Antisymmetry of the Connection


This is easy to see from the attached paper 137, eq. (4). The mu and nu indices of the connection are the same as the mu and nu indices of the commutator. When mu and nu are the same, the commutator and connection vanish. This simple proof is enough to show that any Riemannian geometry based on a symmetric connection is incorrect (paper 139 develops this conclusion). Any work that uses a symmetric connection is incorrect. This is a simple fact of geometry, eq. (4) is easy to understand even by the non-specialist. The connection must be antisymmetric because the commutator is antisymmetric. This means that when mu is replaced by nu and vice versa, the commutator changes sign. This is well known. Similarly the connection is antisymmetric in the sense that when mu is replaced by nu and vice versa, the connection changes sign. All connections of Riemann geometry are antisymmetric in their lower two indices. Further details are given in the attached notes 138(10).




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