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 spinc manifolds; and conversely, in the presence of a
spinc structure. We also show how to obtain an analogue of Kasparov’s fundamental class
for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way
we clarify the bimodule and first-order conditions for spectral triples.