What is the symmetry group of the configuration space of scale factors of pseudo-Riemannian, symmetric metrics on a three-dimensional manifold?
see
Isometric immersions of Riemannian Spaces in Euclidean Spaces
- É. G. Poznyak &
- D. D. Sokolov
Journal of Soviet Mathematics volume 14, pages1407–1428(1980)Cite this article
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That's the wrong question to ask-the scale factor itself doesn't have particularly interesting properties, beyond being a smooth function on the manifold.
The interesting question is what are the manifolds that are invariant under action of the conformal group. The answer to that question is known and is the basis of the AdS/CFT correspondence-and there's a lot, still, to understand here.
Definitely the perception of Riemannian on geometry is fascinated, but it is totally incorrect to apply it on universe. 1) his knowledge on universe was not adequate to write something on universe.
2) our universe with several hundred billions of galaxies can not be described by math, or mechanical perception.
regards
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