# Laplace-Beltrami Operators on Noncommutative Tori

## TL;DR

This paper develops Laplace-Beltrami operators for noncommutative tori with arbitrary metrics, extending classical geometric concepts to noncommutative geometry and accounting for modular automorphisms.

## Contribution

It introduces a construction of Laplace-Beltrami operators on noncommutative tori that generalizes classical properties and incorporates non-trivial modular automorphisms.

## Key findings

- Operators mimic classical Laplace-Beltrami properties
- Introduces noncommutative Riemannian density and volume
- Applicable to tori of any dimension

## Abstract

In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group of modular automorphisms. On the way we introduce notions of Riemannian density and Riemannian volumes for noncommutative tori.