Riemannian manifolds in noncommutative geometry

TL;DR

This paper defines Riemannian manifolds within noncommutative geometry, establishing connections with spin^c structures, Kasparov modules, and Poincaré duality to extend classical geometric concepts to the noncommutative setting.

Contribution

It introduces a new definition of Riemannian manifolds in noncommutative geometry and links them to existing noncommutative spin^c structures using Kasparov modules.

Findings

Constructed Riemannian manifolds from noncommutative spin^c manifolds.

Established an analogue of Kasparov's fundamental class for Riemannian manifolds.

Clarified bimodule and first-order conditions for spectral triples.

Abstract

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fundamental class for a Riemannian manifold, and the associated notion of Poincar\'e duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.