Vector bundle

A rank-$d$ vector bundle $\pi :E\to B$ is locally isomorphic over the base to the projection $U\times R^d\to U$, with linear transition maps on fibers.

On an $n$-dimensional smooth manifold, the density bundle $|{\Lambda }^n{T}^{\ast }M|$ is the real line bundle whose coordinate transformation uses the absolute value of the Jacobian determinant. A density can therefore be integrated without an orientation of a smooth manifold. A Riemannian metric gives the positive density $\sqrt{\det (g_{ij})}|d{x}^1\cdots d{x}^n|$, including on a nonorientable manifold.

// Target: fiber-bundle.bigb

A Riemannian metric on a smooth manifold gives a positive section of the density bundle, locally $d{\mu }_g=\sqrt{\det (g_{ij})}|d{x}^1\cdots d{x}^n|$. The absolute Jacobian determinant makes this independent of coordinates without requiring orientability. With an orientation of a smooth manifold, it corresponds to the Riemannian volume form.

For a smooth manifold $M$, the tensor bundle $T_s^{r}M=(TM{)}^{\otimes r}\otimes (T^{\ast }M{)}^{\otimes s}$ is a vector bundle whose smooth sections of a vector bundle are tensor fields of type $(r,s)$. Its transition functions of a vector bundle are tensor products of the tangent bundle transition matrices and their inverse transposes. Symmetric tensors and differential forms arise from the symmetric and alternating subbundles.

// Target: fiber-bundle.bigb

A G-structure on a real rank-$m$ vector bundle is a principal $G$ reduction of its frame bundle. Equivalently it has local frames whose transition matrices take values in $G$, with the reduction consisting of their right $G$-orbits. An orthogonal structure on a real vector bundle is equivalent to a fiber metric; a special orthogonal reduction is equivalent to that metric together with an orientation of a vector bundle.

A vector bundle is globally generated when its global sections span every fibre. The evaluation map in the display expresses this condition. On a projective curve its space of global sections is finite-dimensional, so a finite collection of sections suffices.

For a vector bundle of finite positive rank let $B$ be an upper degree bound for the line subbundles of $E^{\ast }\otimes K_C$. If $m-1>B$, no nonzero map ${\mathcal{O}}_C(m{p}_0-p)\to E^{\ast }\otimes K_C$ can exist for any point $p$: its saturated image would have degree at least $m-1$. Serre duality then gives $H^1(E(m{p}_0-p))=0$. The evaluation exact sequence shows that $H^0(E(m{p}_0))\to E(m{p}_0){|}_p$ is surjective at every point, proving the assertion with a uniform bound in $p$.

Let $F$ be globally generated of rank $r\ge 2$ on a projective curve, and put $W=H^0(C,F)$. The pairs $(p,s)$ with $s(p)=0$ form the total space of the kernel of the evaluation map over the curve. This incidence variety has dimension $\dim W+1-r<\dim W$. Its image in $W$ is closed because the curve is proper, and is a proper subset by the dimension bound. A section outside it vanishes nowhere and gives a line subbundle ${\mathcal{O}}_C\subset F$ with locally free quotient.

The fiberwise direct sum of two vector bundles over the same base. It is formed from the fiber product of their total spaces, with the direct-sum linear structure in each fiber. Transition matrices are block diagonal.

// Target: geometry-and-topology.bigb

A symplectic vector bundle has a smooth fiberwise symplectic form, making each fiber a symplectic vector space. Its structure group reduces to the symplectic group. A symplectic normal bundle is an important example.

A rank-$n$ vector subbundle of a rank-$2n$ symplectic vector bundle is Lagrangian when each fibre is a Lagrangian subspace. If it exists, a compatible almost complex structure supplies a complementary Lagrangian subbundle by applying $J$. Existence is a global condition: the area-oriented tangent bundle of the two-sphere has no real line subbundle, by triviality of real line bundles over the sphere and the Hairy ball theorem.

For a real vector bundle over a compact base, its Thom space is the quotient of its disk bundle by its sphere bundle, with the collapsed sphere bundle as basepoint. Equivalently it is the one-point compactification of the total space. An oriented bundle's Thom class and Thom isomorphism identify the reduced cohomology of this space with shifted cohomology of the base.

