Connection on a 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 must 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).

This article defines the connection on a vector bundle using one common mathematical notation; however, there are several other more prominent notations in use. One is the notation commonly used in general relativity, using indexed tensors, and another is the notation commonly used in gauge theory, where the endomorphisms of the vector space are emphasized. The differences really are notational, not conceptual, for the most part; the different notations are currently reviewed in the article on the metric connection. That is, the comments made there are generic, applying to vector bundles in general, and not just to metric connections.

Let E → M be a smooth vector bundle over a differentiable manifold M. Denote the space of smooth sections of E by Γ(E). A connection on E is an ℝ-linear map

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