Investigations in noncommutative differential and Riemannian geometry

Abstract

This dissertation consists of several related investigations in noncommutative differential and Riemannian geometry. The motivation is the study of Maxwell’s equations on noncommutative spaces.

First, we study noncommutative U(1)-gauge theory on commutative base spaces by characterizing relative gauge potentials and gauge transformations in terms of group cohomology on Z. Next, we compute noncommutative analogs of diffeomorphism groups for irrational noncommutative 2-tori and the standard Podle´s sphere. Finally, we study an inner differential calculus on the symmetric group S3 and show that the space of Hodge star operators associated with its noncommutative Riemannian geometry is characterized by two independent parameters. Using this result, we make progress towards investigating the existence and uniqueness of the Levi–Civita connection.