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.

Definition

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, 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.

Properties

Unlike the usual exterior derivative, which squares to 0 (that is d² = 0), we have

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

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

See also

• Exterior connections

References

• Kobayashi, Shoshichi and Nomizu, Katsumi (1996 (New edition)). Foundations of Differential Geometry, Vol. 1. Wiley-Interscience. ISBN 0-471-15733-3.