Spectral geometry of functional metrics on noncommutative tori

We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obtained. In particular, the scalar curvature density and the total scalar curvature of functional metrics are explicitly computed in all dimensions for certain classes of metrics including conformally flat metrics and twisted product of flat metrics. Finally a Gauss-Bonnet type theorem for a noncommutative two torus equipped with a general functional metric is proved.

Keywords

Noncommutative curved tori, functional metrics, heat kernel expansion, scalar curvature, Connes pseudodifferential calculus, Gauss-Bonnet theorem

2010 Mathematics Subject Classification

Primary 46L87, 58B34. Secondary 58J42.

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Received 19 February 2021

Accepted 20 March 2024

Published 7 June 2024