What the OP seems to be missing is that light follows the spacetime curvature created by the presence of matter whether normal or dark. Everything follows that same path, even neutrinos, just not as fast as photons.
No, not true. And that is why there is an element of the actual physics/mathematics (accidental or not) in his musings. Photons do not follow the same path through space-time as massive objects - the former follow null-paths and the latter, time-like paths. These can be quite different, especially where curvature is high (good example is the null orbit around a Schwarschild black hole at r=3M.)
I thought the path followed by both matter and energy simply followed the curved shape of space
Yes, it does. But in the pseudo-Riemannian geometry of 4d space-time, there are different types of path for massive and massless particles - we call these time-like geodesics and null (or light-like) geodesics. In our everyday Euclidean geometry, there is only type of path (or geodesic) so this is unfamiliar behaviour.
When you say "massive particle", is that any particle with mass?
Yes, no matter how small a mass.
Do the different paths through [4d]Reimannian space[-time] correspond to different paths in [our everyday 3d] Euclidean space?
Yes, though I don't think we'd ever be able observe the difference without some decent source of curvature, such as a black hole. I have mentioned this photon orbit that exists half again as far from the centre of Schwarzschild black hole as the event horizon. Photons will zip around on this circular orbit quite happily, but anything with mass will spiral into the event horizon.
ABE: ah, SG beat me to it - that's what happens when you write a post then forget to submit it for several hours
Edited by cavediver, : No reason given.
Edited by cavediver, : write not right as it was written right.
CD's answer about needing "some decent source of curvature" hints that we don't yet have experimental or observational confirmation of massive and massless particles following different paths - is that true?
Almost certainly true - we don't have observational confirmation of this. I think you would need a handy black hole and some distant oribital platforms to house a laser, a particle accelerator, and a couple of detectors.