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

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 $q\to 1$.