Exterior covariant derivative

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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 G-bundle on a smooth manifold M. If φ is a tensorial k-form on P, then its exterior covariant derivative is defined by

D\phi(X_0,X_1,\dots,X_k)=\mathrm{d}\phi(h(X_0),h(X_1),\dots,h(X_k))

where h denotes the projection to the horizontal subspace, Hx defined by the connection, with kernel Vx (the vertical subspace) of the tangent bundle of the total space of the fiber bundle. Here Xi 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

D^2\phi=\Omega\wedge\phi

where Ω denotes the curvature form. In particular D2 vanishes for a flat connection.

[edit] References

  • Kobayashi, Shoshichi and Nomizu, Katsumi (1996 (New edition)). Foundations of Differential Geometry, Vol. 1. Wiley-Interscience. ISBN 0471157333. 
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