Curvature form

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. 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. Let G be a Lie group with Lie algebra g {displaystyle {mathfrak {g}}} , and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a g {displaystyle {mathfrak {g}}} -valued one-form on P). Then the curvature form is the g {displaystyle {mathfrak {g}}} -valued 2-form on P defined by Here d {displaystyle d} stands for exterior derivative, [ ⋅ ∧ ⋅ ] {displaystyle } 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 σ ∈ { 1 , 2 } {displaystyle sigma in {1,2}} 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. See also: flat 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: