Connection (fiber bundle)


Connection (fiber bundle) - Definition

In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. That is an application to tangent bundles; there are more general connections, used in differential geometry and other fields of mathematics to formulate intrinsic differential equations . Connection may refer to a connection on any vector bundle, or also a connection on a principal bundle.

Connections give rise to parallel transport along a curve on a manifold. A connection also leads to invariants of curvature (see also curvature tensor and curvature form), and the so-called torsion.

General concept

The general concept can be summarized as follows: given a fiber bundle <math>\eta:E\to B<math> the tangent space at any point of E has canonical "vertical" subspace, the subspace tangent to the fiber. The connection fixes a choice of "horizontal" subspace at each point of E so that the tangent space of E is a direct sum of vertical and horizontal subspaces. Usually more requirements are imposed on the choice of "horizontal" subspaces, but they depend on the type of the bundle.

Given a <math>B'\to B<math> the induced bundle has an induced connection. If <math>B'=I<math> is a segment then connection on B gives a trivialization on the induced bundle over I, i.e. a choice of smooth one-parameter family of isomorphisms between the fibers over I. This family is called parallel displacement along the curve <math>I\to B <math> and it gives an equivalent description of connection (which in case of Levi-Civita connection on a Riemannian manifold is called parallel transport).

There are many ways to describe connection, in one particular approach, a connection can be locally described as a matrix of 1-forms which is the multiplant of the difference between the covariant derivative and the ordinary partial derivative in a coordinate chart. That is, partial derivatives are not an intrinsic notion on a manifold: a connection 'fixes up' the concept and permits discussion in geometric terms.

Possible approaches

There are quite a number of possible approaches to the connection concept. They include the following:

A rather direct module-style approach to covariant differentiation, stating the conditions allowing vector fields to act as differential operators on vector bundle sections.
Traditional index notation specifies the connection by components; see Covariant derivative (three indices, but this is not a tensor).
In Riemannian geometry there is a way of deriving a connection from the metric tensor (Levi-Civita connection).
Using principal bundles and Lie algebra-valued differential forms (see connection form and Cartan connection).
The most abstract approach may be that suggested by Alexander Grothendieck, where a connection is seen as descent data from infinitesimal neighbourhoods of the diagonal.

The connections referred to above are linear or affine connections. There is also a concept of projective connection; the most commonly-met form of this is the Schwarzian derivative in complex analysis.