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 pseudo-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 .

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).

Motivation

Let be a differentiable manifold, such as Euclidean space can be viewed as a section of the trivial vector bundle One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function on .

may be defined by the standard formula

and computes

This means that subtraction of these two terms is not naturally defined.

The problem is resolved by introducing the extra structure of a connection to the vector bundle. There are at least three perspectives from which connections can be understood. When formulated precisely, all three perspectives are equivalent.

1. (Parallel transport a linear isomorphism This is not straightforward to formulate, and so this notion of "parallel transport" is usually derived as a by-product of other ways of defining connections. In fact, the following notion of "Ehresmann connection" is nothing but an infinitesimal formulation of parallel transport.

2. (Ehresmann connection As such, one may consider the pushforward which is an element of the tangent space moves as is varied over a fiber.

3. (Covariant derivative which mimics these properties, together with a form of the product rule.

Unless the base is zero-dimensional, there are always infinitely many connections which exist on a given differentiable vector bundle, and so there is always a corresponding choice of how to differentiate sections. Depending on context, there may be distinguished choices, for instance those which are determined by solving certain partial differential equations. In the case of the tangent bundle, any pseudo-Riemannian metric (and in particular any Riemannian metric) determines a canonical connection, called the Levi-Civita connection.

Formal definition

be a smooth real vector bundle over a smooth manifold. Denote the space of smooth sections. A covariant derivative is either of the following equivalent structures:

1. -linear map such that the product rule holds for all smooth functions

2. an assignment, to any smooth section which depends smoothly on

these two definitions are identical and differ only in the language used.

With this notation, the product rule in the second version of the definition given above is written

Remark. In the case of a complex vector bundle" This places extra restrictions, as not every real-linear map between complex vector spaces is complex-linear. There is some ambiguity in this distinction, as a complex vector bundle can also be regarded as a real vector bundle.

Induced connections

, symmetric and antisymmetric tensor 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.

Dual connection

, the induced dual connection is defined implicitly by

is a smooth vector field a section of the dual bundle so that a natural product rule.

Tensor product connection

, define the tensor product connection by the formula

, one also obtains the tensor power connection.

Direct sum connection

The direct sum connection is defined by

.

Symmetric and exterior power connections

, the definition of the tensor product connection applies in a straightforward manner to this setting. Indeed, since the symmetric and exterior algebras sit inside the tensor algebra to these summands. Explicitly, define the symmetric product connection by

and the exterior product connection by

. Repeated applications of these products gives induced symmetric power and exterior power connections respectively.

Endomorphism connection

, the endomorphism connection also, then the following product rule holds for the endomorphism connection:

By reversing this equation, it is possible to define the endomorphism connection as the unique connection satisfying

, thus avoiding the need to first define the dual connection and tensor product connection.

Any associated bundle

bundle of frames. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the dual representation, the tensor product to the tensor product representation, the direct sum to the direct sum representation, and so on.

Exterior covariant derivative and vector-valued forms

be a vector bundle. An E-valued differential form is a section of the tensor product bundle:

The space of such forms is denoted by

where the last tensor product denotes the tensor product of modules over the ring.

. That is,

is a linear map

A connection may then be viewed as a generalization of the exterior derivative to an exterior covariant derivative

and extended linearly:

.

(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 then their difference is a C^{\infty}(M)-linear operator. That is,

:

.

is an affine space.

Relation to principal and Ehresmann connections

be the frame bundle. Then a (principal) connection are in one-to-one correspondence with right-equivariant. (This can be seen by considering the pullback, which is isomorphic to the trivial bundle is then given by

is the horizontal lift.)

, and these two constructions are mutually inverse.

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

The induced connections discussed in #Induced connections, and the induced connection is precisely that which was described above.

Local expression

admits a local smooth frame of sections

in the frame as

.

as

are a collection of local one-forms. These forms can be put into a matrix of one-forms defined by

called the using the product rule as

as a basis,

then using regular matrix multiplication one can write

is shorthand for applying the exterior derivative. In this sense a connection is locally completely specified by its connection one-form in some trivialisation.

As explained in #Affine properties of the set of connections.

Relationship to Christoffel symbols

In pseudo-Riemannian geometry, the Levi-Civita connection is often written in terms of the Christoffel symbols is also a local chart.

can be expanded in this basis as

, called the Christoffel symbols is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry.

and the Christoffel symbols, to be given by

leads to

locally defined operators

with the property that

Change of local trivialisation

, defined by

is given by

. In matrix notation this may be written

component-by-component.

, the lengthy formulae for the transformation of the Christoffel symbols of the Levi-Civita connection can be recovered from the more succinct transformation laws of the connection form above.

Parallel transport and holonomy

defines a notion of parallel transport be a smooth path is said to be parallel if

. Equivalently, one can consider the pullback bundle is parallel if and only if.

. The above equation defining parallel sections is a first-order ordinary differential equation (cf. local expression. Define a parallel transport map

is a linear isomorphism.

Parallel transport can be used to define the holonomy group consisting of all parallel transport maps coming from loops:

The holonomy group of a connection is intimately related to the curvature of the connection .

pick an integral curve, which may be differentiated. The covariant derivative is recovered as

.

Curvature

The curvature. That is,

It is defined by the expression

is C^{\infty}(M).

As mentioned above 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, then

denotes matrix multiplication, where the components are wedged rather than multiplied.

, then one has

, then

is the full curvature tensor, and in Riemannian geometry would be identified with the Riemannian curvature tensor.

to be wedge product of forms but commutator, and with this alternate notation the Cartan structure equation takes the form

, 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 (see for example ) 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.

Bianchi identity

A version of the second (differential) Bianchi identity-valued two-form, we may apply the exterior covariant derivative to it. The Bianchi identity says that

1. .

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.

There is no analogue in general of the first and a solder form has been chosen, one can use the soldering to interchange the indices and define a notion of torsion for affine connections which are not the Levi-Civita connection.

Gauge transformations

, it is natural to ask when they might be considered equivalent. There is a well-defined notion of an automorphism. Such an automorphism is called a gauge transformation, and the group of all automorphisms is called the gauge group. The group of gauge transformations may be neatly characterised as the space of sections of the capital A adjoint bundle of the frame bundle. This is not to be confused with the lowercase a adjoint bundle is the associated bundle

by

is a connection, one verifies the product rule

It may be checked that this defines a left group action.

, it can be seen that

.

Two connections are said to be gauge equivalent is the moduli space. 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, which is of significant interest in gauge theory and physics.

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.

• can be described explicitly as the operator

• 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 (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.

See also

• D-module

• Connection (mathematics)

References

• Donaldson, S.K. and Kronheimer, P.B., 1997. The geometry of four-manifolds. Oxford University Press.

• Tu, L.W., 2017. Differential geometry: connections, curvature, and characteristic classes (Vol. 275). Springer.

• Taubes, C.H., 2011. Differential geometry: Bundles, connections, metrics and curvature (Vol. 23). OUP Oxford.