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 spinc manifolds; and conversely, in the presence of a spinc structure. We also show how to obtain an analogue...

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Bibliographic Details
Published in:Journal of geometry and physics Vol. 62; no. 7; pp. 1611 - 1638
Main Authors: Lord, Steven, Rennie, Adam, Várilly, Joseph C.
Format: Journal Article
Language:English
Published: Elsevier B.V 01.07.2012
Subjects:
Kasparov product
Noncommutative geometry
Spectral triple
Spectral triple
Noncommutative geometry
Kasparov product
ISSN:0393-0440, 1879-1662
Online Access:Get full text
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Summary: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.
ISSN:0393-0440
1879-1662
DOI:10.1016/j.geomphys.2012.03.004