Curvature form

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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[edit]

Let G be a Lie group with Lie algebra , and PB 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,[1]

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[2]

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).

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.

Curvature form in a vector bundle[edit]

If EB 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.

Bianchi identities[edit]

If is the canonical vector-valued 1-form on the frame bundle, that is, the solder form, 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.

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.

Notes[edit]

  1. ^ since
  2. ^ Proof:

References[edit]

See also[edit]