On the Gauss-Chern-Bonnet theorem for the noncommutative 4-sphere

Abstract

We construct a differential calculus over the noncommutative 4-sphere in the framework of pseudo-Riemannian calculi, and show that for every metric in a conformal class of perturbations of the round metric, there exists a unique metric and torsion-free connection. Furthermore, we find a localization of the projective module corresponding to the space of vector fields, which allows us to formulate a Gauss-Chern-Bonnet type theorem for the noncommutative 4-sphere.

References in corpus (5)

• The Gauss-Bonnet Theorem for the noncommutative two torus
• Modular curvature and Morita equivalence
• *-Compatible Connections in Noncommutative Riemannian Geometry
• Modular curvature for toric noncommutative manifolds
• The Moyal Sphere

Cited by in corpus (8)

• Levi-Civita connections for a class of spectral triples
• A Scalar Curvature Formula For the Noncommutative 3-Torus
• Gauss-Bonnet for matrix conformally rescaled Dirac
• A noncommutative catenoid
• An Atiyah Sequence for Noncommutative Principal Bundles
• Dirac operator on the quantum fuzzy four-sphere $S_{qF}^4$
• Noncommutative localization in smooth deformation quantization
• Projective real calculi over matrix algebras