Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Mathematics

Supervisor

Masoud Khalkhali

Abstract

We introduce a pseudo-Riemannian calculus of modules over noncommutative al- gebras in order to investigate to what extent the differential geometry of classical Riemannian manifolds can be extended to a noncommutative setting. In this frame- work, it is possible to prove an analogue of the Levi-Civita theorem. It states that there exists at most one connection, which satisfies torsion-free condition and metric compatibility condition, on a given smooth manifold with fixed metric. More signif- icantly, the corresponding curvature operator has the same symmetry properties as the classical curvature tensors. We consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and the noncommutative 4-sphere and explicitly determine the torsion-free and metric compatible connection, and we compute its scalar curva- ture. In the case of the noncommutative 4-sphere, we compute the scalar curvature of conformal perturbations of the round metric by localizing the algebra of noncommu- tative 4-sphere, which allows us to formulate and prove a Gauss-Bonnet-Chern type theorem. For the case of the noncommutative 4-torus, the Pfaffian of the curvature form for a conformal class of the flat metric is computed.


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