Notion of Parallelism on a Generic Manifold and Consequent Geometrical Specification of the Riemannian Curvature

TL;DR

This paper from 1916 introduces a geometric concept of parallelism on Riemannian manifolds, offering a new perspective on Riemannian curvature with simplified formalism compared to earlier literature.

Contribution

It presents a novel geometric interpretation of parallelism and curvature in Riemannian geometry, reducing complex formalism of the field.

Findings

Introduces the notion of parallelism on Riemannian manifolds

Provides a geometric explanation for Riemannian curvature

Simplifies the mathematical formalism of the theory

Abstract

With regard to classical differential geometry, this paper written in 1916 by T. Levi-Civita introduces the notion of parallelism for a Riemannian manifold of arbitrary dimensions. It also provides a geometrical explanation for the Riemannian curvature and at the same time significantly reduces the mathematical formalism compared with the scientific literature of the time.