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In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of the curvature tensor in Riemannian geometry. ==Definition== Let ''G'' be a Lie group with Lie algebra , and ''P'' → ''B'' be a principal ''G''-bundle. Let ω be an Ehresmann connection on ''P'' (which is a -valued one-form on ''P''). Then the curvature form is the -valued 2-form on ''P'' defined by : Here stands for exterior derivative, is defined in the article "Lie algebra-valued form" and ''D'' denotes the exterior covariant derivative. In other terms, : where ''X'', ''Y'' are tangent vectors to ''P''. There is also another expression for Ω: : where ''hZ'' means the horizontal component of ''Z'' and on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field).〔Proof: We can assume ''X'', ''Y'' are either vertical or horizontal. Then we can assume ''X'', ''Y'' are both horizontal (otherwise both sides vanish since Ω is horizontal). In that case, this is a consequence of the invariant formula for exterior derivative ''d'' and the fact ω(Z) is a unique Lie algebra element that generates the vector field ''Z''.〕 A connection is said to be flat if its curvature vanishes: Ω = 0. Equivalently, a connection is flat if the structure group can be reduced to the same underlying group but with the discrete topology. See also: flat vector bundle. 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「Curvature form」の詳細全文を読む スポンサード リンク
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