Decompose the ambient vector space as $V\oplus W$, of dimensions $n+1,m+1$. A projective line outside $\mathbb{P}(V)$ is the graph of a unique linear map from a line in $W$ to $V$. Thus the complement is the total space of ${\gamma }^{\ast }\otimes V$, which is complex-isomorphic to ${\bar{\gamma }}^{\oplus (n+1)}$ after choosing a Hermitian metric. Collapsing $\mathbb{P}(V)$ gives its one-point compactification, hence the Thom space because the base is compact.

// Target: geometry-and-topology.bigb

A line subbundle is an injective map from a line bundle into a vector bundle whose quotient is locally free. Equivalently the injection splits locally. On a smooth algebraic curve, saturation of a rank-one coherent subsheaf produces a saturated line subbundle on a smooth curve.

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.

A framing is a continuous ordered basis in every fiber, equivalently a trivialization of the vector bundle. A stable framing trivializes the bundle after adding trivial real summands, up to further stabilization and homotopy. For a smooth map it is the stable normal data that is framed, not necessarily the tangent bundle of either manifold.

The vector bundle whose fibre at $p$ consists of linear maps $E_p\to E_p$. If a local frame changes by $e^{\prime }=eg$, the coefficient matrix of an endomorphism changes by $T^{\prime }=g^{-1}Tg$, giving smooth fibre-linear transitions. For a line bundle this bundle is canonically the trivial scalar bundle, because every fibre endomorphism is scalar multiplication. A connection on a vector bundle induces the endomorphism bundle connection.

For a rank-$k$ smooth vector bundle, take the exterior power ${\Lambda }^r{E}_p$ in each fibre. A local frame gives a trivialization with fibre ${\Lambda }^r{\mathbb{R}}^k$, and a transition matrix $A$ acts by ${\Lambda }^{r}A$. Functoriality preserves the cocycle identities, producing a smooth vector bundle of rank $\left(\genfrac{}{}{0ex}{}{k}{r}\right)$, or the zero bundle when $r>k$. Its top exterior power is the determinant line 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.

For an oriented rank-$k$ bundle over $S^n$, with $k\ge 2$, normalize its clutching function to be based at the identity. A nowhere-zero section exists precisely when the displayed homotopy class vanishes. If it vanishes, extend $\gamma e_k$ across one hemisphere and use the constant section $e_k$ on the other. Conversely normalize any nowhere-zero section using the fiber metric. Its two hemisphere maps satisfy ${\sigma }_{-}=\gamma {\sigma }_{+}$ on the boundary. The first hemisphere makes ${\sigma }_{+}$ homotopic to a constant; the second makes $\gamma {\sigma }_{+}$ null-homotopic. Thus $\gamma e_k$ is null-homotopic. If $k>n$, the lower homotopy group of the sphere vanishes and a section always exists. Rank one is already trivial because $SO(1)$ is trivial.

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.

The top exterior power of a rank-$r$ vector bundle is its determinant line bundle. A short exact sequence of vector bundles gives a canonical isomorphism of the middle determinant with the tensor product of the other two. Locally, a subbundle frame followed by lifts of a quotient frame defines it; changing lifts only adds off-diagonal blocks. This is the determinant step in adjunction for a smooth submanifold.

The zero section sends each base point to the zero vector in its fiber. It is a smooth section of a vector bundle and an embedded copy of the base. The Liouville one-form vanishes as an ambient covector precisely along the zero section of a cotangent bundle.

A fiber metric on a real smooth vector bundle is a positive-definite inner product on each fiber that varies smoothly in vector bundle trivializations. A smooth partition of unity subordinate to trivializing charts averages their Euclidean inner products. Nonnegative weights preserve positivity and local finiteness ensures smoothness. Hence every real vector bundle over a Hausdorff second-countable smooth manifold admits a fiber metric. Applied to the tangent bundle, this is existence of a Riemannian metric.

An orthogonal structure is a reduction of the transition functions of a real rank-$r$ vector bundle to the orthogonal group. Its orthonormal frames define a fiber metric, independent of the chosen frame because orthogonal transitions preserve the inner product. Conversely a fiber metric gives smooth orthonormal frames by the Gram-Schmidt process, with orthogonal transition matrices. This equivalence does not require the vector bundle to be globally trivial.

