Riemannian manifolds in noncommutative geometry

Details Riemannian manifolds in noncommutative geometry Riemannian manifolds in noncommutative geometry

2011-09-10

arxiv.org/abs/1109.2196v2

Mathematics

K-Theory and Homology

Examples and details of some topological issues added

Scientific paper

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.

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