Curvature, area and Gauss-Bonnet formula of the Moyal sphere

TL;DR

This paper investigates the geometric properties of the Moyal sphere, including curvature and area, and demonstrates that the Gauss-Bonnet formula remains valid despite noncommutativity, revealing how these properties evolve with the noncommutative parameter.

Contribution

It provides explicit calculations of curvature, area, and the Gauss-Bonnet formula for the Moyal sphere, including a generalized version with two noncommutative parameters.

Findings

Scalar curvature and area revert to classical values as noncommutative parameter approaches zero.

Area decreases with increasing noncommutative parameter, approaching zero.

Total curvature integral satisfies the Gauss-Bonnet formula regardless of noncommutativity.

Abstract

We studied some geometric properties of the Moyal sphere. Using the conformal metric of the sphere in ordinary space and the matrix basis, we calculated the scalar curvature, total curvature integral and area of the Moyal sphere. We found that when the noncommutative parameter approaches to 0, the scalar curvature and area of the Moyal sphere return to those of the ordinary sphere. As the noncommutative parameter increases, the area of the Moyal sphere will decrease and eventually approach to 0. We found that the total curvature integral of the two-dimensional Moyal sphere still satisfies the usual Gauss-Bonnet formula and does not depend on the noncommutative parameter. We also calculated the approximate expression of the conformal metric with a constant curvature and obtained the corresponding correction function. In addition, we studied a type of generalized deformed Moyal sphere with two noncommutative parameters and obtained similar results.