There are several methods for computing curvatures of manifolds. Most are based upon the construction of a connection, followed by a differential process to produce the concept of curvature.

One method presumes the existence of a metric, and by exterior differential and algebraic processes applied to the (symmetric) metric field, a connection (the Christoffel connection) is produced. A second differential process applied to this connection produces the concept of a metric based curvature.

Another method starts from the assumption that at each point on the manifold there exists a basis frame or matrix of functions, whose columns furnish a linearly independent set of basis vectors. Then by exterior differential and algebraic processes a right Cartan matrix of connection 1-forms is produced, and a second application of exterior differential and algbraic processes to the Cartan connection produces the concept of a Cartan matrix of curvature 2-forms.

The question is "What are the differences and similarities of the two methods?"

The metric point of view is championed by the General Relativity clique. However, the metric induced connection is without Affine Torsion.

The starting point of all of these ideas was based upon those cases where a set of new "coordinate" variables is well defined by a set of functions mapping an old set of "coordinate" variables into the new set. The mapping induces a linear mapping of old differentials into new differentials. The linear mapping, or Frame field, in this case can be chosen to be the Jacobian matrix of the coordinate mapping.

Contravariant and covariant tensor fields can be defined on both the initial and final state. If the final state is constrained by a choice of a metric, mapping the Contravectors onto the Covectors on the final state, then the Frame field, mapping contravectors of the initial state into contravectors of the final state, can be used to induce a compatible (pullback) metric on the initial state from a given metric on the final state.

The induced metric on the initial state may be used to compute a connection on the initial state, and this connection is defined as the Christoffel, metric based, connection on the initial state . The coefficients of the Christoffel connection are not the same as the coefficients of the right Cartan connection, except in special cases. The right Cartan matrix depends only on the the basis frame linking contravectors on the initial state to contravectors on the final state. The Cartan connection is independent from the choice of metric on the final state, the Christoffel connection depends upon the metric.

IF the mapping is to a euclidean space of constant metric coefficients, say [1,1,1], then the Frame induces a special metric on the initial state which can be used, via the Christoffel recipe, to generate a metric connection on the initial state. When the Jacobian matrix is used to play the role of a Frame matrix, then the Right Cartan Connection coefficients defined on the initial state, computed not from the metric but from the Frame field, are exactly equal to the Christoffel Connection coefficients deduced from the pullback metric on the initial state. This is the special case where both connections have identical coefficients.

The Christoffel connection is always free of "Affine Torsion", which depends upon certain anti-symmetries of the connection coefficients. The Cartan connection, based upon the Frame, and not a symmetrical metric, can have such anti-symmetries that lead to "affine torsion". It should be noted that if a metric on the initial state is given, and is diagonal, it is possible to form the "square root" of the diagonal metric to define a Frame. This often used technique is not the same as the method constructed in terms of the JAcobain of a coordinate mapping. The square root of the diagonal metric frame is often NOT integrable. Herein the integral case is studied first, and then perturbations are imposed on both the frame and the metric of the integrable system.

For the integrable case, if the final state has a metric which is a set of constants, the Christoffel connection on the final state must be zero. It follows that the curvature on the final state is zero. However, as the curvature is a tensor field, a zero curvature on the final state must relate to a zero curvature on the initial state. The metric based connection need not be zero on the initial state, even though a metric based connection is zero on the final state. (Connections are not tensor fields.) However, the curvature of the Cartan connection is always zero on the initial state when computed from a basis FRame that has an inversse.
Hence, if the metric curvature is non-zero, then it must be compensated by some other curvature, such that total curvature vanishes on the initial state. These compensation terms will be defined as inertial curvatures.

The result is a principle of equivalence, where metric curvatures are balanced by torsional curvatures. Such is the approach of G. Shipov, and his claim that the physical vacuum should be a parallelizeable An space.

The object of this exercise is to start with the simple case of a coordinate transformation to a euclidean space with a metric equal to the identity matrix. The Frame matrix of the simple case will be assumed to be the Jacobian of the coordinate mapping. The Frame will be a matrix of n x n functions with arguments on the the initial state. The right Cartan connection of 1-forms can be computed based on the Frame, [F], and the Christoffel connection can be computed in terms of the compatible metric [g] = [Ftranspose][F] induced on the initial state. The two methods for computing the connections lead to the same results. The coefficients of the Christoffel connection and the right Cartan connection are identical. The curvatures are also identical and equal to zero.

