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. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

Linear connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them (Koszul 1950).

This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).

Formal definition

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

$\nabla :\Gamma (E)\to \Gamma (E\otimes T^{*}M)$ such that the Leibniz rule

$\nabla (\sigma f)=(\nabla \sigma )f+\sigma \otimes df$ 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

$\nabla _{X}:\Gamma (E)\to \Gamma (E)$ by contracting X with the resulting covariant index in the connection: ∇X σ = (∇σ)(X). The covariant derivative satisfies:

{\begin{aligned}&\nabla \!_{X}(\sigma _{1}{+}\sigma _{2})\ =\ \nabla \!_{X}\sigma _{1}+\nabla \!_{X}\sigma _{2}\\&\nabla \!_{(X_{1}{+}X_{2})}\sigma \ =\ \nabla \!_{X_{1}}\sigma +\nabla \!_{X_{2}}\sigma \\&\nabla \!_{X}(f\sigma )\ =\ f\,\nabla \!_{X}\sigma +X(f)\sigma \\&\nabla \!_{fX}\sigma \ =\ f\,\nabla \!_{X}\sigma .\end{aligned}} 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. An E-valued differential form of degree r is a section of the tensor product bundle:

$E\otimes \bigwedge ^{r}T^{*}M.$ The space of such forms is denoted by

$\Omega ^{r}(E)=\Omega ^{r}(M;E)=\Gamma \left(E\otimes \bigwedge ^{r}T^{*}M\right).$ An E-valued 0-form is just a section of the bundle E. That is,

$\Omega ^{0}(E)=\Gamma (E).$ In this notation a connection on EM is a linear map

$\nabla :\Omega ^{0}(E)\to \Omega ^{1}(E).$ 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

$d^{\nabla }:\Omega ^{r}(E)\to \Omega ^{r+1}(E).$ Unlike the ordinary exterior derivative, one generally has (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,

$(\nabla _{1}-\nabla _{2})(f\sigma )=f(\nabla _{1}\sigma -\nabla _{2}\sigma )$ 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*:

$(\nabla _{1}-\nabla _{2})\in \Omega ^{1}(M;\mathrm {End} \,E).$ 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

$\psi (\nabla _{X}\sigma )=X^{H}(\psi (\sigma ))$ where XH is the horizontal lift of X from M to F(E). (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

$\nabla \sigma =\left(\mathrm {d} \sigma ^{\alpha }+{\omega ^{\alpha }}_{\beta }\sigma ^{\beta }\right)e_{\alpha },$ where

${\omega ^{\alpha }}_{\beta }e_{\alpha }=\nabla e_{\beta }.$ Here ${\omega ^{\alpha }}_{\beta }$ 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 ${\omega ^{\alpha }}_{\beta }$ 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

${\omega ^{\alpha }}_{\beta }={{\omega _{i}}^{\alpha }}_{\beta }\,\mathrm {d} x^{i}.$ Note the mixture of coordinate indices (i) and fiber indices (α,β) in this expression. The coefficient functions ${{\omega _{i}}^{\alpha }}_{\beta }$ 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.:

$f_{\alpha }\ =\ e_{\beta }\,{t^{\beta }}_{\alpha }.$ The connection matrix with respect to frame (fα) is then given by the matrix expression

$\varpi =t^{-1}\omega t+t^{-1}\mathrm {d} t.$ 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

$\nabla _{X}\sigma =X^{i}\left(\partial _{i}\sigma ^{\alpha }+{{\omega _{i}}^{\alpha }}_{\beta }\sigma ^{\beta }\right)e_{\alpha }.$ 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

$\nabla _{{\dot {\gamma }}(t)}\sigma =0$ for all t ∈ [0, 1]. Equivalently, one can consider the pullback bundle 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 (cf. local expression above) 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:

$\mathrm {Hol} _{x}=\{\tau _{\gamma }:\gamma {\text{ is a loop based at }}x\}.\,$ The holonomy group of a connection is intimately related to the curvature of the connection (AmbroseSinger 1953).

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,

$F^{\nabla }\in \Omega ^{2}(\mathrm {End} \,E)=\Gamma (\mathrm {End} \,E\otimes \Lambda ^{2}T^{*}M).$ It is defined by the expression

$F^{\nabla }(X,Y)(s)=\nabla _{X}\nabla _{Y}s-\nabla _{Y}\nabla _{X}s-\nabla _{[X,Y]}s$ 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

$(d^{\nabla })^{2}\sigma =F^{\nabla }\wedge \sigma .$ A flat connection is one whose curvature form vanishes identically.

Examples

• A classical covariant derivative or affine connection defines a connection on the tangent bundle of M, or more generally on any tensor bundle formed by taking tensor products of the tangent bundle with itself and its dual.
• A connection on $\pi :\mathbb {R} ^{2}\times \mathbb {R} \to \mathbb {R}$ can be described explicitly as the operator
$\nabla =d+{\begin{bmatrix}f_{11}(x)&f_{12}(x)\\f_{21}(x)&f_{22}(x)\end{bmatrix}}dx$ where $d$ is the exterior derivative evaluated on vector-valued smooth functions and $f_{ij}(x)$ are smooth. A section $a\in \Gamma (\pi )$ may be identified with a map
${\begin{cases}\mathbb {R} \to \mathbb {R} ^{2}\\x\mapsto (a_{1}(x),a_{2}(x))\end{cases}}$ and then
$\nabla (a)=\nabla {\begin{bmatrix}a_{1}(x)\\a_{2}(x)\end{bmatrix}}={\begin{bmatrix}{\frac {da_{1}(x)}{dx}}+f_{11}(x)a_{1}(x)+f_{12}(x)a_{2}(x)\\{\frac {da_{2}(x)}{dx}}+f_{21}(x)a_{1}(x)+f_{22}(x)a_{2}(x)\end{bmatrix}}dx$ • If the bundle is endowed with a bundle metric, an inner product on its vector space fibers, a metric connection is defined as a connection that is compatible with the bundle metric.
• A Yang-Mills connection is a special metric connection which satisfies the Yang-Mills equations of motion.
• A Riemannian connection is a metric connection on the tangent bundle of a Riemannian manifold.
• A Levi-Civita connection is a special Riemannian connection: the metric-compatible connection on the tangent bundle that is also torsion-free. It is unique, in the sense that given any Riemannian connection, one can always find one and only one equivalent connection that is torsion-free. "Equivalent" means it is compatible with the same metric, although the curvature tensors may be different; see teleparallelism. The difference between a Riemannian connection and the corresponding Levi-Civita connection is given by the contorsion tensor.
• The exterior derivative is a flat connection on $E=M\times \mathbb {R}$ (the trivial line bundle over M).
• More generally, there is a canonical flat connection on any flat vector bundle (i.e. a vector bundle whose transition functions are all constant) which is given by the exterior derivative in any trivialization.