Main Question or Discussion Point
when a space (or spacetime) is said to be maximally symmetric, does this mean that it is homogeneous?
It depends on what you mean by "symmetry." Generally, in this context, a "symmetry" is taken to be an isometry of the metric, i.e., a diffeomorphism which pulls back the metric to itself. If this is taken as the definition, then, as I said above, all maximally symmetric spaces share the same number of symmetries. This includes the maximally symmetric spaces of both closed and open varieties, i.e., hyperboloids and hyperspheres (in general relativity, the analogous spaces are the de Sitter and anti-de Sitter cosmologies).Does this mean a flat Euclidean space is more symmetric than spaces with closed and open curvature (hyperspheric and hyperbolic)?
Thanks for a quick introduction to the jargon in this area.Thus, in addition to a full complement of ordinary Killing vectors, Euclidean space is also blessed with a maximal collection of conformal Killing vectors, making it "uber-maximally symmetric," unlike hyperspheres and hyperboloids. So, indeed, there is a "third kind of invariance," as you say, having to do with changes of scale, that Euclidean space possesses but other so-called maximally symmetric spaces do not.
There are a few problems with attempting to invoke Noether's Theorem to produce a symmetry from conformal Killing vectors. First, in order for Noether's Theorem to apply, you need a differentiable symmetry of an action or Lagrangian of a physical system. Thus, Killing vectors and conformal Killing vectors do not give rise to conserved quantities for just any path through a manifold, but they do if the path can be described via an action proportional to the metric, such as the action defining geodesics:Noether would say for every symmetry there is an inertia or conservation law, so what is law that results from scale symmetry?