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A fiber bundle whose fibers are affine spaces and whose transition functions are affine. Differences of points in one fiber lie in a corresponding model vector bundle, but there need not be a preferred zero in each fiber.

For the vector-bundle curvature matrix $R$ of a metric connection on a vector bundle on a real oriented vector bundle, the A-hat form is the even differential form defined by the displayed finite-degree power series, with square root having constant term one. The expression is independent of the choice of orthonormal frame. Equivalently, for formal paired curvature Chern roots $\pm x_j$, it is ${\prod }_j(x_j/2)/\sinh (x_j/2)$. The Bianchi identity makes it closed, and changing the connection changes it by an exact differential form via Chern-Weil connection transgression. Its degree-four part is $-p_1/24$, where $p_1=-\mathrm{tr}(R\wedge R)/(8{\pi }^2)$ is the first Pontryagin class form.

A linear representation $\rho :G\to GL(V)$ of a principal bundle's structure group defines the quotient of $P\times V$ by $(p,v)\sim (pg,\rho (g{)}^{-1}v)$. It is a vector bundle over the same base. A principal connection induces parallel transport and a local covariant derivative $d+{\rho }_{\ast }(A)$ on its sections.

Every vector bundle on the projective line over a field splits as a direct sum of line bundles. Choose the largest integer $a$ with $H^0(E(-a))\ne 0$. A corresponding map $\mathcal{O}(a)\to E$ is saturated, since saturation otherwise increases its degree. Inductively split its quotient as ${⨁}_i\mathcal{O}(b_i)$. Twisting the exact sequence by $-a-1$ and using $H^1(\mathcal{O}(-1))=0$ gives $b_i\le a$. Hence all extension groups $H^1(\mathcal{O}(a-b_i))$ vanish, and the sequence splits. The multiset of degrees is determined by the dimensions of all twisted global-section spaces.

Let $H$ be the tautological bundle on ${\mathbb{CP}}^1\cong S^2$. One sign convention for the Bott element iswhere $[1]$ is the rank-one trivial vector bundle. It has rank zero, and its Chern character has degree-two term $c_1(H)$, a generator up to sign. Choosing the dual line bundle reverses that sign.

The $n$-fold reduced external product in topological K-theory gives a generator ofThis product uses the smash product of different sphere factors. It is distinct from taking exterior powers of a vector bundle over one fixed base. Multiplication by the Bott element realizes Bott periodicity.^[ref]

Every $C^{\infty }(M)$-linear map between the modules of smooth sections of finite-rank smooth vector bundles is induced by a unique vector bundle morphism. A smooth bump function shows that the map is local. A bumped local frame then proves that a section vanishing at $x$ has image vanishing at $x$. Evaluating the image therefore defines a well-defined fiberwise linear map. The images of the bumped frame give smooth matrix columns, proving the resulting morphism is smooth. This elementary argument is also described in Brian Conrad's bundle notes.

For a complex vector bundle $E\to X$ of rank $r$, its Chern classes are integral cohomology classesThey are natural under pullback and obey the Whitney sum formula for Chern classes. For a complex line bundle, $c_1$ is the Euler class of its underlying oriented real two-plane bundle; generally $c_r(E)$ is the Euler class of the underlying real bundle.^[ref]

For example, if $h=c_1(\mathcal{O}(1))$ on Complex projective space, the Euler sequence gives $c(T{\mathbb{P}}^n)=(1+h{)}^{n+1}$, truncated above degree $2n$. For an algebraic vector bundle on a smooth variety, algebraic Chern classes instead belong to the Chow groups; the cycle class map recovers the topological classes over the complex numbers.

For a vector bundle $E$ of rank $r$ on a smooth algebraic variety, algebraic Chern classes have $c_0(E)=1$, vanish for $i>r$, commute with pullback, and obey the Whitney sum formula for Chern classes for every short exact sequence of vector bundles. A line bundle $\mathcal{O}(D)$ has total class $1+[D]$. The projective-bundle splitting construction makes pullback injective and supplies a filtration by line bundles, so these axioms determine all classes as elementary symmetric functions of their first classes. In particular, $c_i(E^{\ast })=(-1{)}^i{c}_i(E)$ and $c_1(L\otimes M)=c_1(L)+c_1(M)$ in the ordinary Chow ring.

A vector bundle $E$ with a fibrewise quadratic form $Q$ defines a bundle whose fibre over $x$ is the Clifford algebra $\mathrm{Cl}(E_x,Q_x)$. When the polar form is a nondegenerate bilinear form, local orthonormal frames identify its fibres with a fixed Clifford algebra, and orthogonal transition maps induce the algebra transition maps. A Clifford module bundle is a bundle of modules over these algebras, rather than the algebra bundle itself. For a Riemannian manifold, use its cotangent bundle and the metric quadratic form, with the sign convention specified.

