Here is a simple geometric explanation of the redshift in a static Einstein (1917) model which is a scaled . The issue turns on the fact that the obvious frequency-wavelength relation that holds for wave propagation on the plane does not hold for wave propagation on a sphere. Now Einstein’s static universe model is a scaled . The wave equation on an -sphere,

can be solved explicitly by very standard methods in physics, separation of variables and using spherical harmonics (see for example orthoganalpolnomialsinddimesnions for a treatment of spherical harmonics). The frequency of the waves on are

This is in sharp contrast to the frequency of a wave on a circle of radius which is . It is this discrepancy that explains the observed redshift.

An intrinsic curvature for a closed universe has a positive cosmological constant without any need for a dark energy, and of course the universe is static and will not have any expansion since redshift is a geometric phenomenon. De Sitter relativity is one approach that can apply, but more interesting is the fact that quantization of energy can be explained easily also as a geometric feature.