This is a frame of a real vector bundle with a positive-definite fiber metric for which the basis vectors are orthonormal. The Gram-Schmidt process applied to any smooth local frame produces such frames smoothly. They are distinguished from Lorentzian orthonormal frames, whose metric has mixed signature.

A vector bundle trivialization is orthogonal when its coordinate frame is orthonormal for the fiber metric, so each fiber is identified isometrically with Euclidean space. Overlap changes then take values in the orthogonal group. Local orthogonal trivializations exist even when no global frame does.

For a smooth complex vector bundle endomorphism $A$ and a smooth eigenvalue function $\lambda$, a kernel of constant rank is a smooth vector subbundle, called its eigenbundle. Without a constant-rank hypothesis these kernels need not form a vector bundle.

A Hermitian vector bundle is a complex vector bundle with a smoothly varying positive definite Hermitian form on each fibre. When the bundle is holomorphic, this is a Hermitian metric on a holomorphic vector bundle. Orthogonal projection to a smooth subbundle produces its induced metric and the quotient Hermitian metric.

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.

This is a smoothly varying positive-definite Hermitian form on each fiber, without requiring a holomorphic base or bundle. A partition of unity glues local positive metrics into a global one. Applying smooth Gram-Schmidt to local frames then makes the transitions unitary, reducing the structure group of a vector bundle to $U(k)$. For a line bundle its unit vectors form a principal bundle for the circle group.

For an even-rank real vector bundle $\xi$, a stable complex structure is a complex structure on $\xi \oplus {\mathbb{R}}^{2s}$, considered up to adding trivial complex bundles and homotopy. This gives a stable complex class but does not necessarily make $\xi$ itself a complex vector bundle. Stable Chern classes therefore need not recover the unstable Euler class: $T{S}^4\oplus {\mathbb{R}}^2$ can be made trivial complex, while $e(T{S}^4)$ evaluates to $2$.

The conjugate bundle has the same underlying real bundle and the opposite complex scalar action. Its transitions are the complex conjugates of the original transitions. For a holomorphic bundle on $X$, these transitions are smooth and generally antiholomorphic on $X$; it is not automatically a holomorphic bundle there. It is naturally holomorphic over the conjugate complex manifold instead.

A complex bundle connection induces a connection on its conjugate vector bundle by the displayed rule on real tangent fields. The conjugate scalar action gives the correct Leibniz rule. In conjugate frames the matrix is the complex conjugate of the original connection matrix, with evaluation of one-forms on real tangent vectors understood.

The rank of a vector bundle at a point is the dimension of its fiber as a vector space. A vector bundle trivialization identifies nearby fibers with the same vector space, so this rank is locally constant. A rank-$r$ bundle has fibers of dimension $r$ everywhere.

A morphism of vector bundles over one base is a smooth map between their total spaces that commutes with projection to the base and is linear on each fiber. In local vector bundle trivializations it is multiplication by a smoothly varying matrix.

A fiberwise surjective morphism of constant-rank vector bundles has a kernel vector subbundle. For a morphism from a trivial rank-$N$ bundle, a local full-row-rank matrix $F(x)$ gives the continuous kernel projection $I-F(x{)}^T(F(x)F(x{)}^T{)}^{-1}F(x)$. Changing the local frame in the target leaves this projection unchanged. Thus its kernel defines a continuous map into the appropriate Grassmannian.

A smooth vector bundle morphism from a vector bundle $E$ to itself, covering the identity on the base, is a vector bundle endomorphism. It is fibrewise linear and is equivalently a smooth section of $\mathrm{End}E$. In a local frame its matrix is smooth and changes by conjugation under changes of frame. An almost complex structure is a real endomorphism of the tangent bundle satisfying $J^2=-I$.

A holomorphic bundle map is a holomorphic map between total spaces of holomorphic vector bundles that covers a holomorphic base map and is complex-linear on each fibre. Over a fixed base, its matrix in holomorphic local frames has holomorphic entries.

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.

A vector bundle morphism is an isomorphism if it admits an inverse of the same kind. A smooth fiberwise bijective bundle morphism automatically has a smooth inverse, because inverse matrices in vector bundle trivializations depend smoothly on their entries wherever their determinants are nonzero.

