Main Question or Discussion Point
What are the most widely accepted geometries of the universe?
I'm not sure what you mean by this. We have one current best-fit model for the spacetime geometry of the universe: it's an expanding universe with spatial slices of constant comoving time that are flat (Euclidean). The error bars in measurements still allow a small chance that the spatial geometry of slices of constant comoving time is a 3-sphere or an open "3-hyperbolic" geometry, either way with an extremely large radius of curvature (i.e., very close to flat). Is that what you're asking about?What are the most widely accepted geometries of the universe?