Quantum Riemannian geometry of quantum projective spaces

TL;DR

This paper develops the quantum Riemannian geometry of quantum projective spaces, computing key geometric tensors and establishing a quantum Einstein condition with scalar curvature, extending classical geometric concepts into the quantum realm.

Contribution

It introduces explicit calculations of Riemann and Ricci tensors for quantum projective spaces and demonstrates a quantum Einstein condition, advancing quantum geometric theory.

Findings

Riemann tensor is a bimodule map

Ricci tensor is proportional to the quantum metric

Scalar curvature computed for quantum projective spaces

Abstract

We study the quantum Riemannian geometry of quantum projective spaces of any dimension. In particular we compute the Riemann and Ricci tensors, using previously introduced quantum metrics and quantum Levi-Civita connections. We show that the Riemann tensor is a bimodule map and derive various consequences of this fact. We prove that the Ricci tensor is proportional to the quantum metric, giving a quantum analogue of the Einstein condition, and compute the corresponding scalar curvature. Along the way we also prove several results for various objects related to those mentioned here.