# Curvature in Noncommutative Geometry

## TL;DR

This paper reviews recent progress in defining and understanding curvature within noncommutative geometry, highlighting key theoretical advances and computational techniques inspired by spectral geometry and heat kernel methods.

## Contribution

It provides an overview of the latest developments in noncommutative curvature, including the extension of classical concepts and new computational approaches.

## Key findings

- Proof of Gauss-Bonnet theorem for noncommutative two torus
- Development of local curvature invariants using spectral methods
- Introduction of new ideas for explicit curvature computations

## Abstract

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.