To Horst Eckardt

This is an interesting and fundamental problem of Riemann geometry itself. All these equations come from the metric compatibility condition when using a diagonal metric. Solutions could be as follows:

1) A symmetric connection is mathematically allowed but not meaningful in physics.

2) The metric cannot be diagonal, could for example be antisymmetric off diagonal.

The overall problem is to find a metric from the commutator equation, metric compatibility and identity. I think we are making great progress by use of logic without any preconceptions or dogma. A symmetric connection is not allowed in the commutator method, but on a purely mathematical level could exist when the commutator method is not applied. Then it would a purely mathematical construct that is needed for self consistency in metric compatibility, but has no physical meaning. If a fundamental problem is found in Riemann geometry, it is also present in Cartan geometry and indeed all differential geometries. Without an antisymmetric connection there is no torsion. The computer can crunch out a mass of equations and is very useful. This could not have been done of course in 1900 when the subject began. So they probably made several errors that are being uncovered 110 years later.

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