Connection (vector bundle)
In mathematics, and especially differential geometry and gauge theory, 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.
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).
A section of a vector bundle generalises the notion of a function on a manifold, in the sense that a standard vector-valued function can be viewed as a section of the trivial vector bundle . It is therefore natural to ask if it is possible to differentiate a section in analogy to how one differentiates a vector field. When the vector bundle is the tangent bundle to a Riemannian manifold, this question is answered naturally by the Levi-Civita connection which is the unique torsion-free connection compatible with the Riemannian metric on the tangent bundle. In general there is no such natural choice of a way to differentiate sections.
The model case is to differentiate an -component vector field on Euclidean space . In this setting the derivative at a point in the direction may be simply defined by
Notice that for every , we have defined a new vector so the derivative of in the direction of has yielded a new -component vector field on .
When passing to a section of a vector bundle on a manifold , one encounters two key issues with this definition. Firstly, since the manifold has no linear structure, the term makes no sense on . Instead one takes a path such that and computes
However this still does not make sense, because is a vector in the fibre over , and , the fibre over , which is a different vector space. This means there is no way to make sense of the subtraction of these two terms lying in different vector spaces.
The goal is to resolve the above conundrum by coming up with a way of differentiating sections of a vector bundle in the direction of vector fields, and getting back another section of the vector bundle. There are three possible resolutions to this problem. All three require making a choice of how to differentiate sections, and only in special settings like the tangent bundle on a Riemannian manifold is there a natural such choice.
- (Parallel transport) Since the problem is that the vectors and lie in different fibres of , one solution is to define an isomorphism for all close to zero. Using this isomorphism one can transport to the fibre and then take the difference. Explicitly
- This is the parallel transport, and the choice of the isomorphisms for all curves in can be taken as the definition of how to differentiate a section.
- (Ehresmann connection) Use the notion of differential of a map of smooth manifolds. A section is by definition a smooth map such that . This has a differential , with the property that for a vector field . However, one would like instead for to be a section of itself. In fact, since the tangent space to a vector space is isomorphic to the space itself, the vertical bundle of all tangent spaces to the fibres of is naturally isomorphic to a copy of itself, or more precisely the pullback of along the projection . If one chooses a projection from to this vertical subbundle which is compatible with the linear structure of the fibres, composing with this projection would land back in . This is called a linear Ehresmann connection on the vector bundle . There are many choices of projection operators so in general there are many different ways of differentiating a vector field.
- (Covariant derivative) The third solution is to abstract the properties that a derivative of a section of a vector bundle should have and take this as an axiomatic definition. This is the notion of a connection or covariant derivative described in this article. The other two approaches above can both be shown to be equivalent to this axiomatic definition of differentiation.
such that the Leibniz rule
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
by contracting X with the resulting covariant index in the connection: ∇X σ = (∇σ)(X). The covariant derivative satisfies:
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.
Given a vector bundle , there are many associated bundles to which may be constructed, for example the dual vector bundle , tensor powers , symmetric and antisymmetric tensor powers , and the direct sums . A connection on induces a connection on any one of these associated bundles. The ease of passing between connections on associated bundles is more elegantly captured by the theory of principal bundle connections, but here we present some of the basic induced connections.
Given a connection on , the induced dual connection on is defined by
Here is a smooth vector field, is a section of , and a section of the dual bundle, and the natural pairing between a vector space and its dual (occurring on each fibre between and ). Notice that this definition is essentially enforcing that be the connection on so that a natural product rule is satisfied for pairing .
Given connections on two vector bundles , define the tensor product connection by the formula
Here we have . Notice again this is the natural way of combining to enforce the product rule for the tensor product connection. Similarly define the direct sum connection by
Since the exterior power and symmetric power of a vector bundle may be viewed as subspaces of the tensor power, , the definition of the tensor product connection applies in a straightforward manner to this setting. Namely, if is a connection on , one has the tensor power connection by repeated applications on the tensor product connection above. We also have the symmetric product connection defined by
and the exterior product connection defined by
for all . Repeated applications of these products gives induced symmetric power and exterior power connections on and respectively.
