, where $E'=E\smallsetminus\pi^{-1}(B')$ and $B'=B\smallsetminus\varSigma$, the singularity very often carries the most important part of the information encoded in the specific degeneracy of the cocycle automorphisms $\{H_{\alpha\beta}(\cdot,b)\}$.

Morphisms and sections

The "triangular" structure (fibers parametrized by points of the base) dictates necessarily restrictions on the morphisms in the category of bundles, but also the possible operations with bundles.

Fibered maps

If $\pi_i:E_i\to B_i$ are two bundles, $i=1,2$, then a morphism between the two bundles is a map between the total spaces, which sends fibers to fibers. Formally, such morphism is defined by a pair of maps $h:B_1\to B_2$ between the bases and $H:E_1\to E_2$ between the total spaces, such that $$ \pi_2\circ H=h\circ \pi_1, $$ and such that the restriction of $H$ on each fiber $F_b=\pi_1^{-1}(b_1)$ preserves the possible additional structures which may exist on the fibers $\pi_1^{-1}(b_1)$ and $\pi_2^{-1}(h(b_1))$. E.g., if $\pi_1,\pi_2$ are vector bundles, then the restriction of $H$ on each fiber should be a linear map.

Two bundles are equivalent (or isomorphic), if there exist two mutually inverse morphisms $(H,h)$ and $(H^{-1},h^{-1})$ between them in the two opposite directions.

Gauge transforms

A fibered self-map of the bundle which covers the identical map of the base is called also gauge transformation because of the connections with the various field theories in Physics. In each trivializing map $F\times U_\alpha$ it is defined by a map $G_\alpha:U_\alpha\to\operatorname{Hom}(F,F)$; two trivializations $G_\alpha,G_\beta$ both defined in a common domain $U_{\alpha\beta}$ are conjugated by the corresponding cocycle, $$ H_{\alpha\beta}\circ G_\beta=G_\alpha\circ H_{\alpha\beta}\qquad\forall \alpha,\beta. $$

Induced bundle

If $\pi:E\to B$ is a topological bundle and $h:B'\to B$ a continuous map, then one can construct the induced fibre bundle $\pi':E\to B'$ with the same generic fiber $F$. By construction (pullback), the fibers of the new bundle, $\pi'^{-1}(b')$ coincide with the fibers $\pi^{-1}(h(b'))$ for all $b'\in B'$. Formally one defines the total space $E'$ as a subset of $E\times B'$ which consists of pairs $(x,b')$ such that $\pi(x)=h(b')$. Then the map $\pi':E'\to B'$ is well defined by the tautological identity $\pi'(x,b')=b'$. Simple checks show that this construction allows to carry all additional fiber structures from one bundle to another.

If $B'\subseteq B$ and $h$ is the inclusion map, $h:B'\hookrightarrow B$, then the induced bundle is simply the restriction of $\pi$ on $B'$, usually denoted as $\pi|_{B'}$.

Sections

A section of a bundle $\pi:E\to B$ is a regular (continuous, smooth, analytic) selector map which chooses for each point $b\in B$ of the base a single element from the corresponding fiber $F_b=\pi^{-1}(b)$. Formally, a section is a map $s:B\to E$, such that $\pi\circ s:B\to B$ is the identity map. A frequent notation for the space of sections of the bundle $E$ is $\Gamma(E)$.

Examples. A "scalar" ($\Bbbk$-valued) function $f:B\to\Bbbk$ is a section of the trivial line bundle $\pi:\Bbbk\times B\to B$. A section of the tangent bundle of a manifold $M$ is called the vector field on $M$. A section of the cotangent bundle is a differential 1-form.

Not every bundle admits sections. For instance, the principal bundle associated with the tangent bundle $T\mathbb S^2$ to the 2-sphere, admits no smooth sections (if it would, then one would be able to construct a nonvanishing vector field on the 2-sphere, which is impossible).

