The Semi-Symmetric Metric Connection – Part II: Mathematical Preliminaries

In the previous post in this series, I gave the rationale for undertaking this extended (re-)examination of the geometry of the semi-symmetric metric connection (SSMC): essentially, it represents (to my mind) the most ultra-minimalist extension to General Relativity (GR) at all possible – or so I thought back in the early 1990s – given that it introduces precisely one new object – a vector field – as part of the connection.

In gauge field theories the “connection” carries the gauge field, while the “curvature” corresponds to the field strength, a view that was argued in a book by Göckeler and Schücker (1989), which I had also been reading at that time. Since electromagnetism is often introduced as the archetypal gauge field in mathematical treatments of differential geometry (such as that by Göckeler & Schücker), it seemed to make intuitive sense to me that introducing electromagnetism into an extension of GR intended to model electromagnetism by way of a geometrical object might require it to enter by way of the connection, rather than as an additional field just lying around in spacetime, as it is in Einstein-Maxwell Theory (EMT). Hence, in this view, the SSMC is an obvious candidate.

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The Semi-Symmetric Metric Connection – Part I: The Background

Many years ago (getting close to 30 now), while doing my PhD (Voros 1996) in theoretical physics on mathematical extensions to General Relativity – and in particular, on Einstein’s own “unified field theory” – I happened across a book by Jan Schouten (1954) called Ricci-Calculus, which was an introduction (by a mathematician) to tensors and their applications, especially to geometrical thinking and analysis.

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