A local trivialization of a rank-$r$ vector bundle $\pi :E\to M$ is a bundle isomorphism ${\pi }^{-1}(U)\cong U\times {\mathbb{R}}^r$ over an open set $U$, linear on each fiber. Overlapping trivializations differ by $(p,v)\mapsto (p,g(p)v)$ for a smooth $G{L}_r(\mathbb{R})$-valued function $g$. A tangent bundle chart uses the derivative of its base manifold chart for this fiber identification.

A frame over an open set $U$ is a list of smooth sections $e_1,\dots ,e_r$ of a rank-$r$ vector bundle whose values form a basis in every fibre over $U$. It is equivalent to a vector bundle trivialization by $(x,v)\mapsto {\sum }_a{e}_a(x)v^a$. Two local frames differ by a smooth map $g:U\to \mathrm{GL}(r)$, written $e^{\prime }=eg$.

For a vector bundle carrying a smooth nondegenerate symmetric bilinear form of constant metric signature, a pseudo-orthonormal frame is a smooth local frame of a vector bundle whose Gram matrix is the constant diagonal signature matrix $\eta$. It identifies the metric bundle locally with a fixed model ${\mathbb{R}}^{p,q}$. Two such frames differ by a smooth map into the corresponding orthogonal group. Positive-definite signature is included as the ordinary orthonormal case. A global pseudo-orthonormal frame is a section of a fiber bundle of the orthonormal frame bundle and trivializes that principal bundle.

A coframe dual to a local frame of the tangent bundle is the corresponding local frame of the cotangent bundle. Its elements are differential one-forms. Applying the coframe to covariant derivatives of the frame extracts the connection one-forms; applying it to the torsion tensor extracts the torsion forms.

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.

The projective bundle of a real or complex vector bundle $E\to X$ has fiber the space of one-dimensional subspaces of $E_x$. It carries a tautological bundle $\lambda \subset p^{\ast }E$; in the real case this is the projectivization of a real vector bundle.

The same construction applies to an algebraic vector bundle over an algebraic variety: local trivializations glue products with projective space. In the lines convention its universal tautological bundle is a line bundle; the alternative projectivization by quotients convention parametrizes one-dimensional quotient spaces. Both conventions give dimension $\dim X+\mathrm{rank}(E)-1$ and a proper morphism to the base.

This convention for a projective bundle parametrizes one-dimensional quotients of the fibres of a vector bundle $E$. Its universal quotient is ${\pi }^{\ast }E\twoheadrightarrow {\mathcal{O}}_{\mathbb{P}(E)}(1)$, and ${\pi }_{\ast }\mathcal{O}(1)=E$. Tensoring $E$ by a line bundle on the base leaves the projective bundle unchanged but tensors its universal quotient by the pullback of that line bundle. The alternative convention of holomorphic projectivization by lines replaces $E$ by its dual and reverses the tautological sign.

On the projectivization by quotients, the fibre over a one-dimensional quotient of $E_x$ is that quotient line itself. These lines glue to the universal quotient. Its restriction to a projective fibre has degree one. It is the dual of the relative tautological line after identifying quotient projectivization of $E$ with line projectivization of $E^{\ast }$.

For this smooth projective bundle, put $L={\pi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(1)$ and $M={\mathcal{O}}_{\mathbb{F}}(1)$, using projectivization by quotients. If every $a_i>0$, the complete system $|M|$ embeds it as a rational normal scroll. If every $a_i\ge 0$, the same system defines a scroll image which may be singular. Allowing arbitrary integers gives an abstract scroll; a common twist makes an embedding available. Its intersection numbers are $L^2=0$, $L{M}^{n-1}=1$, $M^n={\sum }_i{a}_i$, and its canonical divisor is $-nM+({\sum }_i{a}_i-2)L$.

The divisor class of a fibre of the ruling of a rational normal scroll has square zero. Its complete pencil recovers the projection to ${\mathbb{P}}^1$, and its intersection with $M^{n-1}$ is one.