The set of all sections forms a topological space with additional structures inherited from that on the generic fiber, e.g., sections of the vector bundle form a module over the ring of "scalar" functions.

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Fiberwise operations

For topological bundles with generic fibers having extra structure, almost every construction which makes sense in this structure, can be implemented "fiberwise".

Example. Let $\pi:E\to B$ be a topological bundle with a generic fiber $F$, and $A\subset F$ is a topological subspace. The map $\pi': E'\to B$ is a subbundle of $\pi$, if $E'\subset E$ is a subset and the trivializing maps $H_\alpha:\pi^{-1}(U_\alpha)\to F\times U_\alpha$ can be chosen in such a way that they map $\pi'^{-1}(U_\alpha)$ homeomorphically onto $A\times U_\alpha$. In other words, a subbundle of $\pi$ is a subspace $E'\subset E$ which is itself a bundle with respect to the restriction of $\pi|_{E'}$.

A subbundle of the tangent bundle $TM$ of a smooth manifold is called distribution of tangent subspaces.

Note. A subbundle of a trivial bundle may well be nontrivial.

Whitney sum of bundles

If $\pi_i:E_i\to B$, $i=1,2$, are two bundles with generic fibers $F_1,F_2$ over the same base, then one can construct a bundle $\pi$ with the generic fiber $F=F_1\times F_2$ over the same base. In case of the vector bundles one usually says about the direct sum, or Whitney sum and denoted by $\pi_1\oplus \pi_2$.

Intuitively this means that the fibers $\pi^{-1}(b)$ of new bundle for all $b\in B$ are Cartesian products $\pi^{-1}_1(b)\times\pi^{-1}_2(b)\simeq F_1\times F_2=F$. Formally the construction goes through the intermediate step of the bundle $\pi'=\pi_1\times \pi_2$ with the total space $E'=E_1\times E_2$ and the base $B'=B\times B$: $$ \pi'(x_1,x_2)=(b_1,b_2),\qquad b_i=\pi_i(x_i)\in B,\quad x_i\in E_i. $$ The Whitney sum $\pi_1\oplus\pi_2$ is the restriction (see above) of the bundle $\pi'$ on the diagonal $B\simeq\{(b_1,b_2):\ b_1=b_2\}\subset B\times B=B'$.

Predictably, if both $\pi_1$ and $\pi_2$ are subbundles of some common ambient vector bundle $\varPi:\mathbf E\to B$, and the fibers $\pi_i^{-1}(b)\subset\varPi^{-1}(b)$ are disjoint, then their sum $\pi_1\oplus\pi_2$ is isomorphic to the subbundle of $\varPi$ with the fibers $\pi_1^{-1}(b)+\pi_2^{-1}(b)$ for all $b\in B$.

In terms of the trivializing coordinates, if the matrix cocycles of the two vector bundles are $M^1_{\alpha\beta}(\cdot)$ and $M^2_{\alpha\beta}(\cdot)$, defined in the pairwise intersections $U_{\alpha\beta}=U_\alpha\cap U_\beta\subseteq B$, then the matrix cocycle associated with the Whitney sum is the cocycle of the block diagonal matrix functions $$ M_{\alpha\beta}(\cdot)=\begin{pmatrix}M^1_{\alpha\beta}(\cdot)&\\& M^2_{\alpha\beta}(\cdot)\end{pmatrix}:U_{\alpha\beta}\to \operatorname{GL}(d_1+d_2,\Bbbk),\tag{WS} $$ where $d_{1,2}$ are the dimensions (ranks) of the vector bundles $\pi_{1,2}$. Moreover, the Whitney sum can be directly build from the cocycle (WS) using the "patchworking" construction.

