  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.
