Connection on a principal bundle
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1 Definition
Let be a Lie group with Lie algebra and a principal bundle for over a
smooth manifoldTex syntax error. 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).
2 Examples
Tex syntax errordefines 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 inTex syntax error.
Another type of example is the canonical connection on the principal bundle of a reductive homogeneous space .
For further information see [Kobayashi&Nomizu1963].
References
- [Kobayashi&Nomizu1963] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR1393940 (97c:53001a) Zbl 0508.53002