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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 ''P'' → ''M'' be a principal ''G''-bundle on a smooth manifold ''M''. Suppose there is a connection on ''P'' so that it gives a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces. Let be the projection. If ϕ is a ''k''-form on ''P'' with values in a vector space ''V'', then its exterior covariant derivative ''D''ϕ is a form defined by : where ''v''''i'' are tangent vectors to ''P'' at ''u''. Suppose ''V'' is a representation of ''G''; i.e., there is a Lie group homomorphism ρ: ''G'' →''GL''(''V''). If ''φ'' is equivariant in the sense: : where , then ''D''ϕ is a tensorial (''k'' + 1)-form on ''P'' of the type ρ: it is equivariant and horizontal (a form ψ is horizontal if ψ(''v''0, …, ''v''k) = ψ(''hv''0, …, ''hv''''k'').) *Example: if ω is the connection form on ''P'', then Ω = ''D''ω is called the curvature form of ω. Bianchi's second identity says the exterior covariant derivative of Ω is zero; i.e., ''D''Ω = 0. We also denote the differential of ρ at the identity element by ρ: : If φ is a tensorial ''k''-form of type ρ, then :〔If ''k'' = 0, then, writing for the fundamental vector field (i.e., vertical vector field) generated by ''X'' in on ''P'', we have: :, since φ(''gu'') = ρ(''g''−1)φ(''u''). On the other hand, ''D''φ(''X''#) = 0. If ''X'' is a horizontal vector field, then and . In general, by the invariant formula for exterior derivative, we have: for any vector fields ''X''''i'''s, since φ takes the same values at ''hX''''i'''s and ''X''''i'''s, : where the hat means the term is omitted. This equals .〕 where is a -valued form, and : *Example: Bianchi's second identity (''D''Ω = 0) can be stated as: . Unlike the usual exterior derivative, which squares to 0 (that is d2 = 0), we have: :〔Proof: We have: : (cf. the example at Lie algebra-valued differential form#Operations), which is by E. Cartan's structure equation.〕 where ''F'' = ρ(Ω). In particular ''D''2 vanishes for a flat connection (i.e., Ω = 0). If ρ: ''G'' →''GL''(''R''n), then one can write : where is the matrix with 1 at the (''i'', ''j'')-th entry and zero on the other entries. The matrix whose entries are 2-forms on ''P'' is called the curvature matrix. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「exterior covariant derivative」の詳細全文を読む スポンサード リンク
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