A Clifford module bundle is a vector bundle over a Riemannian manifold with a fibrewise Clifford multiplication. A compatible connection on a vector bundle satisfies $[{\nabla }_X,c(\alpha )]=c({\nabla }_X\alpha )$ for the Levi-Civita connection on covectors. On a graded Clifford module bundle, Clifford multiplication exchanges the two grading summands. The associated Dirac operator is $D={\sum }_{i}c(e^i){\nabla }_{e_i}$. Its fibres are modules over the fibres of the Clifford algebra bundle.

A vector bundle over $S^n$ can be constructed by gluing two trivial bundles over its hemispheres with a transition map $\gamma :S^{n-1}\to GL(k,\mathbb{R})$. Extend this map constantly in an equatorial collar to obtain local vector bundle trivializations. A transition map into the special orthogonal group gives compatible fiber orientations and a fiber metric. Homotopies of transition maps give isomorphic bundles.

A clutching function is the equatorial transition map used in the clutching construction. For an oriented real vector bundle one may choose a fiber metric and orthonormal trivializations, so its clutching function takes values in the special orthogonal group.

In a proper flat family of algebraic curves, if the first sheaf cohomology of a line bundle vanishes on every fibre of an open set, its direct image there is a vector bundle of rank given by the Riemann-Roch theorem and commutes with arbitrary base change. Locally a finite free complex computes both fibre sheaf cohomology groups. Vanishing of the last cokernel makes its last map surjective; splitting that map leaves a finite locally free kernel in degree zero. This is the mechanism behind charts of the Picard scheme of a curve and evaluation descriptions of the theta divisor.

Complexification replaces each fiber of a real vector bundle by its tensor product with $\mathbb{C}$ over $\mathbb{R}$. The complexification of a real vector bundle of rank one is a complex line bundle. For the tautological bundle over Real projective space, this complex line bundle squares to the trivial vector bundle.

A complex line bundle is a vector bundle whose fibers are one-dimensional complex vector spaces and whose transition maps are complex linear. Its underlying real bundle has rank two and a canonical orientation, and its Euler class equals its First Chern class.

A complex vector bundle is a vector bundle with complex-linear local trivializations and transition maps. A rank-$r$ complex bundle has an underlying oriented real bundle of rank $2r$, using the orientation of complex bases. Rank-one bundles are complex line bundles, and their higher-rank counterparts have integral Chern classes.

Write a local frame of a rank-$r$ vector bundle as the row $e=(e_1,\dots ,e_r)$. The connection one-form of a connection on a vector bundle is the matrix of one-forms $A$ determined byThe placement of $A$ matters: its entry $A^i{}_j$ is the coefficient of $e_i$ in $\nabla e_j$. If $e^{\prime }=eg$, substitution in the Leibniz rule givesThe derivative term means that $A$ is not itself a global endomorphism-valued one-form. The curvature form $F=dA+A\wedge A$ instead changes by $F^{\prime }=g^{-1}Fg$, and the difference of two connections on a vector bundle has the same homogeneous change law. These formulas fix the row-frame convention consistently.

A local trivialization of a smooth vector bundle gives a local connection by differentiating its component functions. Weight these connections by a subordinate locally finite partition of unity and extend the weighted terms by zero. The weights sum to one, so the connection Leibniz rule survives. This proves existence on a paracompact smooth manifold. Averaging connections is valid with weights summing to one because the collection is the affine space of vector-bundle connections.

For an oriented rank-$r$ vector bundle, let $U\in H^r(D(E),S(E);\mathbb{Z})$ be its Thom class. Forgetting relative supports sends $U$ to ${\pi }^{\ast }e(E)$, where $e(E)$ is the Euler class. Compatibility of relative and mixed cup products givesFor odd $r$, graded commutativity of the cup product makes $2U^2=0$. The Thom isomorphism theorem is injective, so $2e(E)=0$. This explains why an odd-rank Euler class can have only two-torsion.

Choose an effective Cartier divisor $E$ of degree at least $g$ on a smooth projective genus-$g$ algebraic curve. For a family $M$ of degree-$g-1$ line bundles, the evaluation map from ${\pi }_{\ast }M(E)$ to ${\pi }_{\ast }(M(E){|}_E)$ is between vector bundles of the same rank. Its determinant vanishes exactly when $H^0(C,M)$ is nonzero. Changing $E$ changes the presentation by cancelling an invertible determinant factor; the zero Cartier divisor is the theta divisor. At a point with $h^0(M)=h^1(M)=1$, the derivative is the cup-product map $H^1({\mathcal{O}}_C)\to \mathrm{Hom}(H^0(M),H^1(M))$. By Serre duality it is nonzero, since the product of nonzero sections of $M$ and $K_C\otimes M^{-1}$ is nonzero. Hence this Cartier divisor is generically reduced.