Curvature form - Alchetron, The Free Social Encyclopedia

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 g , and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a g -valued one-form on P).

Then the curvature form is the g -valued 2-form on P defined by

Ω = d ω + 1 2 [ ω ∧ ω ] = D ω .

Here d stands for exterior derivative, [ ⋅ ∧ ⋅ ] is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,

Ω ( X , Y ) = d ω ( X , Y ) + 1 2 [ ω ( X ) , ω ( Y ) ]

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 Ω ( X , Y ) = − ω ( [ X , Y ] ) = − [ X , Y ] + h [ X , Y ]

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

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:

Ω = d ω + ω ∧ ω ,

where ∧ is the wedge product. More precisely, if ω j i and Ω j i denote components of ω and Ω correspondingly, (so each ω j i is a usual 1-form and each Ω j i is a usual 2-form) then

Ω j i = d ω j i + ∑ k ω k i ∧ ω j k .

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.

R ( X , Y ) = Ω ( X , Y ) ,

using the standard notation for the Riemannian curvature tensor.

Bianchi identities

If θ is the canonical vector-valued 1-form on the frame bundle, that is, the solder form, the torsion Θ of the connection form ω is the vector-valued 2-form defined by the structure equation

Θ = d θ + ω ∧ θ = D θ ,

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

D Θ = Ω ∧ θ .

The second Bianchi identity takes the form

D Ω = 0

and is valid more generally for any connection in a principal bundle.

References

Curvature form Wikipedia

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Contents

Definition

Curvature form in a vector bundle

Bianchi identities

References