Two rank-$n$ abstract rational normal scrolls are isomorphic exactly when their splitting degrees agree as multisets after a common integer shift. For $n\ge 3$, the primitive nef class $L$ of the ruling is intrinsically identified by $L^2=0$: restriction to a projective fibre forces every other divisor class with square zero to have zero coefficient of $M$. An isomorphism therefore preserves the ruling, and its effect on $M$ is a twist by $cL$. Pushforward gives an isomorphism of the defining bundles up to this twist; uniqueness in the Birkhoff–Grothendieck theorem recovers their splitting degrees. For surfaces the unique nef square-zero ray identifies the ruling except on ${\mathbb{P}}^1\times {\mathbb{P}}^1$, where exchanging the two rulings changes no normalized splitting type.

For the line-projectivization of a rank-$n$ complex bundle, $p^{\ast }E=S\oplus S^{\perp }$. The Whitney sum formula for Chern classes gives $c_t(S^{\perp })=p^{\ast }{c}_t(E)/(1+xt)$ as a formal power series. Comparing coefficients proves the formula. Since $S^{\perp }$ has rank $n-1$, its degree-$n$ Chern class vanishes and yields the monic projective-bundle relation.

// Target: geometry-and-topology.bigb

For a holomorphic vector bundle, projectivize each fibre using lines. The transition $g_{ij}(x)$ acts holomorphically by $[v]\mapsto [g_{ij}(x)v]$. Thus local products $U_i\times {\mathbb{P}}^{r-1}$ define a complex manifold with holomorphic projection to the base. The lines convention gives the relative tautological line bundle of fibre degree $-1$, and its dual has degree $+1$.

On the holomorphic projectivization by lines, take the line itself as the fibre of $S$. The evaluation inclusion makes it a holomorphic line subbundle of $p^{\ast }E$. Its fibre restriction is $\mathcal{O}(-1)$; the dual is the relative hyperplane line bundle.

Dualize the relative tautological line bundle to obtain the positive relative hyperplane line bundle. It restricts to $\mathcal{O}(1)$ on each projective fibre. This sign depends on the convention that projectivization parametrizes lines; fixing the convention avoids confusing it with the tautological line itself.

A complex line subbundle $L\subset E$ defines a section of a fiber bundle $s:X\to \mathbb{P}(E)$ selecting $L_x$ at each point. Pullback of the tautological bundle along $s$ is $L$, so if $t$ is its Euler class, then $s^{\ast }t=e(L)$. For a decomposition $E={⨁}_i{L}_i$, the resulting sections and the open sets where coordinate projection is nonzero identify the tautological line with ${\pi }^{\ast }{L}_i$. These identifications support the projective-bundle factorization via a vanishing cup product from an open cover.

For $n\ge 2$, ordered orthogonal complex lines in ${\mathbb{C}}^n$ form a projective bundle over ${\mathbb{CP}}^{n-1}$, with fiber ${\mathbb{CP}}^{n-2}$. Here $L$ is the first tautological line and the complement is formed using the standard Hermitian inner product. Its real dimension is $4n-6$. It is also the partial complex flag space of a line contained in a two-plane: send an orthogonal pair to $({\ell }_1,{\ell }_1\oplus {\ell }_2)$, and recover the second line by orthogonal complement inside the two-plane.

For the orthogonal complex line flag manifold, take $x,y$ to be the first Chern classes of the duals of its two tautological lines, both in degree two. The Whitney sum formula for Chern classes gives $c_i(L^{\perp })=x^i$. The projective bundle definition of Chern classes gives the second relation, and Leray-Hirsch theorem gives the integral basis $x^i{y}^j$ with $0\le i<n$, $0\le j<n-1$. Polynomial division by the monic relation in $y$ proves that there are no further relations. Multiplying that relation by $y-x$ also gives $y^n=0$.

For a rank-$r$ complex vector bundle and $u=c_1({\lambda }^{\ast })$, the cohomology of its projective bundle is free over the cohomology of the base on $1,u,\dots ,u^{r-1}$. Consequently pullback is injective, which yields the splitting principle for complex vector bundles by iteration.

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.

On a smooth manifold, the sequence of sheaves of abelian groupsis exact. The additive smooth-function sheaf is a fine sheaf, so the connecting map identifies $H^1(X,{\mathcal{C}}_{\mathbb{C}}^{\infty ,\ast })$ with $H^2(X,\mathbb{Z})$. The former classifies smooth complex line bundles and the connecting class is their First Chern class.

