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 inJournal of geometry and physics Vol. 62; no. 7; pp. 1611 - 1638
Main Authors Lord, Steven, Rennie, Adam, Várilly, Joseph C.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.07.2012
Subjects
Kasparov product
Noncommutative geometry
Spectral triple
Spectral triple
Noncommutative geometry
Kasparov product
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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