On the Koszul formula in noncommutative geometry

TL;DR

This paper extends the Koszul formula to noncommutative geometry, constructs a spectral triple on a fuzzy sphere, and computes scalar curvature for a noncommutative Levi-Civita connection.

Contribution

It proves a Koszul formula for noncommutative Levi-Civita connections and applies it to spectral triples on fuzzy spheres.

Findings

Established a Koszul formula for noncommutative Levi-Civita connections

Constructed a spectral triple on a fuzzy sphere

Computed scalar curvature for the noncommutative Levi-Civita connection

Abstract

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.