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 connection.

Given a tensor-valued differential k-form φ its exterior covariant derivative is defined by

Dφ(X₀,X₁,...,X_k) = dφ(h(X₀),h(X₁),...,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 E.

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

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