Wikipedia Entry on Riemannian Geometry

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This entry is obsolete, incomplete and erroneous. Modern scholarship has shown that in Riemannian geometry, the curvature and torsion must coexist, that the torsion is identically non-zero, and that the connection is antisymmetric. The wikipedia entry on Riemannian geometry makes no mention of torsion, and the omission of torsion is incorrect. Except for the tiny minority of zealots who still adhere to the standard model, this conclusion is accepted by professionals worldwide as we know accurately from the feedback book attached to many of these recent blog entries. Wikipedia still adheres to an incorrect symmetric connection. With reference to the blog entry below to the Cartan structure equations, the omission of torsion (or equivalently the use of a symmetric connection), means:

T =? 0

This means that the Cartan Bianchi identity reduces incorrectly to

R ^ q =? 0

This is what is referred to incorrectly on wikipedia as “the first Bianchi identity”. It is obviously not an identity because it is true if and only if the torsion is zero. The correct identity is the Cartan Bianchi identity:

D ^ T := q ^ R

which is proven in many papers on this site, and which is given in chapter three of a standard text such as that by Carroll cited below in a blog posting. The connection is antisymmetric because there is a very simple one to one relation between the connection and the commutator of covariant derivatives. The commutator and connection are antisymmetric by definition, which means that T is never zero. Another consequence of the incorrect omission of torsion is that

R tilde ^ q =? 0

The correct identity is the Cartan Evans identity

D ^ T tilde := q ^ R tilde

which is a special case of the Cartan Bianchi identity. In many papers on this site it has been shown through the use of computer algebra that the metrics of the Einstein field equation produce, in general

R tilde ^ q not zero

even though T tilde is zero. This violates basic geometry and is incorrect, a disaster for the standard model. The reason is that the Einstein field equation incorrectly uses zero torsion, i.e. incorrectly uses a symmetric connection. The latter is incorrectly related to the metric in standard physics. In consequence the whole of twentieth century cosmology is meaningless, and in modern scholarship has been replaced by a cosmology based on torsion and ECE theory as described in great detail on this site.

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