Clickable: Goedecke - Global embedding via coordinate basis vectors; http://doi.org/10.1140/epjp/i2011-11032-x. I am skeptical of the idea that the need for dark matter can be "conventioned away", but, while I am a mathematician, I'm not even glancingly familiar with the physics.

I don't have access to the paper right now, but the abstract is off to a very bad start.

> any curved Riemannian space must be a subspace of a larger flat host space

This is not even wrong. Yes, you can always embed a Riemannian manifold in a Euclidean space. In fact, there are infinitely many ways to do so. But this is extra structure that you can choose to impose, if you want to - it's not intrinsic.

You might as well say that every string must be a substring of a larger "host" string.

> the 4D Riemann-Christoffel curvature tensor is identically equal to a geometrical tensor associated with the complementary subspace of the host space

Impossibly vague. Yes, sometimes things are equal to other things. What tensor, and associated how?

> Einstein’s field equations are automatically geometrized, with the stress-energy tensor expressed in terms of the contracted complementary tensor

This is a tautology. The Einstein field equations relate the Ricci tensor to the metric and the stress energy tensor, and the Ricci tensor is a contraction of the Riemann curvature tensor. This remains true even if you rename the Riemann curvature tensor "the complementary tensor".

But the deeper problem is this: the entire point of general relativity is that physics does not, in fact, care what coordinates we use. And this is built into the mathematical structure of the theory - there can't be any such thing as the "the coordinate basis vector approach to tensor calculus", because tensors (properly speaking: tensor fields) are, by definition, coordinate-independent objects.

Sounds like a misunderstanding, I’d recommend reading the paper before jumping to conclusions. Certainly we covered the various tensors mentioned here in the class I took - if you want details, you have to read past the summary.

General Relativity largely describes how geometry is nature - what may seem obvious from a mathematical perspective isn’t always so from a physical one. The results of using basis vectors for derivations aren’t riddled with artifacts of any particular coordinate system, as it would become immediately clear when expressing Christoffel symbols, for example - they wouldn’t be equal with other approaches.

One of the other students who was getting her PhD specializing in GR said that using a vector approach gave her intuition about concepts she had previously considered incomprehensible.

> The results of using basis vectors for derivations aren’t riddled with artifacts of any particular coordinate system, as it would become immediately clear when expressing Christoffel symbols, for example - they wouldn’t be equal with other approaches.

But the Christoffel symbols are artifacts of a particular coordinate system - the real object is the connection, of which the Christoffel symbols are just basis-dependent components.

I must admit I haven’t touched the subject for twelve years. It sounds like you have a vested interest in dark matter. I think you’d have better luck talking to the author or reading the paper.

The arguments against it sounded plausible at the time.