Riemannian geometry was created by Riemann as the evolutionary step from Gauss’ work on surfaces as intrinsic spaces. But Riemann did not come up with the Riemannian manifold without direct experience with electromagnetism and the conception of higher dimensional spaces for grand unification. Yes, Kaluza-Klein and Weyl produced some concrete possibilities, as does the simple S4 model but the project of Riemann and Gauss are still active because they were interested in discovering the laws governing nature as much as the physicists.

When it comes to the genius of the project undertaken by Riemann but I think not completed, it is the right way to understand Gauss’s discovery of intrinsic geometry of surfaces. It is natural to expect that Riemann’s ultimate goal would be to attempt to provide the support for the geometry of nature. The ambition of human capacity to decode nature geometrically was subject of controversy from Kant’s positioning of Euclidean geometry. Riemann’s solution, of clear pure mathematical interest as shown by the analytic work with bounds on Ricci curvature we have seen throughout the past century, was originally introduced to describe the geometry of the universe and to explain electromagnetic laws.

The empirical facts surrounding the distribution of heat in the universe starting with Penzias and Wilson’s discovery of the CBR (cosmic background radiation) in the 1960s was not available to Riemann and therefore the analysis of heat equation could not be applied to show compactness of the universe. I think the conclusion is extremely useful to understand the extent to which Riemann’s approach to description of the universe is right. One cannot forget that Riemann knew about Gauss’ doubts about applicability of Euclidean geometry to describe space. I am fairly lucky in fitting the redshift slope on a relatively tractable shape for the universe, a four-sphere. The issue had obviously laid at the foundations of geometry from the Riemannian view. Of course orthodox directions in physics had produced a different type of geometry for the universe based on the Einsteinian picture. I still believe that four spatial dimensions, compactness and four-sphere are defensible deductions for their relative simplicity.

Morris Kline says that Riemann followed Gauss for geometry and Cauchy and Abel for function theory, and he was also influenced by teachings of psychologist Friedrich Herbart (1776-1841). Learning something about these teachings seems like an interesting endeavor for me because I am keenly aware of Nietzsche’s prediction that psychology would be crowned as the queen of science along with the reality of a four dimensional universe where for me it is not even clear what a photon should be, geometrically characterized with minimal assumptions (removing even the orthodox assumptions about the force of electromagnetism).

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