Curvature of differentiable Hilbert modules and Kasparov modules

TL;DR

This paper introduces a new notion of curvature for universal connections on Hilbert C*-modules relative to spectral triples, providing a framework that generalizes classical geometric curvature to noncommutative geometry.

Contribution

It defines the curvature of universal connections on Hilbert modules relative to spectral triples and refines this concept for Kasparov modules, connecting to classical Riemannian submersion curvature.

Findings

Curvature depends only on the represented form modulo junk forms.

The refined curvature captures all data of Riemannian submersion curvature.

Provides a new operator-theoretic approach to noncommutative geometric curvature.

Abstract

In this paper we introduce the curvature of densely defined universal connections on Hilbert $C^{*}$-modules relative to a spectral triple (or unbounded Kasparov module), obtaining a well-defined curvature operator. Fixing the spectral triple, we find that modulo junk forms, the curvature only depends on the represented form of the universal connection. We refine our definition of curvature to factorisations of unbounded Kasparov modules. Our refined definition recovers all the curvature data of a Riemannian submersion of compact manifolds, viewed as a $KK$-factorisation.