On the Koszul formula in noncommutative geometry

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.

To appear in Reviews in Mathematical Physics

Cited by in corpus (6)

• Gauge theories on quantum spaces
• Levi-Civita connections and vector fields for noncommutative differential calculi
• Cartan structure equations and Levi-Civita connection in noncommutative geometry
• Levi-Civita connections on quantum spheres
• Quantum Riemannian geometry of quantum projective spaces
• Levi-Civita connections from toral actions