Curvature form
Category:Differential geometry
In
differential geometry, the
curvature form describes
curvature of
principal bundle with
connection. \nIt 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 the Lie algebra of G.
Let denotes the connection form, a 1-form on E with values in g. \nThen the curvature form is the 2-form with values in g defined by
-
here stands for
exterior derivative, is the
Lie bracket and
D denotes the exterior covariant derivative\nMore precisely,
-
If is a fiber bundle with structure group
G one can repeat the same for\nthe associated principal
G bundle.
If is a vector bundle then one can also think of as \nabout matrix of 1-forms then the above formula takes the following form:
-
where \nis the
wedge product. \nMore precisely, if and denote \ncomponents of and corespondently, \n(so each is a usual 1-form and \neach is a usual 2-form) then
-
For example, the
tangent bundle of a
Riemannian manifold we have as the structure group and is the 2-form with values in (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 connection with tosion on the frame bundle) takes form
- ,
here D denotes the exterior covariant derivative and the torsion.
The second Bianchi identity holds for general bundle with connection and takes form
-
See also
