Levi-Civita simplifies Einstein. The Ricci rotation coefficients and unified field theories
Résumé
This paper concerns late 1920 s attempts to construct unitary theories of gravity and
electromagnetism. A first attempt using a non-standard connection—with torsion and
zero-curvature—was carried out by Albert Einstein in a number of publications that
appeared between 1928 and 1931. In 1929, Tullio Levi-Civita discussed Einstein’s
geometric structure and deduced a new system of differential equations in a Rieman-
nian manifold endowed with what is nowadays known as Levi-Civita connection.
He attained an important result: Maxwell’s electromagnetic equations and the grav-
itational equations were obtained exactly, while Einstein had deduced them only as
a first order approximation. A main feature of Levi-Civita’s theory is the essential
use of the Ricci’s rotation coefficients, introduced by Gregorio Ricci Curbastro many
years before. We trace the history of Ricci’s coefficients that are still used today, and
highlight their geometric and mechanical meaning.