Exterior covariant derivative
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with 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" 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" 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. For vector bundles. Given a smooth real vector bundle with a connection and rank , the exterior covariant derivative is a real-linear map on the vector-valued differential forms which are valued in : The covariant derivative is such a map for . The exterior covariant derivatives extends this map to general . There are several equivalent ways to define this object: In the case of the  trivial real line bundle with its standard connection, vector-valued differential forms and differential forms can be naturally identified with one another, and each of the above definitions coincides with the standard exterior derivative. Given a principal bundle, any linear representation of the structure group defines an associated bundle, and any connection on the principal bundle induces a connection on the associated vector bundle. Differential forms valued in the vector bundle may be naturally identified with fully anti-symmetric tensorial forms on the total space of the principal bundle. Under this identification, the notions of exterior covariant derivative for the  principal bundle and for the vector bundle coincide with one another. The curvature of a connection on a vector bundle may be defined as the composition of the two exterior covariant derivatives and , so that it is defined as a real-linear map . It is a fundamental but not immediately apparent fact that only depends on , and does so linearly. As such, the curvature may be regarded as an element of . Depending on how the exterior covariant derivative is formulated, various alternative but equivalent definitions of curvature (some without the language of exterior differentiation) can be obtained. It is a  well-known fact that the composition of the standard exterior derivative with itself is zero: . In the present context, this can be regarded as saying that the standard connection on the trivial line bundle has zero curvature.