Connection

Connection may refer to:

Technology

Database connection

Telecommunication circuit, the complete path between two terminals

Virtual connection, also known as a virtual circuit

Mathematics

Connectedness

Connected sum

Connection (algebraic framework)

Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold

Connection (affine bundle)
Connection (composite bundle)
Connection (fibred manifold)

Connection (affine bundle)

Connection (composite bundle)

Connection (fibred manifold)

Connection (principal bundle), gives the derivative of a section of a principal bundle

Connection (vector bundle), differentiates a section of a vector bundle along a vector field

Cartan connection, achieved by identifying tangent spaces with the tangent space of a certain model Klein geometry

Ehresmann connection, gives a manner for differentiating sections of a general fibre bundle

Electrical connection, allows the flow of electrons

Galois connection, a type of correspondence between two partially ordered sets

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Connection (principal bundle)

In mathematics, a connection 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. A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G.

A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.

Formal definition

Let π:P→M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra $\mathfrak g$ of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Connection_(principal_bundle)

Connection (affine bundle)

Let $Y\to X,$ be an affine bundle modelled over a vector bundle $\overline Y\to X$ . A connection $\Gamma$ on $Y\to X$ is called the affine connection if it as a section $\Gamma:Y\to J^1Y$ of the jet bundle $J^1Y\to Y$ of $Y$ is an affine bundle morphism over $X$ . In particular, this is the case of an affine connection on the tangent bundle $TX$ of a smooth manifold $X$ .

With respect to affine bundle coordinates $(x^\lambda,y^i)$ on $Y$ , an affine connection $\Gamma$ on $Y\to X$ is given by the tangent-valued connection form

An affine bundle is a fiber bundle with a general affine structure group $GA(m,\mathbb R)$ of affine transformations of its typical fiber $V$ of dimension $m$ . Therefore, an affine connection is associated to a principal connection. It always exists.

For any affine connection $\Gamma:Y\to J^1Y$ , the corresponding linear derivative $\overline\Gamma:\overline Y\to J^1\overline Y$ of an affine morphism $\Gamma$ defines a unique linear connection on a vector bundle $\overline Y\to X$ . With respect to linear bundle coordinates $(x^\lambda,\overline y^i)$ on $\overline Y$ , this connection reads

Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.

If $Y\to X$ is a vector bundle, both an affine connection $\Gamma$ and an associated linear connection $\overline\Gamma$ are connections on the same vector bundle $Y\to X$ , and their difference is a basic soldering form on $\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i$ . Thus, every affine connection on a vector bundle $Y\to X$ is a sum of a linear connection and a basic soldering form on $Y\to X$ .