Definitions

# Connection (vector bundle)

In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear. Such a connection is equivalently specified by a covariant derivative, which is an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Connections in this sense generalize, to arbitrary vector bundles, the concept of a linear connection on the tangent bundle of a smooth manifold, and are sometimes known as linear connections.

Connections on vector bundles are also sometimes called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them .

## Formal definition

Let EM be a smooth vector bundle over a smooth manifold M. Denote the space of smooth sections of E by Γ(E). A connection on E is an R-linear map

$nabla : Gamma\left(E\right) to Gamma\left(Eotimes T^*M\right)$
such that the Leibniz rule
$nabla\left(sigma f\right) = \left(nablasigma\right)f + sigmaotimes df$
holds for all smooth functions f on M and all smooth sections σ of E.

If X is a tangent vector field on M (i.e. a section of the tangent bundle TM) one can define a covariant derivative along X

$nabla_X : Gamma\left(E\right) to Gamma\left(E\right)$
by contracting X with the resulting covariant index in the connection ∇ (i.e. ∇Xσ = (∇σ)(X)). The covariant derivative satisfies the following properties:
begin\left\{align\right\}&nabla_X\left(sigma_1 + sigma_2\right) = nabla_Xsigma_1 + nabla_Xsigma_2
&nabla_{X_1 + X_2}sigma = nabla_{X_1}sigma + nabla_{X_2}sigma &nabla_{X}(fsigma) = fnabla_Xsigma + X(f)sigma &nabla_{fX}sigma = fnabla_Xsigmaend{align} Conversely, any operator satisfying the above properties defines a connection on E and a connection in this sense is also known as a covariant derivative on E.

## Vector-valued forms

Let EM be a vector bundle. A E-valued differential form of degree r is a section of the tensor product bundle E⊗ΛrT*M. The space of such forms is denoted by

$Omega^r\left(E\right) = Gamma\left(EotimesLambda^rT^*M\right).$
An E-valued 0-form is just a section of the bundle E. That is,
$Omega^0\left(E\right) = Gamma\left(E\right).,$

In this notation a connection on EM is a linear map

$nabla:Omega^0\left(E\right) to Omega^1\left(E\right).$
A connection may then be viewed as a generalization of the exterior derivative to vector bundle valued forms. In fact, given a connection ∇ on E there is a unique way to extend ∇ to a covariant exterior derivative or exterior covariant derivative
$d^\left\{nabla\right\}: Omega^r\left(E\right) to Omega^\left\{r+1\right\}\left(E\right).$
Unlike the ordinary exterior derivative one need not have (d)2 = 0. In fact, (d)2 is directly related to the curvature of the connection ∇ (see below).

## Affine properties

Every vector bundle admits a connection. However, connections are not unique. If ∇1 and ∇2 are two connections on EM then their difference is a C-linear operator. That is,

$\left(nabla_1 - nabla_2\right)\left(fsigma\right) = f\left(nabla_1sigma - nabla_2sigma\right)$
for all smooth functions f on M and all smooth sections σ of E. It follows that the difference ∇1 − ∇2 is induced by a one-form on M with values in the endomorphism bundle End(E) = EE*:
$\left(nabla_1 - nabla_2\right) in Omega^1\left(mathrm\left\{End\right\},E\right).$
Conversely, if ∇ is a connection on E and A is a one-form on M with values in End(E), then ∇+A is a connection on E.

In other words, the space of connections on E is an affine space for Ω1(End E).

## Relation to principal and Ehresmann connections

Let EM be a vector bundle of rank k and let F(E) be the principal frame bundle of E. Then a (principal) connection on F(E) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant maps F(E) → Rk. (This can be seen by considering the pullback of E over F(E) → M, which is isomorphic to the trivial bundle F(E) × Rk.) Given a section σ of E let the corresponding equivariant map be ψ(σ). The covariant derivative on E is then given by

$psi\left(nabla_Xsigma\right) = X^H\left(psi\left(sigma\right)\right)$
where XH is the horizontal lift of X (recall that the horizontal lift is determined by the connection on F(E)).

Conversely, a connection on E determines a connection on F(E), and these two constructions are mutually inverse.

A connection on E is also determined equivalently by a linear Ehresmann connection on E. This provides one method to construct the associated principal connection.

