Exterior covariant derivative

In mathematics, the exterior covariant derivative is an analog of an exterior derivative that takes into account the presence of a connection.
Definition.
Let "G" be a Lie group and be a principal "G"-bundle on a smooth manifold "M". Suppose there is a connection on "P"; this yields a natural direct sum decomposition formula_1 of each tangent space into the horizontal and vertical subspaces. Let formula_2 be the projection to the horizontal subspace.
If "ϕ" is a "k"-form on "P" with values in a vector space "V", then its exterior covariant derivative "Dϕ" is a form defined by
where "v""i" 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 formula_5, then "Dϕ" is a tensorial -form on "P" of the type "ρ": it is equivariant and horizontal (a form "ψ" is horizontal if .)
By abuse of notation, the differential of "ρ" at the identity element may again be denoted by "ρ":
Let formula_7 be the connection one-form and formula_8 the representation of the connection in formula_9 That is, formula_8 is a formula_11-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 formula_11 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 "D"2 vanishes for a flat connection (i.e. when ).
If , then one can write
where formula_17 is the matrix with 1 at the -th entry and zero on the other entries. The matrix formula_18 whose entries are 2-forms on "P" is called the curvature matrix.
Exterior covariant derivative for vector bundles.
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 (sometimes called the exterior connection) 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" (see "").
Requiring ∇ to satisfy Leibniz's rule, ∇ also acts on any "E"-valued form; thus, it is given on decomposable elements of the space formula_20 of formula_21-valued "k"-forms by 
For a section "s" of "E", we also set
where formula_24 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" (depending on a connection). 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.