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 must 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. Nonlinear connections are connections that are not necessarily linear in this sense.

Connections on vector bundles are also sometimes 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 one common mathematical notation; however, there are several other more prominent notations in use. One is the notation commonly used in general relativity, using indexed tensors, and another is the notation commonly used in gauge theory, where the endomorphisms of the vector space are emphasized. The differences really are notational, not conceptual, for the most part; the different notations are currently reviewed in the article on the metric connection. That is, the comments made there are generic, applying to vector bundles in general, and not just to metric connections.

Formal definition

Let E → M 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 ∇ (i.e. ∇_Xσ = (∇σ)(X)). The covariant derivative satisfies the following properties:

{\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 E → M be a vector bundle. An E-valued differential form of degree r is a section of the tensor product bundle E ⊗ Λ^rT*M. The space of such forms is denoted by

\Omega ^{r}(E)=\Gamma (E\otimes \textstyle \bigwedge ^{r}T^{*}M).

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 E → M 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 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 E → M 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) = E⊗E*:

(\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 E → M 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) → R^k. (This can be seen by considering the pullback of E over F(E) → M, which is isomorphic to the trivial bundle F(E) × R^k.) 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 X^H 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 E → M 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, …,e_k) 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 =({\mathrm d}\sigma ^{\alpha }+\omega ^{\alpha }\!{}_{\beta }\sigma ^{\beta })e_{{\alpha }}

where

\omega ^{\alpha }\!{}_{\beta }\,e_{\alpha }=\nabla e_{\beta }.

Here ω^α_β 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 ω^α_β 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 and fiber indices in this expression. The coefficient functions ω_i^α_β 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, …,f_k) be another smooth local frame over U and let the change of coordinate matrix be denoted t (i.e. f_α = e_βt^β_α). The connection matrix with respect to frame (f_α) 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 (e_α) is given by the expression

\nabla _{X}\sigma =X^{i}(\partial _{i}\,\sigma ^{\alpha }+{\omega _{i}}^{\alpha }\!{}_{\beta }\sigma ^{\beta })e_{{\alpha }}.

Parallel transport and holonomy

A connection ∇ on a vector bundle E → M 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]. More formally, one can consider the pullback bundle γ*E of E by γ. This is a vector bundle over [0, 1] with fiber E_γ(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 E_x 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 τ_γ(v) = σ(1). It can be shown that τ_γ 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(E_x) 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 E → M is a 2-form F^∇ on M with values in the endomorphism bundle End(E) = E⊗E*. 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. For example, given a section $a(x)\in \Gamma (\pi )$ ,

$\nabla (a)=\nabla {\begin{bmatrix}a_{1}(x)\\a_{2}(x)\end{bmatrix}}={\begin{bmatrix}{\frac {da_{1}(x)}{dx}}+f_{11}(x)a_{1}(x)+f_{12}(x)a_{2}(x)\\{\frac {da_{2}(x)}{dx}}+f_{21}(x)a_{1}(x)+f_{22}(x)a_{2}(x)\end{bmatrix}}dx$

If the base space of the bundle has a metric, then it can be extended to the entire vector bundle, giving a bundle metric. Then, the metric connection is defined as a connection on the vector bundle that is compatible with the underlying bundle metric.
The Yang-Mills connection is a special case of a metric connection: it is a metric connection that satisfies the Yang-Mills equations of motion.
The Riemannian connection is a special case of a metric connection: it is a metric connection on the tangent bundle of a Riemannian manifold.
The Levi-Civita connection is a special case of a Riemann connection: it is the (unique) connection that is also torsion-free. It is unique, in the sense that given some Riemannian connection, one can always find an (equivalent) connection that is torsion-free, and there is only one such connection. It is equivalent in the sense that it is compatible with the same metric, although, in general, the curvature tensors will be different; see, for example, teleparallelism for a discussion of the general case. 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 × 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.

References

Chern, Shiing-Shen (1951), Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes
Darling, R. W. R. (1994), Differential Forms and Connections, Cambridge, UK: Cambridge University Press, ISBN 0-521-46800-0
Kobayashi, Shoshichi; Nomizu, Katsumi (1996) [1963], Foundations of Differential Geometry, Vol. 1, Wiley Classics Library, New York: Wiley Interscience, ISBN 0-471-15733-3
Koszul, J. L. (1950), "Homologie et cohomologie des algebres de Lie", Bulletin de la Société Mathématique, 78: 65–127
Wells, R.O. (1973), Differential analysis on complex manifolds, Springer-Verlag, ISBN 0-387-90419-0
Ambrose, W.; Singer, I.M. (1953), "A theorem on holonomy", Transactions of the American Mathematical Society, 75: 428–443, doi:10.2307/1990721

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