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. 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.
(Parallel transport) 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 (in the definition) 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. (Ehresmann connection) 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. (Covariant derivative) 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.