Finally, one obtains the induced connection on the vector bundle , the endomorphism connection. This is simply the tensor product connection of the dual connection on and on . If and , so that the composition also, then the following product rule holds:
Exterior covariant derivative and vector-valued forms
The space of such forms is denoted by
An E-valued 0-form is just a section of the bundle E. That is,
In this notation a connection on E → M is a linear map
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 an exterior covariant derivative
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 of the set of connections
Every vector bundle over a manifold admits a connection, which can be proved using partitions of unity. 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,
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*:
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). This affine space is commonly denoted .
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) → 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
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.
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, ..., ek) of E over U, any section σ of E can be written as (Einstein notation assumed). A connection on E restricted to U then takes the form
Here 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 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
Note the mixture of coordinate indices (i) and fiber indices (α,β) in this expression. The coefficient functions 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.:
The connection matrix with respect to frame (fα) is then given by the matrix expression
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
Parallel transport and holonomy
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
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:
The holonomy group of a connection is intimately related to the curvature of the connection (AmbroseSinger 1953).
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at . For each we will write for the parallel transport map traveling along from to . In particular for every , we have . Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
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,
It is defined by the expression
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
A flat connection is one whose curvature form vanishes identically.
Local form and Cartan's structure equation
The curvature form has a local description called Cartan's structure equation. If has local form on some trivialising open subset for , then
on . To clarify, where is an endomorphism-valued one-form. For simplicity let us suppose for a one-form and an endomorphism . Then we use the conventions
where is another endomorphism valued one-form. In general will be a sum of simple tensors of this form, and the operators and are extended linearly.
It can be checked that if we define to be wedge product of forms but commutator of endomorphisms as opposed to composition, then , and with this alternate notation the Cartan structure equation takes the form
This alternate notation is commonly used in the theory of principal bundle connections, where the connection form is a Lie algebra-valued one-form for which there is no notion of composition (unlike in the case of endomorphisms), but there is a notion of a Lie bracket.
In some references the Cartan structure equation may be written with a minus sign:
This different convention uses an order of matrix multiplication that is different from the standard Einstein notation in the wedge product of matrix-valued one-forms.
A version of the Bianchi identity from Riemannian geometry holds for a connection on any vector bundle. Recall that a connection on a vector bundle induces an endomorphism connection on . This endomorphism connection has itself an exterior covariant derivative, which we ambiguously call . Since the curvature is a globally defined -valued two-form, we may apply the exterior covariant derivative to it. The Bianchi identity says that
This succinctly captures the complicated tensor formulae of the Bianchi identity in the case of Riemannian manifolds, and one may translate from this equation to the standard Bianchi identities by expanding the connection and curvature in local coordinates.
Given two connections on a vector bundle , it is natural to ask when they might be considered equivalent. There is a well-defined notion of an automorphism of a vector bundle . A section is an automorphism if is invertible at every point . Such an automorphism is called a gauge transformation of , and the group of all automorphisms is called the gauge group, often denoted or . The group of gauge transformations may be neatly characterised as the space of sections of the capital A adjoint bundle of the frame bundle of the vector bundle . This is not to be confused with the lowercase a adjoint bundle , which is naturally identified with itself. The bundle is the associated bundle to the principal frame bundle by the conjugation representation of on itself, , and has fibre the same general linear group where . Notice that despite having the same fibre as the frame bundle and being associated to it, is not equal to the frame bundle, nor even a principal bundle itself. The gauge group may be equivalently characterised as
A gauge transformation of acts on sections , and therefore acts on connections by conjugation. Explicitly, if is a connection on , then one defines by
for . To check that is a connection, one verifies the product rule
It may be checked that this defines a left group action of on the affine space of all connections .
Since is an affine space modelled on , there should exist some endomorphism-valued one-form such that . Using the definition of the endomorphism connection induced by , it can be seen that
which is to say that .
Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space is the moduli space of all connections on . In general this topological space is neither a smooth manifold or even a Hausdorff space, but contains inside it the moduli space of Yang–Mills connections on , which is of significant interest in gauge theory and physics.
- 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 can be described explicitly as the operator
- where is the exterior derivative evaluated on vector-valued smooth functions and are smooth. A section may be identified with a map
- and then
- 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 (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.
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