Let G be a Lie group and be a principal G-bundle on a smooth manifoldM. Suppose there is a connection on P; this yields a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces. Let be the projection to the horizontal subspace. If ϕ is a k-form on P with values in a vector spaceV, then its exterior covariant derivative Dϕ is a form defined by where vi are tangent vectors to P at u. Suppose that is a representation of G on a vector space V. If ϕ is equivariant in the sense that where, then Dϕ is a tensorial -form on P of the type ρ: it is equivariant and horizontal By abuse of notation, the differential of ρ at the identity element may again be denoted by ρ: Let be the connection one-form and the representation of the connection in That is, is a -valued form, vanishing on the horizontal subspace. If ϕ is a tensorial k-form of type ρ, then where, following the notation in, we wrote Unlike the usual exterior derivative, which squares to 0, the exterior covariant derivative does not. In general, one has, for a tensorial zero-form ϕ, where is the representation in of the curvature two-form Ω. The form F is sometimes referred to as the field strength tensor, in analogy to the role it plays in electromagnetism. Note that D2 vanishes for a flat connection. If, then one can write where is the matrix with 1 at the -th entry and zero on the other entries. The matrix whose entries are 2-forms on P is called the curvature matrix.
When is a representation, one can form the associated bundle. Then the exterior covariant derivative D given by a connection on P induces an exterior covariant derivative on the associated bundle, this time using the nabla symbol: Here, Γ denotes the space of local sections of the vector bundle. The extension is made through the correspondence between E-valued forms and tensorial forms of type ρ Requiring ∇ to satisfy Leibniz's rule, ∇ also acts on any E-valued form; thus, it is given on decomposable elements of the space of -valued k-forms by For a sections of E, we also set where is the contraction by X. Conversely, given a vector bundle E, one can take its frame bundle, which is a principal bundle, and so obtain an exterior covariant differentiation on E. Identifying tensorial forms and E-valued forms, one may show that which can be easily recognized as the definition of the Riemann curvature tensor on Riemannian manifolds.
Example
Bianchi's second identity, which says that the exterior covariant derivative of Ω is zero can be stated as:.
