Journal of Noncommutative Geometry

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Volume 11, Issue 2, 2017, pp. 507–536
DOI: 10.4171/JNCG/11-2-3

Published online: 2017-06-29

Riemannian curvature of the noncommutative 3-sphere

Joakim Arnlind[1] and Mitsuru Wilson[2]

(1) Linköping University, Sweden
(2) University of Western Ontario, London, Canada

In order to investigate to what extent the calculus of classical (pseudo-)Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civita’s theorem, which states that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivating example, we consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and explicitly determine the torsion-free and metric connection, as well as the curvature operator together with its scalar curvature.

Keywords: Noncommutative geometry, 3-sphere, Riemannian curvature, Levi-Civita connection

Arnlind Joakim, Wilson Mitsuru: Riemannian curvature of the noncommutative 3-sphere. J. Noncommut. Geom. 11 (2017), 507-536. doi: 10.4171/JNCG/11-2-3