Curvature and geometric modules of noncommutative spheres and tori

TL;DR

This paper introduces noncommutative analogues of projection operators for spheres and tori, defining noncommutative tangent spaces as projective modules and computing their scalar curvature, extending classical geometric concepts.

Contribution

It presents a novel formulation of noncommutative tangent spaces using projection operators, and calculates the scalar curvature within this framework.

Findings

Existence of noncommutative projection operators for spheres and tori

Definition of noncommutative tangent spaces as projective modules

Calculation of scalar curvature for these modules

Abstract

When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connection from Euclidean space, and we compute its scalar curvature.