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Concept
Curvature form
Summary
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.
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
(In another convention, 1/2 does not appear.) 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: 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 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.
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 is the wedge product. More precisely, if and denote components of ω and Ω correspondingly, (so each is a usual 1-form and each 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.
Contracted Bianchi identities
Riemann curvature tensor#Symmetries and identities
If is the canonical vector-valued 1-form on the frame bundle, the torsion of the connection form is the vector-valued 2-form defined by the structure equation
where as above D denotes the exterior covariant derivative.
Official source
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