Exterior covariant derivative
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In mathematics, the covariant exterior derivative, sometimes also exterior covariant derivative, is a very useful notion for calculus on manifolds, which makes it possible to simplify formulas which use a connection.
Given a tensor-valued differential k-form <math>\phi </math> its exterior covariant derivative is defined by
- <math>D\phi(X_0,X_1,...,X_k)=d\phi(h(X_0),h(X_1),...,h(X_k))</math>
where h denotes the projection to the horizontal subspace, <math>H_x</math> defined by the connection, with kernel <math>V_x</math> (the vertical subspace) of the tangent bundle of the total space of the fiber bundle. Here <math>X_i</math> are any vector fields on E.
Unlike the usual exterior derivative, which squares to 0, we have
- <math>D^2\phi=\Omega\wedge\phi</math>
where <math>\Omega</math> denotes the curvature form. In particular <math>D^2</math> vanishes for a flat connection.