Noncommutative tori are ubiquitous examples of noncommutative spaces. Following the seminal work of Connes-Tretkoff, Connes-Moscovici, and others a differential geometric apparatus is currently being built. So far the main focus has been on conformal deformation of the (flat) Euclidean metric or product of such metrics. A new challenge is the accounting of the non-triviality of the modular automorphism group due to the lack of commutativity.
This talk will report on ongoing work to deal with general Riemannian metrics on NC tori (in the sense of J. Rosenberg). After explaining the construction of the Laplace-Beltrami operator in this setting, three main results will be presented. The first main result is a topological version of the Gauss-Bonnet theorem for NC tori. This extends the Gauss-Bonnet theorem of Connes-Tretkoff for conformally flat metrics. The second result is a microlocal Weyl law for noncommutative tori. This can be seen as a first step toward Quantum Ergocity on NC tori. The third result is a local index formula for NC 2-tori.