This is very interesting. I think that the trace antisymmetry constraint in its original form must be the right one to use. So I will proceed like this.
To: EMyrone@aol.com
Sent: 11/12/2017 14:58:12 GMT Standard Time
Subj: Re: ExamplesThe examples were developed with the Riemannian elements Gamma^rho_mu nu (without tangent space indices). We can see later if these can be rewritten to the form Gamma^a_mu nu.
Horst
Am 11.12.2017 um 15:47 schrieb EMyrone:
These examples would be very useful. What about gamma sup a sub mu nu, do non zero diagonal elemnets exist for this? These are very interesting developments. So I will revert to the original Lindstrom constraint, which is a fundamental equation of physics and mathematics.
To: EMyrone, Sent: 11/12/2017 14:22:02 GMT Standard Time
Subj: Re: Eq. (42) of UFT298I will give some examples for non-vanishing diagonal elements of the Christoffel connection in the text book I am preparing. I will see that I can finish the first chapter over the holidays and then will send it to you as a draft.
Horst
Am 10.12.2017 um 13:14 schrieb EMyrone:
This is where the trace antisymmetry constraint first appears, and uses the fact that the trace of an antisymmetric tensor or matrix is zero. The excellent Eckardt Lindstrom papers UFT292 – UFT299 are by now all heavily studied classics. This can be seen in the daily report I send out every morning. In the case of ECE2 field tensors each element of the trace is zero as is well known. The fundamental property of a square matrix is that it is made up of a sum of symmetric and antisymmetric square matrices. This is described by Stephenson, “Mathematical Methods for Science Students”, pp 292 ff. (fifth impression 1968). Every one of the antisymmetric matrices used by Stephenson has diagonal elements which are all zero. This was my undergraduate text book and it is signed by me with the address: “Brig y Don”, Sea View Place, Aberystwyth. I would like to ask Horst where the antisymmetric connection with non zero elements but zero trace first makes an appearance. I guess that it is the classic UFT354, but I cannot find it there. This is an interesting idea which I would like to study further. If the connection is defined by the commutator as starting point, instead of metric compatibility, the connection is antisymmetric with all diagonal elements zero, so is the curvature, so is the commutator of covariant derivatives.