- journal article metadata
European Mathematical Society Publishing House
2017-07-02 23:45:01
Journal of Noncommutative Geometry
J. Noncommut. Geom.
JNCG
1661-6952
1661-6960
Global analysis, analysis on manifolds
General
10.4171/JNCG
http://www.ems-ph.org/doi/10.4171/JNCG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
11
2017
2
Riemannian curvature of the noncommutative 3-sphere
Joakim
Arnlind
Linköping University, Sweden
Mitsuru
Wilson
University of Western Ontario, London, Canada
Noncommutative geometry, 3-sphere, Riemannian curvature, Levi-Civita connection
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.
Functional analysis
Global analysis, analysis on manifolds
507
536
10.4171/JNCG/11-2-3
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-3