How Many Structure Constants Do Exist in Riemannian Geometry

Abstract

After reading such a question, any mathematician will say that, according to a well known result of L.P. Eisenhart found in 1926, the answer is " One " of course, namely the only constant allowing to describe the so-called " {\it 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 {\it still totally unknown today} after more than a century. In fact, we shall compute locally the {\it Vessiot structure equations} and prove that there are indeed " Two " {\it Vessiot structure constants} satisfying a single {\it linear Jacobi condition} showing that one of them must vanish while the other one must be equal to the known one. This result depends on deep mathematical reasons in the formal theory of Lie pseudogroups, which are involving both the Spencer $δ$-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 " {\it 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 in a recent arXiv preprint.