 Exterior covariant derivative

In mathematics, the exterior covariant derivative, sometimes also covariant exterior derivative, is a very useful notion for calculus on manifolds, which makes it possible to simplify formulas which use a principal connection.
Let P → M be a principal Gbundle on a smooth manifold M. If φ is a tensorial kform on P, then its exterior covariant derivative is defined by
where h denotes the projection to the horizontal subspace, H_{x} defined by the connection, with kernel V_{x} (the vertical subspace) of the tangent bundle of the total space of the fiber bundle. Here X_{i} are any vector fields on P. Dφ is a tensorial k+1 form on P.
Unlike the usual exterior derivative, which squares to 0, we have
where Ω denotes the curvature form. In particular D^{2} vanishes for a flat connection.
See also
References
 Kobayashi, Shoshichi and Nomizu, Katsumi (1996 (New edition)). Foundations of Differential Geometry, Vol. 1. WileyInterscience. ISBN 0471157333.
This geometryrelated article is a stub. You can help Wikipedia by expanding it.