Alpha Institute for Advanced Studies

The precise interpretation is that the commutator of covariant derivatives acting on any tensor produces the torsion and curvature tensors in any space of any dimension. The connection is not a tensor because it does not transform as a tensor under the general coordinate transformation, but is always antisymmetric in its lower two indices (mu and nu). The curvature tensor is always antisymmetric in mu and nu, the commutator is always antisymmetric in mu and nu. This means that of mu and nu are switched to nu and mu, the tensor changes sign. If mu is the same as nu the tensor is zero (all elements of it are zero). In general a non zero tensor may contain non-zero and zero elements. This is true in Riemann geometry itself and there exists a Riemannian torsion which is always non-zero., i.e. it is a non-zero tensor in general. Cartan’s geometry reduces to Riemann’s geometry, and in Cartan’s geometry the torsion is a vector valued two-form. A two-form is antisymmetric in mu and nu. The Cartan curvature is a tensor valued two-form. There can be other geometries too, but ECE uses Cartan’s elegant geometry because it is simple and profound. There seems to be no need for another geometry in natural philosophy. In the 166 ECE papers to date there are many proofs of all details of Cartan geometry, many of them check themselves. At first, tensor analysis and form analysis is difficult, looking like a blizzard of subscripts and superscripts, but I developed a basic notation which reveals its basic simplicity. This is

T = D ^ q; R = D ^ omega;
D ^ T := R ^ q

as given by Cartan, and my own identity (proven precisely in UFT 137):

D ^ T tilde := R tilde ^ q