Every smooth complex line bundle on ${\mathbb{CP}}^1$ is isomorphic to exactly one twisting bundle $\mathcal{O}(k)$. Trivializing on the two standard affine charts leaves a transition function on ${\mathbb{C}}^{\ast }$; its winding number determines $k$, and a partition of unity removes the zero-winding factor.

A real line bundle is a rank-one real vector bundle. It is orientable exactly when it is trivial.

The Möbius line bundle over the circle iswith projection to $t$ modulo $2\pi$. It is a smooth rank-one vector bundle. A section is represented by a function $s$ satisfying $s(t+2\pi )=-s(t)$, so the intermediate value theorem forces every continuous section to vanish somewhere. Thus it has no nowhere-zero section and is not isomorphic to the trivial vector bundle of rank one.

Real line bundles over a paracompact space are classified up to isomorphism by their first Stiefel–Whitney class in $H^1(X;{\mathbb{F}}_2)$. Tensor product corresponds to addition of classes.

For a real line bundle, the mod-two Euler class is its top Stiefel–Whitney class, namely $w_1(L)\in H^1(X;{\mathbb{F}}_2)$. Its evaluation on a closed curve records whether parallel transport reverses the fiber orientation.

A vector subbundle $F\subseteq E$ is a subset whose fibers are vector subspaces $F_x\subseteq E_x$ of constant dimension and which admits local frames extending to local frames of $E$.

A fiber metric on a smooth vector bundle $E$ gives a smooth orthogonal projection $P:E\to F$ onto any vector subbundle. In a local frame of $E$, let $G$ be the metric matrix and $A$ the columns of a frame of $F$. Then $P=A(A^{t}GA{)}^{-1}{A}^{t}G$, proving smoothness. The complementary subbundle $F^{\perp }=\ker P$ maps isomorphically onto the quotient vector bundle. Explicitly $v\mapsto (Pv,[v])$ is a vector bundle isomorphism $E\to F\oplus (E/F)$, with inverse $(f,[v])\mapsto f+(I-P)v$. This inverse is independent of the representative $v$, but the splitting depends on the chosen metric.

For a smooth vector subbundle $F\subset E$, the quotient has fibers $(E/F{)}_p=E_p/F_p$. Extend a local frame of a vector bundle of $F$ to one of $E$; the classes of the complementary frame vectors trivialize the quotient. Adapted transition functions of a vector bundle are block upper triangular, and their lower-right blocks give the quotient transitions. These are smooth and satisfy the cocycle identity, defining a smooth vector bundle. The projection $E\to E/F$ is a smooth vector bundle morphism.

Over ${\mathrm{Gr}}_k({\mathbb{R}}^N)$, quotient the trivial bundle by the tautological bundle. The quotient is a vector bundle because it identifies continuously with the orthogonal-complement bundle via $[v]\mapsto P_{W^{\perp }}v$. Its rank is $N-k$.

Over ${\mathrm{Gr}}_1({\mathbb{R}}^2)={\mathbb{RP}}^1$, the universal quotient is the orthogonal-complement line bundle, a Möbius line bundle. Writing a section as $a(\theta )(-\sin \theta ,\cos \theta )$ forces $a(\theta +\pi )=-a(\theta )$. Continuity then forces a zero, so no nowhere-zero section exists.

On a compact base, finitely many global sections can be selected to span every fiber when evaluations of global sections are surjective. They give a surjective morphism $M\times {\mathbb{R}}^N\to E$. Its kernel is a kernel bundle of a surjective vector bundle morphism, and its fibers give the displayed map $\varphi$. The first isomorphism theorem on each fiber identifies the resulting pullback quotient with $E$, with continuously varying inverses. The finite selection follows by choosing a frame's values at each point, extending each value to a global section, then taking a finite cover of the regions where those sections remain independent.

A rank-$d$ vector bundle is trivial when it is isomorphic over its base $B$ to the product bundle $B\times {\mathbb{R}}^d\to B$. Equivalently, it admits a global frame of $d$ pointwise linearly independent sections.

This is a section of a fiber bundle whose fibers are vector spaces.

A section of a vector bundle $\pi :E\to B$ is a map $s:B\to E$ satisfying $\pi \circ s={\mathrm{id}}_B$.

