How Many Structure Constants do Exist in Riemannian Geometry?

Abstract

After reading such a question, any mathematician or physicist will say that, according to a well known result of L.P. Eisenhart found in 1926, the answer is surely ”One”, namely the constant allowing to describe the so-called “ constant Riemannian curvature ” condition. The purpose of this paper is to prove the contrary by studying the case of two dimensional Riemannian geometry in the light of an old work of E. Vessiot published in 1903 but still totally unknown today after more than a century. In fact, we shall compute locally the Vessiot structure equations and prove that there are indeed “ Two ” Vessiot structure constants satisfying a single linear Jacobi condition showing that one of them must vanish while the other one must be equal to the known one or that both must be equal. This result depends on deep mathematical reasons in the formal theory of Lie pseudogroups, involving both the Spencer delta -cohomology and diagram chasing in homological algebra. Another similar example will illustrate and justify this comment out of the classical tensorial framework of the famous “ equivalence problem ”. The case of contact transformations will also be studied. Though it is quite unexpected, we shall reach the conclusion that the mathematical foundations of both classical and conformal Riemannian geometry must be revisited. We have treated the case of conformal geometry and its application in recent papers (Pommaret in J Mod Phys 12:829–858, 2021. https://doi.org/10.4236/jmp.2020.1110104; The conformal group revisited. arxiv:2006.03449; Nonlinear conformal electromagnetism. arxiv:2007.01710).