Riemannian Geometry of Noncommutative Surfaces

TL;DR

This paper develops a Riemannian geometric framework for noncommutative surfaces, introducing metrics, connections, and curvature, aiming to lay groundwork for a noncommutative theory of gravity.

Contribution

It introduces a consistent Riemannian geometry for noncommutative surfaces, including metric, connections, and curvature, with examples and discussion on covariance.

Findings

Connections are metric-compatible, leading to noncommutative Riemann curvature.

Noncommutative analogues of classical surfaces are explicitly constructed.

The framework satisfies noncommutative Bianchi identities.

Abstract

A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the underlying structure of Einstein's theory of general relativity and led to further developments of the latter. The notions of metric and connections on such noncommutative surfaces are introduced and it is shown that the connections are metric-compatible, giving rise to the corresponding Riemann curvature. The latter also satisfies the noncommutative analogue of the first and second Bianchi identities. As examples, noncommutative analogues of the sphere, torus and hyperboloid are studied in detail. The problem of covariance under appropriately defined general coordinate transformations is also discussed and commented on as compared with other treatments.