The zero locus of a holomorphic section of a vector bundle of a rank-$r$ bundle is regular when its derivative has rank $r$ along the locus. Its kernel is then the tangent bundle of the smooth zero set. Smoothness of the underlying set alone is insufficient: the square of a reduced defining function has the same set of zeros but zero derivative there.

The smooth sections of a vector bundle form a module over the commutative ring $C^{\infty }(M)$ by pointwise multiplication. A vector bundle morphism over the identity induces a $C^{\infty }(M)$-linear map between these modules. Bundle morphisms from maps of smooth sections gives the converse, without assuming the bundle has a global frame.

A nowhere-zero section of a vector bundle is a section $s$ with $s(x)\ne 0$ in every fiber. Every trivial vector bundle of positive rank has one.

Identify ${\mathbb{R}}^{2m+2}$ with ${\mathbb{C}}^{m+1}$ and let $J$ be multiplication by $i$. On the unit sphere $S^{2m+1}$, the vector field $V(x)=Jx$ has norm one and satisfies $⟨x,Jx⟩=0$. It is therefore a continuous tangent nowhere-zero section, including on $S^1$.

A unit nowhere-zero section $\sigma$ of a real vector bundle with a fiber metric defines a rank-one trivial vector bundle. Its fiberwise orthogonal complement is a subbundle: locally complete $\sigma$ to an orthonormal frame. The vector bundle isomorphism $(v,t)\mapsto v+t\sigma$ has inverse $w\mapsto (w-⟨w,\sigma ⟩\sigma ,⟨w,\sigma ⟩)$. For an oriented bundle, orient the complement so that its positive frame followed by $\sigma$ is positive.

The hairy ball theorem says that every continuous tangent vector field on an even-dimensional sphere vanishes somewhere. Equivalently, the tangent bundle of an even-dimensional sphere has no nowhere-zero section.

A nowhere-zero tangent vector field on $S^n$ can be normalized to a unit field $w(x)$ perpendicular to $x$. Then $\cos (\pi t)x+\sin (\pi t)w(x)$ is a homotopy from the identity to the antipodal map. Their mapping degrees are $1$ and $(-1{)}^{n+1}$, respectively, contradicting homotopy invariance of mapping degree when $n$ is positive and even. For $S^0$ the tangent spaces are zero, so every tangent field is zero directly.

After choosing a fiber metric on a rank-$d$ vector bundle $E\to B$, its sphere bundle consists of the unit vectors and has fiber $S^{d-1}$. Different choices of metric give isomorphic sphere bundles.

For an oriented rank-four real vector bundle $E\to S^4$, its sphere bundle has fibre $S^3$. Let $m=⟨e(E),[S^4]⟩$ be its Euler class evaluated on the orientation class. The Gysin sequence gives integral homology $\mathbb{Z}$ in degrees $0,7$, $\mathbb{Z}/m\mathbb{Z}$ in degree $3$, and a copy of $\mathbb{Z}$ in degree $4$ exactly when $m=0$, with all other groups zero. Here $\mathbb{Z}/0\mathbb{Z}=\mathbb{Z}$. A transition function $v\mapsto q^{k}v{q}^l$ on the unit quaternions has Euler number $k+l$, up to an overall orientation sign: evaluating it at a fixed unit vector gives the power map $q\mapsto q^{k+l}$, whose mapping degree is $k+l$. The same homology calculation follows by applying the Mayer–Vietoris sequence to the two trivializations over the hemispheres.

The unit tangent bundle of a Riemannian manifold $M$ is the sphere bundle of its tangent bundle. Its points are pairs $(x,v)$ with $x\in M$, $v\in T_{x}M$, and $\Vert v\Vert =1$.

On an oriented Riemannian surface, set $\alpha (\xi )=⟨v,d\pi \xi ⟩$, $\theta (\xi )=⟨iv,d\pi \xi ⟩$, and let $\omega$ be the Levi-Civita connection form with $\omega (V)=1$. The dual frame is $(X,H,V)$ andIn a local oriented frame, $\omega =d\vartheta +{\omega }_0$ and $\alpha ,\theta$ are the base coframe rotated through angle $\vartheta$; the torsion-free and curvature equations give these identities. Evaluating exterior derivatives on dual fields yields $[V,X]=H$, $[V,H]=-X$, $[X,H]=KV$.