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 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.

Motivation

Let be a differentiable manifold, such as Euclidean space. A vector-valued function 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.
The model case is to differentiate a function on Euclidean space. In this setting the derivative at a point in the direction may be defined by the standard formula
For every, this defines a new vector
When passing to a section of a vector bundle over 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 and are elements of the distinct vector spaces and 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.

A connection can be viewed as assigning to every differentiable path a linear isomorphism for all Using this isomorphism one can transport to the fibre and then take the difference; explicitly, In order for this to depend only on and not on the path extending it is necessary to place restrictions on the dependence of on 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.
The section may be viewed as a smooth map from the smooth manifold to the smooth manifold As such, one may consider the pushforward which is an element of the tangent space In Ehresmann's formulation of a connection, one chooses a way of assigning, to each and every a direct sum decomposition of into two linear subspaces, one of which is the natural embedding of With this additional data, one defines by projecting to be valued in In order to respect the linear structure of a vector bundle, one imposes additional restrictions on how the direct sum decomposition of moves as is varied over a fiber.
The standard derivative in Euclidean contexts satisfies certain dependencies on and the most fundamental being linearity. A covariant derivative is defined to be any operation 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 determines a canonical connection, called the Levi-Civita connection.

Formal definition

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

an -linear map such that the product rule holds for all smooth functions on and all smooth sections of
an assignment, to any smooth section and every, of an -linear map which depends smoothly on and such that for any two smooth sections and any real numbers and such that for every smooth function, is related to by for any and

Beyond using the canonical identification between the vector space and the vector space of linear maps these two definitions are identical and differ only in the language used.
It is typical to denote by with being implicit in 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, the above definition is still meaningful, but is usually taken to be modified by changing "real" and "" everywhere they appear to "complex" and "" 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

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.

Dual connection

Given a connection on, the induced dual connection on is defined implicitly 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, i.e.,. Notice that this definition is essentially enforcing that be the connection on so that a natural product rule is satisfied for pairing.

Tensor product connection

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. By repeated application of the above construction applied to the tensor product, one also obtains the tensor power connection on for any and vector bundle.

Direct sum connection

The direct sum connection is defined by
where.

Symmetric and exterior power connections

Since the symmetric power and exterior power of a vector bundle may be viewed naturally as subspaces of the tensor power,, 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 as direct summands, and the connection respects this natural splitting, one can simply restrict to these summands. Explicitly, define the symmetric product connection by
and the exterior product connection by
for all. Repeated applications of these products gives induced symmetric power and exterior power connections on and respectively.

Endomorphism connection

Finally, one may define the induced connection on the vector bundle of endomorphisms, 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 for the endomorphism connection:
By reversing this equation, it is possible to define the endomorphism connection as the unique connection satisfying
for any, thus avoiding the need to first define the dual connection and tensor product connection.

Any associated bundle

Given a vector bundle of rank, and any representation into a linear group, there is an induced connection on the associated vector bundle, where is the principal bundle of frames of. 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

Let be a vector bundle. An -valued differential form of degree 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 of smooth functions on.
An -valued 0-form is just a section of the bundle. That is,
In this notation a connection on 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 there is a unique way to extend to an exterior covariant derivative
This exterior covariant derivative is defined by the following Leibniz rule, which is specified on simple tensors of the form and extended linearly:
where so that, is a section, and denotes the -form with values in defined by wedging with the one-form part of. Notice that for -valued 0-forms, this recovers the normal Leibniz rule for the connection.
Unlike the ordinary exterior derivative, one generally has. In fact, is directly related to the curvature of the connection .