Curvature form


In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.

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


(In another convention, 1/2 does not appear.) 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, on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field), and   is the inverse of the normalization factor used by convention in the formula for the exterior derivative.

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.

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, 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.

The Bianchi identities can be written in tensor notation as:  

The contracted Bianchi identities are used to derive the Einstein tensor in the Einstein field equations, the bulk of general theory of relativity.[clarification needed]

Notes edit

  1. ^ since  . Here we use also the   Kobayashi convention for the exterior derivative of a one form which is then  
  2. ^ Proof:  

References edit

See also edit