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 PM be a principal G-bundle on a smooth manifold M. If \phi 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.

Unlike the usual exterior derivative, which squares to 0, we have

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

where \Omega denotes the curvature form. In particular D^2 vanishes for a flat connection.

See also

References