Connection on a principal bundle
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Let be a Lie group with Lie algebra and a principal bundle for over a smooth manifold . A connection on is a distribution (a subbundle of the tangent bundle) on , called the "horizontal distribution", which is -invariant and complementary to the vertical distribution on .
The decomposition can be given by the projection onto the vertical distribution. Since each vertical space can be identified with (see Principal bundle), this map can be viewed as a -valued 1-form on , a linear map ; this is called the connection form.
The -valued 2-form is called curvature form and measures the non-integrability of the distribution , see the theory page Connections for details.
A connection on a -principal bundles induces a distribution on any associated bundle (see Principal bundle) since passes trivially to and by -invariance to . The induced distribution is called a connection on . If is a vector bundle (the action of on is linear), the connection on is closely related to a covariant derivative (see Connections).
A (semi-)Riemannian metric on defines a connection on the orthonormal frame bundle , the Levi-Civita connection: The horizontal space at some orthonormal basis of consists of the derivatives of all curves in which are parallel along their base point curve in .
Another type of example is the canonical connection on the principal bundle of a reductive homogeneous space .
For further information see [Kobayashi&Nomizu1963].