Non-Zero Trace of curvature 2-forms

It is a remarkable result that for any frame field [F] with inverse [Finverse], which implies that the manifold under study is parallelizable, it is possible to compute the matrix of curvature 2-forms and find that the trace of the matrix is zero independent of any group constraint place upon the Frame. (This is related to the idea that the first Chern number vanishes.)

Actually the situation is even more dire, in the sense that every matrix element of the matrix of curvature 2-forms vanishes for all frame fields with inverse.

If the connection generated from the Frame field is considered independent from any metric that can be imposed on the manifold, then it is possible to construct a connection in terms of metric functions alone and their partial derivatives. This connection is defined as the connection of Christoffel.

Now the curvature of the Christoffel metric based connection need not be the same as the Cartan Frame based connection. The two connections are precisely the same if the Frame matrix generates a set of integrable 1-forms, thereby mapping an initial set of variables y into a final set of coordinate variables, x, and the metric on the final state is a set of constants. The pullback metric on the intial state need not be a set of constants, but the two connections are identical and generate the same set of curvature 2-forms, with each matrix element equal to zero.

Now suppose that the metric on the final state is not a set of constants and in addition the Frame field generates a set of 1-forms on the final state that are not integrable. These are two separate concepts. The compatible pullback metric on the initial state may be used to compute a Christoffel connection on the initial state, which can be compared to the Frame based (non-integrable) connection defined as the Cartan connection. The two connections are no longer the same. Moreover, the formulas used to generate the zero curvature (on the initial state based upon the Cartan connection) can be decomposed into two parts: a metric based part based on [Gamma] and another part, based upon

[W] = [Gamma] - [C].

It is possible to define a "metric" curvature matrix of 2-forms from the formula

metric curvature = d[Gamma] + [C]^[Gamma]

and "inertial" curvature matrix of 2-forms from the formula

inertial curvature = d[W] + [C]^[W].

Although [W] is not at all the same as [Gamma], the two matrices of curvature 2-forms generated from [W] and from [Gamma] are identical.

THat is

Metric curvature = inertial curvature

thereby establishing a principle of equivalence.

Moreover, the trace of the identical matrices of curvature 2-forms so generated need not be ZERO.

A certain class of examples have been studied at

It is remarkable that in the case that the Frame field is not integrable, or if the final state metric is not constant, then the trace of the metric (or inertial) matrix of metric ( or inertial) curvature 2-forms is not Zero.

In the case that the Torsion coefficients of the Cartan connection is not zero, then it is even more remarkable that not only is the trace of matrix of metric curvature 2-forms not zero, but the trace as a 2-form has a finite exterior derivative.

Hence the trace of the curvature 2-forms has the properties of the G(D,H) 2-form of Maxwell EM theory and there exists a possibility of a 4-current, J, generated by the affine torsion coefficients.


MOREOVER, the example indicates THAT FINITE TORSION IS NECESSARY to produce the "current".
(This is a conjecture not proven in general as yet).

THESE RESULTS ARE TO BE COMPARED TO THE work of Scofield WHO CONJECTURED THE EXISTENCE OF a NON-ZERO TRACE OF A closed MATRIX OF CURVATURE 2-FORMS (without precisely defining how to get such a matrix). The difference between the the two formalisms is that Scofield's trace is assumed to be exact, where the example construction mentioned herein, has a trace which is not exact.

More over, the exterior derivative of the inertial (or metric) curvature is NOT zero.
But the "covariant with respect to the Cartan connection" derivative does vanish.

d[inertial curvature] + [C]^[inertial curvature]

= {d[C] + [C]^[C]}^[W]

= {Cartan curvature}^[W] = Zero.


d(trace[inertial curvature]) = - trace( [C]^[inertial curvature]) = non-zero 3 form = J.

The exterior derivative of the inertial (or metric) matrix of curvature 2-forms, as constructed above, is not zero.

Click here for Maple Example

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Last update 07/12/2001
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