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Encyclopedia > Bianchi identity


In differential geometry, the curvature form describes curvature of principal bundle with connection. It can be considered as an alternative or generalization of curvature tensor.


Definition

Let G be a Lie group and be a principal G bundle. Let us denote by g the Lie algebra of G.


Let ω denotes the connection form, a 1-form on E with values in g. Then the curvature form is the 2-form with values in g defined by

here d stands for exterior derivative, [ * , * ] is the Lie bracket and D denotes the exterior covariant derivative More precisely,

If is a fiber bundle with structure group G one can repeat the same for the associated principal G bundle.


If is a vector bundle then one can also think of ω as about matrix of 1-forms then the above formula takes the following form:

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

For example, the tangent bundle of a Riemannian manifold we have O(n) as the structure group and is the 2-form with values in o(n) (which can be thought of as antisymmetric matrices, given an orthonormal basis). In this case the form is an alternative description of the curvature tensor, namely in the stadard notation for curvatur tensor we have

Bianchi identities

The first Bianchi identity (for a connection with torsion on the frame bundle) takes the form

,

here D denotes the exterior covariant derivative and Θ the torsion.


The second Bianchi identity holds for general bundle with connection and takes the form

DΩ = 0.

See also


  Results from FactBites:
 
Luigi Bianchi - Wikipedia, the free encyclopedia (413 words)
Luigi Bianchi was born on January 18, 1856 in Parma, Italy, and died on June 6, 1928 in Pisa, Italy.
Bianchi was also greatly influenced by the geometrical ideas of Bernhard Riemann and by the work on transformation groups of Sophus Lie and Felix Klein.
In 1890, Bianchi and Dini supervised the dissertation of the noted analyst and geometer Guido Fubini.
A Unified Field Theory by N.F.J. Matthews (10618 words)
All 24 field equations are identities in affine connections in the same sense that Bianchi's equations of Riemannian geometry are identities.
Bianchi's tensor identities (2.26) of Riemannian geometry can be generalized to yield intensor identities in Weyl's geometry.
Since the Bianchi equations of (4.30) are identities in the Weyl affine connections, all field equations and conservation laws derived from (4.30) are also identities in the Weyl affine connections.
  More results at FactBites »

 
 

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