Curvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition. Let "G" be a Lie group with Lie algebra formula_1, and "P" → "B" be a principal "G"-bundle. Let ω be an Ehresmann connection on "P" (which is a formula_1-valued one-form on "P"). Then the curvature form is the formula_1-valued 2-form on "P" defined by Here formula_5 stands for exterior derivative, formula_6 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: if "X", "Y" are horizontal vector fields on "P", then where "hZ" means the horizontal component of "Z", on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field), and formula_9 is the inverse of the normalization factor used by convention in the formula for the exterior derivative. 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. Curvature form in a vector bundle. If "E" → "B" is a vector bundle, then one can also think of ω as a matrix of 1-forms and the above formula becomes the structure equation of E. Cartan: where formula_11 is the wedge product. More precisely, if formula_12 and formula_13 denote components of ω and Ω correspondingly, (so each formula_12 is a usual 1-form and each formula_13 is a usual 2-form) then For example, for the tangent bundle of a Riemannian manifold, the structure  group is O("n") and Ω is a 2-form with values in the Lie algebra of O("n"), i.e. the antisymmetric matrices. In this case the form Ω is an alternative description of the curvature tensor, i.e. using the standard notation for the Riemannian curvature tensor. Bianchi identities. If formula_18 is the canonical vector-valued 1-form on the frame bundle, the torsion formula_19 of the connection form formula_20 is the vector-valued 2-form defined by the structure equation where as above "D" denotes the exterior covariant derivative. The first Bianchi identity takes the form The second Bianchi identity takes  the form and is valid more generally for any connection in a principal bundle.