From this starting point it is possible to perturb the metric and the Frame separately. Perturbation of the Frame can lead to the concept of Torsion; perturbation of the (symmetric) metric does not lead to Torsion. When the Cartan connection has certain anti-symmetric coefficients, the perturbed Frame acting on the differentials of the initial state produces non-exact 1-forms on the final state (with arguments on the initial state). This result implies that there does not exist a unique map of functions producing the coordinate variables of the final state in terms of functions of the initial state.

Both metrical and Frame perturbtions will lead to the fact that the right Cartan connection coefficients based on the Frame are no longer identical to the Christoffel connection coefficients based on the metric -- all in terms of functions on the initial state.

To demonstrate these features, a Maple program has been used to create examples of a matrix Frame of functions, [ F ] , acting algebraically on a column vector of coordinate differentials of an initial domain, R = {r,theta,phi}, to produce a column vector of 1-forms |sigma> on a final range.

The starting point is the linear map [ F ] of 1-forms, and not the non-linear map of coordinates. If the 1-forms on the final state are exact, then there exist integrals uniquely defining the coordinates {x,y,z} of the final state in terms of functions of the variables of the initial state. The Frame field, whether integrable or not, also induces, via differential and algebraic processes, a (right) Cartan matrix [ C ] of connection 1-forms, such that the differential of any column vector of the Frame of basis vectors is a linear combination of all of the column vectors of the Frame. The differential process is then said to be closed. If any component of the column vector of 1-forms |sigma> is not integrable, the associated Cartan matrix of connection 1-forms is said to admit Affine Torsion. The lack of integrability implies that there does not exist a unique function of the independent variables that maps to the associated coordinate on the final state. There may be more than one such function (!) or none.

A metric [metricfinal] will be imposed arbirarily on the final range, and its compatible preimage [pullbackmetric] on the initial domain will be computed relative to the linear map,[ F ] .

A special symmetric Christoffel Connection, { Christoffel }, can be deduced on the initial state in terms of algebraic processes and exterior differentiations of the [pullbackmetric]. Christoffel (Riemannian) Curvatures on the initial state can also be deduced in terms of another exterior differential process applied to the Christoffel connection.

These concepts will be compared to the right Cartan Connection deduced from differential and algebraic processes applied to the Frame matrix, not the metric. The right Cartan matrix of Connection 1-forms is also defined on the initial state, and also can be used to compute curvatures.

In general, the Christoffel Connection is not the same as the Cartan connection. They are equivalent only in special situations.

The Christoffel connection is always free of Affine Torsion, the Cartan connection can support certain anti-symmetries defined as Affine Torsion.

The Cartan Curvatures, based on the Cartan connection, are not always the same as the Riemann curvatures based upon the Christoffel connection.

It is classic to decompose the Cartan Connection coefficients into the Christoffel connection coefficients, and another piece usually defined as the Ricci rotation coefficients, [T] . For purposes that will be obvious below, herein it is convenient to define the decomposition with a minus sign:

As the Christoffel connections are symmetric in the lower two indices, any anti-symmetries of the Cartan connection must be associated with the RIcci coefficients [T]. Following Shipov, it is assumed that the Ricci coefficients (via their induced curvatures) are the source of inertia forces (accelerations). The Christoffel coefficients (via their induced curvatures) are the source of gravitational metric forces (accelerations).

The Ricci Rotation coefficients [T] can have both a symmetric and an anti-symmetric part. The anti-symmetries are equivalent to the anti-symmetries of the Right Cartan connection, and are defined as the Affine Torsion coefficients appearing in the matrix of 1 forms defined as [Affine Torsion]. Hence, the Cartan connection is decomposed into three parts, as:

The matrix of Affine Torsion connection components is a matrix of 1-forms with certain anti-symmetric coefficients.

The same anti-symmetric set of coefficients can appear in a column vector of Affine Torsion 2-forms, which is defined by the matrix exterior product of the Right Cartan connection and the differential position vector on the initial state.

Note that the Ricci coefficients can contain symmetric as well as anti-symmetric parts. The symmetric parts can have contributions which depend upon the existence of the Affine Torsion coefficients, and contributions which are independent from the existence of Affine Torsion coefficients

The examples (given in the Maple program) will start with the classic map of spherical coordinates mapped into the Cartesian coordiantes {x,y,z}. The Jacobian matrix of partial derivatives of the mapping function will serve as the primitive definition of a Basis Frame, [F]. The Right Cartan connection matrix for any basis frame is defined as [ C ] = - [dG][F] = + [G][dF] , where [G] is the inverse of [ F ]. Realize that the Right Cartan matrix is defined entirely on the initial state in terms of initial state independent variables. The curvature and the Affine Torsion of the initial state based upon an integrable set of mapping functions are both zero.

