Fiber bundle

A fiber bundle is a map $\pi :E\to B$ locally isomorphic over each open set $U\subseteq B$ to the projection $U\times F\to U$ for a fixed fiber $F$.

A circle bundle is a fiber bundle with fiber $S^1$. Oriented circle bundles over a paracompact base are classified by their Euler class in $H^2(B;\mathbb{Z})$.

Every proper surjective submersion is a locally trivial smooth fiber bundle. In particular, the fibers in a proper smooth family are diffeomorphic.

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.

The tangent bundle of a smooth manifold $M$ is the vector bundle whose fiber over $p$ is the tangent space $T_{p}M$.

The projectivization $\mathbb{P}(E)\to B$ has fiber over $b$ equal to the space of one-dimensional linear subspaces of $E_b$.

The dual bundle $E^{\ast }$ has fibre $\mathrm{Hom}(E_x,\mathbb{F})$ over $x$. Its transition matrices are the inverse transposes of those of $E$.

For a real or complex line bundle $L$, the evaluation isomorphism $L^{\ast }\otimes L\cong \mathrm{End}(L)$ identifies the identity endomorphisms with a nowhere-zero global section. Consequently $L^{\ast }\otimes L$ is trivial.

For a smooth map $f:M\to B$ and a vector bundle $\pi :E\to B$, the pullback bundle isPulling back local trivializations of $E$ makes it a vector bundle over $M$ whose transition functions are the original transition functions composed with $f$.

The tensor product of vector bundles is formed fiberwise. Tensor products of local trivializations have transition functions given by tensor products of the original transition matrices.

A connection is a linear map $\nabla :\Gamma (E)\to {\Omega }^1(B;E)$ satisfying $\nabla (fs)=df\otimes s+f\nabla s$. In a local frame it has the form $d+A$ for a matrix-valued one-form $A$.

The covariant exterior derivative extends a connection to bundle-valued forms byLocally it is $d_A=d+A\wedge$.

At $v\in E$, the vertical subspace is $V_{v}E=\ker (d\pi {)}_v$. A horizontal subspace is a complement $H_{v}E$ to $V_{v}E$. A linear connection is equivalently a smooth choice of such complements compatible with the vector-space structure in the fibres.

If a connection on $E\to B$ has connection matrix $A=A_i(y)d{y}^i$ in a local frame and $f:M\to B$, its pullback connection has matrixIt is characterized by ${\nabla }_X^{f^{\ast }A}(f^{\ast }s)=f^{\ast }({\nabla }_{df(X)}^{A}s)$.

The curvature is $F(A)=d_A^2$ and in a local frame satisfies $F(A)=dA+A\wedge A$.

The curvature of a connection satisfies $d_{A}F(A)=0$. In a local frame this follows directly by expanding $d(dA+A\wedge A)+A\wedge F-F\wedge A$.

A connection on $E$ induces one on $E^{\ast }$ byIts connection matrices are the negatives of the transposes of those for $E$.

A connection on the tangent bundle is torsion-free when ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$. Its Christoffel symbols are symmetric in their two lower indices.

A horizontal lift of a base curve is a curve in the bundle projecting to it and tangent to the horizontal distribution. In a local frame its fiber coordinate solves a linear ordinary differential equation.

An $R$-orientation of a rank-$d$ vector bundle is a coherent choice of generator of $H^d(E_b,E_b\setminus \{0\};R)$ for every fiber, equivalently a Thom class with the corresponding fiberwise restriction.

A Thom class of an $R$-oriented rank-$d$ vector bundle is a class $u_E\in H^d(D(E),S(E);R)$ restricting to the chosen generator on every fiber pair $(D^d,S^{d-1})$.

For an $R$-oriented rank-$d$ vector bundle, multiplication by the Thom class gives isomorphisms

The Euler class of an oriented rank-$d$ vector bundle is the pullback $e(E)=s^{\ast }{u}_E\in H^d(B;R)$ of its Thom class along the zero section. Over ${\mathbb{F}}_2$ it equals the top Stiefel-Whitney class.

For a compact oriented smooth manifold $M$, the Euler class of its tangent bundle satisfiesEquivalently, the sum of the indices of the isolated zeros of a vector field equals the Euler characteristic.

The complex orientation turns a complex line bundle $L$ into an oriented real rank-two bundle, andMoreover $e(L^{\ast })=-e(L)$ and $e(L_1{\otimes }_{\mathbb{C}}{L}_2)=e(L_1)+e(L_2)$.

For the unit sphere bundle $S(E)\to B$ of an oriented rank-$d$ vector bundle, the Gysin sequence contains

Let $a,b$ be the standard basis of $H^2(S^2\times S^2;\mathbb{Z})$ and let $E\to S^2\times S^2$ be an oriented circle bundle with Euler class $pa+qb\ne 0$. If $d=\gcd (|p|,|q|)$, its Gysin sequence givesFor zero Euler class the bundle is trivial and its cohomology is that of $(S^2\times S^2)\times S^1$.

The tautological bundle over a projective space has as its fiber over a line precisely that line. The real tautological line bundle over ${\mathbb{RP}}^m$ has first Stiefel-Whitney class equal to the degree-one generator.