Notion of Parallelism on a General Manifold and Consequent Geometrical Specification of the Riemannian Curvature (Excerpts)

Abstract

Einstein ' s theory of relativity (now corroborated by the explanation of the famous secular inequality, revealed by observations on Mercury ' s perihelion, which was not predicted by Newton ' s law) considers the geometrical structure of space as very tenuously, but nonetheless intimately, dependent on the physical phenomena taking place in it, differently from classical theories, which assume the whole physical space as given a priori. The mathematical development of Einstein ' s grandiose conception (which finds in Ricci ' s absolute differential calculus its natural algorithmic instrument) utilizes as an essential element the curvature of a certain four-dimensional manifold and the Riemann symbols relative to it. Meeting these symbols---or, better said, continuously using them---in questions of such a general interest, led me to investigate whether it would be possible to somewhat reduce the formal apparatus commonly used in order to introduce them and to establish their covariant behaviour.1 Some progress in this direction is actually possible, and essentially forms the content of sections 15 and 16 of the present paper, which, initially conceived with this only purpose, gradually expanded to make some room for the geometric interpretation too.