The Semi-Symmetric Metric Connection series

Part I – The Background
The intention is to describe the work I did for many years (starting about 1989/1990 or so) on the semi-symmetric metric connection (SSMC) as a possible extension to the (Riemannian) geometry of GR, which was done in order to see whether it might be able to model the addition of the electromagnetic field to GR in a geometrically-unified way, given that the SSMC adds only a single new object to Riemann geometry – to wit, an object of precisely a most very-highly suggestive form, namely, a 4-vector. I spent a fair bit of time trying to nut this idea out over the years, but I have never quite got there… It always seems as though there is some clever trick or an important insight that is dangling just beyond reach… I hope that by making these explorations public and open – and having to clarify and explain to others what I had tried to do all those years ago (and since) – it might lead to someone else examining these ideas and might possibly nudge them to have a try, and yield a coherent mathematical theory which, one hopes, could be tested – both for internal consistency and for empirical validity.
Part II: Mathematical Preliminaries
This post establishes some basic definitions for some mathematical operations that will be required in the next post (Part III) where we will examine the geometry of the SSMC as a precursor to seeking field equations that may be implicitly contained in the geometry. I will work using the component notation (as opposed to the more elegant component-free form) for tensors, as well as working within a co-ordinate frame (as opposed to non-coordinate frames) which simplifies the number of symbols used.
Part III: The Geometry
OK, with the mathematical preliminaries suitably dealt with, let’s go…
As noted previously in Part II, we’ll be using the tensor component notation and assuming a co-ordinate basis (together with the misuses of geometrically precise language that comes with that choice; my bad).
Part IV: General Relativity (pending…)
With the underlying geometry of the space defined by the semi-symmetric metric connection (SSMC) having been explored, we’re now in a position to examine how Einstein derived his field equations for GR. We will be seeking to follow similar physically-motivated reasoning, such as he used for GR, in our search for candidate field equations which might add electromagnetism to GR based on the geometrical properties of the SSMC.
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