Connection (vector Bundle)

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 is required to 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).

Read more about Connection (vector Bundle):  Formal Definition, Vector-valued Forms, Affine Properties, Relation To Principal and Ehresmann Connections, Local Expression, Parallel Transport and Holonomy, Curvature, Examples

Famous quotes containing the word connection:

    Children of the same family, the same blood, with the same first associations and habits, have some means of enjoyment in their power, which no subsequent connections can supply; and it must be by a long and unnatural estrangement, by a divorce which no subsequent connection can justify, if such precious remains of the earliest attachments are ever entirely outlived.
    Jane Austen (1775–1817)