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 Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.
Original language | English |
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Pages (from-to) | 1611-1638 |
Number of pages | 28 |
Journal | Journal of Geometry and Physics |
Volume | 62 |
Issue number | 7 |
DOIs | |
Publication status | Published - Jul 2012 |