**Subject:** Ineluctable Link between Torsion and Curvature

**Date:** Wed, 30 Apr 2008 06:57:43 EDT

This is well known to scholars of geometry and topology (e.g. Carroll chapter three), and comes from the action of an operator, called the commutator of covariant derivatives, on the general vector in n dimensions. The most rigorous application of this theorem is to be found in the well read paper 109, posted earlier this month. The commutator is sometimes called the round trip operator, and is well known in field theory. Gauge theory was developed from it as in Ryder’s book, “Quantum Field Theory” (Cambridge 1996). The commutator defines the torsion tensor and the curvature tensor. The torsion differential form and the curvature differential form of Cartan are defined from the tensors by multiplying them with a tetrad. EH was developed before this was known in geometry, and EH used the symmetric Christoffel connection, which is defined on the basis of the metric compatibility condition and the symmetric metric. However, the commutator acting on a vector does not assume metric compatibility and does not assume a symmetric metric, so does not assume the Christoffel connection. Prior to paper 93 on _www.aias.us_ (http://www.aias.us) it was thought that the torsion could be omitted from gravitational theory. However papers 93 to 110 prove that it cannot be omitted without violating the Hodge dual of the Bianchi identity. The complete argument is given in paper 109. The duality invariance in four dimensions of the Bianchi identity means that in four dimensions, the original identity and its Hodge dual have the same mathematical structure. So all field equations of ECE are duality invariant. The fundamental reason for this is that the Hodge dual in four dimensions of a two-form (rank two antisymmetric tensor) is another two-form. The Cartan torsion is a vector valued two-form, and the Cartan curvature is a tensor valued two-form. Each is antisymmetric in its last two indices.

Finally, the Riemann tensor is a type of curvature tensor in which the connection is the Christoffel connection. In general neither the metric nor the connection are symmetric.

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