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Gauss-Bonnet-Chern Type Theorem For The Noncommutative Four-Sphere, Mitsuru Wilson

Electronic Thesis and Dissertation Repository

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 …