Riemannian manifolds in noncommutative geometry
  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. 
Steven Lord, Adam Rennie, Joseph C. Varilly
math.KT math.DG math.OA