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.

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 $d_A=d+A\wedge$.

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

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.