Riemannian manifolds in noncommutative geometry
Date
2012
Authors
Lord, S.
Rennie, A.
Varilly, J.
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Journal article
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Journal of Geometry and Physics, 2012; 62(7):1611-1638
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Steven Lord, Adam Rennie, Joseph C. Várilly
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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. © 2012 Elsevier B.V.
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Copyright © 2012 Elsevier B.V. All rights reserved.