Curvature form

In differential geometry, the curvature form describes curvature of a connection form on a principal bundle. It can be considered as an alternative or generalization of curvature tensor in Riemannian geometry. ==Definition== Let ''G'' be a Lie group and

E\to B

be a principal bundle. Let us denote the Lie algebra of ''G'' by

g

. Let

\omega

denotes the connection form on ''E'' (which is a ''g''-valued Differential form on ''E''). Then the curvature form is the ''g''-valued 2-form on ''E'' defined by :

\Omega=d\omega +{1\over 2}[\omega,\omega]=D\omega.

Here

d

stands for exterior derivative,

[*,*]

is the Lie bracket and ''D'' denotes the Connection form#Exterior covariant derivative. More precisely, :

\Omega(X,Y)=d\omega(X,Y) +{1\over 2}[\omega(X),\omega(Y)].

E\to B

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

E\to B

is a vector bundle then one can also think of

\omega

as about matrix of 1-forms then the above formula takes the following form: :

\Omega=d\omega +\omega\wedge \omega,

where

\wedge

is the Exterior power. More precisely, if

\omega^i_j

and

\Omega^i_j

denote components of

\omega

and

\Omega

correspondingly, (so each

\omega^i_j

is a usual 1-form and each

\Omega^i_j

is a usual 2-form) then :

\Omega^i_j=d\omega^i_j +\sum_k \omega^i_k\wedge\omega^k_j.

For example, the tangent bundle of a Riemannian manifold we have

O(n)

as the structure group and

\Omega^{}_{}

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

\Omega^{}_{}

is an alternative description of the curvature tensor, namely in the standard notation for curvature tensor we have :

R(X,Y)Z=\Omega^{}_{}(X\wedge Y)Z.

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

D\Theta=\Omega\wedge\theta={1\over 2}[\Omega,\theta]

, here ''D'' denotes the Connection form#Exterior covariant derivative and

\Theta

the Connection form#torsion. The second Bianchi identity holds for general bundle with connection and takes the form :

D\Omega=0.

==See also== *Chern-Simons form *Curvature of Riemannian manifolds *Gauge theory Differential geometry

See other meanings of words starting from letter:

C

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Words begining with Curvature_form:

Curvature_form