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 2forms 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 2forms 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 1forms, 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 2forms, 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 1forms 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 (nonintegrable) 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 2forms from the formula
metric curvature = d[Gamma] + [C]^[Gamma]
and "inertial" curvature matrix of 2forms from the formula
inertial curvature = d[W] + [C]^[W].
Although [W] is not at all the same as [Gamma], the two matrices of curvature
2forms 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 2forms 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 2forms 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 2forms not zero, but the trace as a 2form has a finite exterior
derivative.
Hence the trace of the curvature 2forms has the properties of the G(D,H) 2form
of Maxwell EM theory and there exists a possibility of a 4current, J, generated
by the affine torsion coefficients.
IN OTHERWORDS, AFFINE TORSION OF THE CARTAN CONNECTION APPEARS TO BE ASSOCIATED
WITH A MASS or charge (??) CURRENT DENSITY.
MOREOVER, the example indicates THAT FINITE TORSION IS NECESSARY to produce the
"current".
THESE RESULTS ARE TO BE COMPARED TO THE work of
Scofield WHO CONJECTURED THE EXISTENCE OF a NONZERO TRACE OF A closed
MATRIX OF CURVATURE 2FORMS (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.
d[inertial curvature] + [C]^[inertial curvature]
= {d[C] + [C]^[C]}^[W]
= {Cartan curvature}^[W] = Zero.
HEnce
d(trace[inertial curvature]) =  trace( [C]^[inertial curvature]) = nonzero 3
form = J.
The exterior derivative of the inertial (or metric) matrix of curvature 2forms,
as constructed above, is not zero.
