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 $$\delta $$ 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 “ 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).

From Control Theory to Gravitational Waves

“…and refer the reader to [LAP] for more details about the link between this result and the deformation theory of algebraic structures. We also notice that R 1 is formally integrable and thus R 2 is involutive if and only if ω has constant Riemannian curvature along the well known result of L. P. Eisenhart in 1926 [31]. The only structure constant c appearing in the corresponding Vessiot structure equations is such that c c a = (See also [31] for details).…”

“…Using the Vessiot structure equations [31], we notice that α is not invariant by the contact Lie pseudogroup. The associated invariant geometric object is a 1-form density ω leading to the system of infinitesimal Lie equations in Medolaghi form:…”

Killing Operator for the Kerr Metric

“…Let us recall a few facts from Riemannian geometry. A metric ω = (ω ij ) ∈ S 2 T * with det(ω) = 0 is providing the Christoffel symbols γ = (γ k ij ) as geometric objects according to the forgotten work of E. Vessiot in 1903 [7,24,27]. The Riemann tensor…”

“…Both are depending on a certain number of constants like the single geometric structure constant of the constant Riemannian curvature for the first and the many algebraic structure constants of Lie algebra for the second. However, Cartan and followers never acknowledged the existence of another approach which is therefore still totally ignored today, in particular by physicists ( [22][23][24]). Now, it is well known that the structure constants of a Lie algebra play a fundamental part in the Chevalley-Eilenberg cohomology of Lie algebras and their deformation theory ( [14]).…”

“…However, as we shall see, there are even simple academic systems depending on parameters but such that a convenient system (say involutive) may no longer depend on the parameters. Also, in a totally independent way still not acknowledged, E. Vessiot has shown that certain operators may depend on geometric objects satisfying non-linear structure equations that are depending on certain Vessiot structure constants c. The simplest example is the condition of constant Riemannian curvature [7,24] which is necessary in order that the Killing system becomes FI but we shall prove that the case of contact structures is similar. In such situations, we shall prove that the extension modules only depend on these constants.…”