Curvature and Weitzenbock formula for the Podle\'{s} quantum sphere

TL;DR

This paper establishes a unique Levi-Civita connection and computes the curvature, Ricci, and scalar curvature for the Podleś quantum sphere, revealing quantum-classical transition properties.

Contribution

It introduces a unique Levi-Civita connection on the Podleś sphere and derives a generalized Weitzenbock formula, extending classical geometric results to the quantum setting.

Findings

Scalar curvature is constant and converges to 2 as q approaches 1.

Derived a generalized Weitzenbock formula for the quantum sphere.

Computed the full curvature tensor, Ricci, and scalar curvature.

Abstract

We prove that there is a unique Levi-Civita connection on the one-forms of the Dabrowski-Sitarz spectral triple for the Podle\'{s} sphere $S^{2}_{q}$. We compute the full curvature tensor, as well as the Ricci and scalar curvature of the Podle\'{s} sphere using the framework of \cite{MRLC}. The scalar curvature is a constant, and as the parameter $q\to 1$, the scalar curvature converges to the classical value $2$. We prove a generalised Weitzenbock formula for the spinor bundle, which differs from the classical Lichnerowicz formula for $q\neq 1$, yet recovers it for $q\to 1$.