Initially it will be assumed that the metric on the final state is a set of constants equal to the identity matrix. Then a perturbation of this finalstate metric will be defined and its effect on curvatures and torsion will be examined. Then a pertubation will be made on the original Frame Field, such that the perturbed Frame Field will admit Affine Torsion. The effect of the combined perturbations on curvature and Torsion also will be computed.

For the Maple program which gives 3D examples of the concepts described above, see

CARTAN CONNECTIONS, RICCI ROTATION, AFFINE TORSION, METRIC, CHRISTOFFEL CONNECTIONS, CURVATURE and CARTAN's EQUATIONS of STRUCTURE

A 4D version will appear shortly.

Epilogue:

If the total matrix of Cartan curvature 2-forms is zero, then the metric induced curvature is balanced by the Ricci rotation induced curvature. Hence Torsion induced curvature can influence, or be related to, the metric induced curvature, when the Ricci coefficients are not zero. An An space is a space of total curvature equal to zero. The simplest example is when the metric curvature and the Cartan curvature are equivalent. However, it is the suggestion of G. Shipov that the physical vacuum be not so simple. The Shipov vacuum is an A4 space, with a zero total curvature, as deduced from the connection. However, the associated metrical curvature need not be zero, and can be cancelled by the Torsion induced curvature from perturbed non-integrable Frame fields. Note that there can be other modes of cancellation due to metric perturbations (It is not clear that Shipov recognizes the possibilities due to metric perturbations).

In any case, the cancellation of metrical curvature by other curvature components of the Ricci rotations (including torsion) may be interpreted as the equivalence between gravitational (metric) mass and inertial (Ricci Rotation) mass.

See The Equivalence between Inertial and Gravitational Mass

A coordinate transformation from an initial domain of variables R = {r,theta,phi,tau} will be given to a final range of {x,y,z,t}. An unperturbed metric will be prescribed on the final state as [eta] = {-1,-1,-1,+1}. The Jacobian matrix of the map will be used as the unperturbed (invertible) Frame matrix of functions with arguments on the initial state. The Frame matrix will be used to compute the right Cartan matrix of 1-forms [C] on the initial state. The pullback metric [g], induced on the initial state by the compatibility condition [g] = [Ftranspose][eta][F], will be used to compute the Christoffel connection [Gamma] in terms of the metric on the initial state. The two connections will be compared and will be identical: [C] = [Gamma]. The curvatures induced by the two connections are also identical and equal to zero. Then two types of pertubations will be applied. The first perturbation will be to the metric on the final state. Again the Cartan Frame based connection [C] and the Christoffel metrical based connection [Gamma] will be computed on the initial state. The connections are no longer the same. The Cartan connection can be decomposed into two parts, a metrical part [Gamma] and a Ricci rotation part [T]:

[ C ] = [ Gamma ] - [ T ].

The Cartan matrix of curvature 2-forms will be computed by the formula

[Theta] = [dC]+[C]^[C].

As the Frame manifold has an inverse [G], the Cartan matrix of curvature 2-forms vanishes [Theta]. Such spaces are called A4 spaces, and are "flat". However, the total curvature formula can be written as

[Theta] = [dC]+[C]^[C] = {[dGamma]+[C]^[Gamma]} - {[dT]+[C]^[T]}.

The first term will be defined as metric curvature relative to the Cartan connection [C], and the second term will be defined as inertial curvature relative to the Cartan connection [C].

As the total curvature [Theta] is zero, it is apparent that the metric curvature is balanced by the inertial curvature. It is conjectured that this result is in effect the proper definition of what intuitively is known as the principle of equivalence.

The second type of perturbation will be on the Frame matrix, and will introduce torsion to the Cartan connection. There are two species of Torsion (besides left and right handedness). One species of Torsion will be due to "vorticity" around a space-space axis, and the other will be due to a "vorticity" around a space time axis. The Cartan connection again will be compared to the Christoffel connection and the Ricci rotation terms [T] are not zero. The conclusion holds even though the perturbation of the final state metric is zero. Again, the metrical curvature is balanced by the inertial curvature to produce a total zero Cartan curvature.

When both types of perturbations are present simultaneously interactions between the two pertubations can occur, but the fundamental result of metric - inertial equivalence still holds.

{[dGamma]+[C]^[Gamma]} = {[dT]+[C]^[T]}.

{[ metric curvature ]} = {[ inertial curvature ]}.