Notes 137(15) and 137(16) below are definitive, self-checking, proofs of the Cartan identity and its dual identity. These are identities of the Riemannian manifold (S. P. Carroll, “Spacetime and Geometry: an Introduction to General Relativity” (Addison Wesley, New York, 2004), chapter 3, and downloadable notes). With reference to his chapter three, Carroll leaves the proof in 137(15) as an exercise for graduate students in theoretical physics. I have written it out in full and first gave it in paper 15. It was again given in paper 102. This proves the correctness of Cartan geometry, and the correctness of ECE theory. The huge success of ECE theory (attached) is based directly on this geometry, developed in the Riemannian manifold used in physics. Note 137(16) repeats note 137(15) for the dual identity (Cartan Evans identity) and note 137(16) is again self checking. This is all I need to say mathematically, any “criticisms” of self checking proofs can be discarded immediately as unscientific. The ECE theory can be developed in other manifolds, for example non-orientable manifolds, but that kind of development must be reduced to vector format in order to be useful for engineers. This has been done for ECE theory, which my colleagues and I have developed into a new school of thought in physics. The ECE theory is rigorously correct in the Riemannian manifold, and the torsion is defined in the Riemannian manifold in ECE theory. It is therefore the well known Riemannian torsion, omitted incorrectly in the standard model.

## Torsion in the Riemannian Manifold

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