## Local expression

Let EM be a vector bundle of rank k, and let U be an open subset of M over which E is trivial. Given a local smooth frame (e1, …,ek) of E over U, any section σ of E can be written as $sigma = sigma^alpha e_alpha$ (Einstein notation assumed). A connection on E restricted to U then takes the form

$nablasigma = \left(mathrm dsigma^alpha + \left\{omega^alpha\right\}_beta sigma^beta\right)e_\left\{alpha\right\}$
where
$\left\{omega^alpha\right\}_beta,e_alpha = nabla e_beta.$
Here ωαβ defines a k × k matrix of one-forms on U. In fact, given any such matrix the above expression defines a connection on E restricted to U. This is because ωαβ determines a one-form ω with values in End(E) and this expression defines ∇ to be the connection d+ω, where d is the trivial connection on E over U defined by differentiating the components of a section using the local frame. In this context ω is sometimes called the connection form of ∇ with respect to the local frame.

If U is a coordinate neighborhood with coordinates (xi) then we can write

$\left\{omega^alpha\right\}_beta = \left\{\left\{omega_i\right\}^alpha\right\}_beta,mathrm dx^i.$
Note the mixture of coordinate and fiber indices in this expression. The coefficient functions ωiαβ are tensorial in the index i (they define a one-form) but not in the indices α and β. The transformation law for the fiber indices is more complicated. Let (f1, …,fk) be another smooth local frame over U and let the change of coordinate matrix be denoted t (i.e. fα = eβtβα). The connection matrix with respect to frame (fα) is then given by the matrix expression

$varpi = t^\left\{-1\right\}omega t + t^\left\{-1\right\}mathrm dt.$

Here dt is the matrix of one-forms obtained by taking the exterior derivative of the components of t.

The covariant derivative in the local coordinates and with respect to the local frame field (eα) is given by the expression

$nabla_Xsigma = X^i\left(partial_i,sigma^alpha + \left\{\left\{omega_i\right\}^alpha\right\}_betasigma^beta\right)e_\left\{alpha\right\}.$

## Parallel transport and holonomy

A connection ∇ on a vector bundle EM defines a notion of parallel transport on E along a curve in M. Let γ : [0, 1] → M be a smooth path in M. A section σ of E along γ is said to be parallel if

$nabla_\left\{dotgamma\left(t\right)\right\}sigma = 0$
for all t ∈ [0, 1]. More formally, one can consider the pullback γ*E of E by γ. This is a vector bundle over [0, 1] with fiber Eγ(t) over t ∈ [0, 1]. The connection ∇ on E pulls back to a connection on γ*E. A section σ of γ*E is parallel if and only if γ*∇(σ) = 0.

Suppose γ is a path from x to y in M. The above equation defining parallel sections is a first-order ordinary differential equation and so has a unique solution for each possible initial condition. That is, for each vector v in Ex there exists a unique parallel section σ of γ*E with σ(0) = v. Define a parallel transport map

$tau_gamma : E_x to E_y,$
by τγ(v) = σ(1). It can be shown that τγ is a linear isomorphism.

Parallel transport can be used to define the holonomy group of the connection ∇ based at a point x in M. This is the subgroup of GL(Ex) consisting of all parallel transport maps coming from loops based at x:

$mathrm\left\{Hol\right\}_x = \left\{tau_gamma : gamma text\left\{ is a loop based at \right\} x\right\}.,$
The holonomy group of a connection is intimately related to the curvature of the connection.

## Curvature

The curvature of a connection ∇ on EM is a 2-form F on M with values in the endomorphism bundle End(E) = EE*. That is,

$F^nabla in Omega^2\left(mathrm\left\{End\right\},E\right) = Gamma\left(mathrm\left\{End\right\},EotimesLambda^2T^*M\right).$
It is defined by the expression
$F^nabla\left(X,Y\right)\left(s\right) = nabla_Xnabla_Y s- nabla_Ynabla_X s- nabla_\left\{\left[X,Y\right]\right\}s$
where X and Y are tangent vector fields on M and s is a section of E. One must check that F is C-linear in both X and Y and that it does in fact define a bundle endomorphism of E.

As mentioned above, the covariant exterior derivative d need not square to zero when acting on E-valued forms. The operator (d)2 is, however, strictly tensorial (i.e. C-linear). This implies that it is induced from a 2-form with values in End(E). This 2-form is precisely the curvature form given above. For an E-valued form σ we have

$\left(d^nabla\right)^2sigma = F^nablawedgesigma.$

A flat connection is one whose curvature form vanishes identically.

## References

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