Levi-Civita connections from toral actions

TL;DR

This paper develops a framework for differential calculus on certain C*-algebras influenced by toral symmetries, establishing a Levi-Civita connection, a Bianchi identity, and a Gauss-Bonnet theorem for specific cases.

Contribution

It introduces a new class of tame differential calculi from toral actions and proves the existence of a Levi-Civita connection and related geometric identities.

Findings

Existence of a unique Levi-Civita connection on the constructed calculus

Proof of a Bianchi identity in this noncommutative setting

A Gauss-Bonnet theorem for rank two tame calculus

Abstract

We construct tame differential calculi coming from toral actions on a class of $\mathrm{C}^*$-algebras. Relying on the existence of a unique Levi-Civita connection on such a calculus, we prove a version of the Bianchi identity. A Gauss-Bonnet theorem for the canonical tame calculus of rank two is studied.