Other constructions with bundles

Besides the Whitney sum, one can use most of (linear algebraic) "continuous" functorial constructions to produce new bundles from existing ones. The formal way to do this is by applying the constructions in the trivializing charts and use the patchworking method to piece the results together. A partial list of such constructions is as follows:

1. Dual bundle $\pi^*:E^*\to B$ with the generic fiber being the dual vector space $F^*\simeq \R^{n*}$ and the matrix cocycle $\{(M_{\alpha\beta}^*)^{-1}\}$.
2. Tensor product $\pi_1\otimes\pi_2$ of two bundles $\pi_1,\pi_2$ (always over the same base) with the matrix cocycle $\{M^1_{\alpha\beta}\otimes M^2_{\alpha\beta}\}$;
3. The cocycle $\operatorname{Hom}(\pi_1,\pi_2)$ with the generic fiber being the space of linear operators from $F_1$ to $F_2$. The dual bundle is the particular case of this construction, $\pi^*=\operatorname{Hom}(\pi,\epsilon)$, where $\epsilon:\Bbbk\times B\to B$ is the trivial scalar bundle. As with the linear spaces, $\operatorname{Hom}(\pi_1,\pi_2)=\pi_1^*\otimes\pi_2$.
4. The exterior products, e.g., powers $\pi\land\cdots\land\pi$, including the determinant bundle (the highest exterior power). Especially important are wedge powers of the tangent and cotangent bundle, $T^pM=\bigwedge^pTM$, resp., ${T^*}^q M=\bigwedge^q T^*M$ of a smooth manifold $M$: sections of these bundles are $p$-polyvector fields, resp., exterior (differentiable) $q$-forms.

Clearly, this approach works (with necessary minimal modifications) also in the categories of bundles with other structure of the generic fiber.

Comments

Vector bundles over differentiable manifolds may carry a special geometric structure, called connection. In terms of these connections one can introduce certain cohomology classes of the base manifold, which in fact depend only on the on the bundle and not on the connection.

Connections on bundles

Although the fibers $F_b=\pi^{-1}(b)$ of a bundle vary "in a regular way" in the total space $E$ together with the base point $b\in B$, in general there is no canonical way to compare (identify) points on two (even close) fibers. One can introduce an additional structure on the bundle, which allows for any two fibers $F_{b_0},F_{b_1}$ over two different points $b_0,b_1\in B$ connected by a piecewise-smooth curve $\gamma:[0,1]\to B$, $\gamma(0)=b_0$, $\gamma(1)=b_1$, to construct a linear parallel transport map $T_\gamma:F_{b_0}\to F_{b_1}$ describing the way vectors from the fibers are moved along the curve $\gamma$. The infinitesimal generator of this "parallel translation" construction is called the covariant derivative and is formalized by a family of operators allowing to differentiate sections of the vector bundle in the direction of the velocity vector $w=\dot \gamma(0)$. The result of a parallel transport (translation) along a closed loop with $\gamma(0)=\gamma(1)$ may well be nonzero and its quantitative measure is the curvature of the connection. Flat connections (with zero curvature) are similar to coverings: they admit a special class of locally constant sections.

Riemannian geometry

This huge subject covers the local and global study of manifolds whose tangent bundle is equipped with the Euclidean structure (positive definite quadratic form, also known as the metric tensor). This additional structure allows to define lengths or smooth arcs on the manifold, introduce the extremals of this length etc.

In the Riemannian geometry (of manifolds whose tangent spaces are equipped with a scalar product) the isometric parallel transport can be introduced in a unique way leading to the notion of the Levi-Civita connection. For this connection the abstract curvature is closely related with the Gauss curvature.

Characteristic classes

Using connections on real and complex bundles, one can define special cohomology classes of the manifold $B$ (with coefficients in $\Z_2$ or $\Z$) which turn out to be independent of a specific connection used for their construction and measure different aspects of nontriviality of vector bundles. These classes behave naturally with respect to the pullback operation (induced connections) and obey some simple rules for the Whitney sums. These classes are called characteristic classes (there are four main types of them, -- Stiefel-Whitney class, Pontryagin class, Euler class, Chern class).

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Literature

The classical expositions are still the most popular sources for references.

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