Riemannian manifolds in noncommutative geometry

Authors

Steven Lord, University of Adelaide
Adam Rennie, Australian National UniversityFollow
Joseph C. Varilly, Universidad De Costa Rica

RIS ID

76327

Publication Details

Lord, S., Rennie, A. & Varilly, J. C. (2012). Riemannian manifolds in noncommutative geometry. Journal of Geometry and Physics, 62 (7), 1611-1638.

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 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.