Antisymmetry of the Connection in Riemann Geometry


The antisymmetry of the connection is easy to understand at A level or advanced O level in the British system. It comes about because the lower two indices of the connection, mu and nu, are the same as the indices of the commutator. The latter is zero if mu is the same as nu, so the connection is zero if mu is the same as nu. Anyone who tries to reject such a simple deduction is unreliable and should be viewed as a pseudoscientist. This is the overwhelming verdict of the scientific and related communities. The error in the obsolete Einsteinian physics was to neglect the Riemannian torsion. Why this was done is not clear, and needs historical scholarship. If the Riemannian torsion is neglected, then only the Riemannian curvature tensor can take the commutator symmetry, and as is well known, the curvature tensor is antisymmetric in mu and nu and vanishes if mu is the same as nu. The symmetry of the connection within the curvature tensor is not however defined if the torsion is incorrectly neglected, (i.e. incorrectly assumed to be zero), and so the mythology was propagated for over ninety years (1915 to present) that the connection can be symmetric, or even worse must be symmetric. This has become a catastrophe for the subject of physics, because it has become empty propaganda using the media as any political party would do. This procedure has nothing to do with Baconian science, which is the investigation of nature. The connection in Riemannian and Cartan geometries is antisymmetric in its lower two indices.


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