Riemannian manifolds in noncommutative geometry (Q423691)

The work is devoted to a definition of closed noncommutative Riemannian manifold. The construction relies on a choice of orientation and metric and the definition itself is modelled on the Hodge-de Rham operator. Such noncommutative Riemannian manifolds are compared to those known from the spectral triple construction for spin\(^{\mathcal{C}}\) manifolds. The authors summarize their approach as consisting in replacing the condition of ``noncommutative spin\(^{\mathcal{C}}\)'' for spectral triples with a ``noncommutative Riemann'' condition. At the beginning, the reader is introduced to the Hermitian modules and Morita equivalences, and some tools for studying operators on finitely generated projective modules are developed. Then the Kasparov product of unbounded Kasparov modules is used to relate modules and spectral triples and to generate new spectral triples. The subsequent section is devoted to the description of manifold structures on spectral triples: the spin\(^{\mathcal{C}}\) case is presented and a new Riemannian spectral triple is introduced. In the last section are given main theorems connecting spin\(^{\mathcal{C}}\) manifolds and Riemannian manifolds. The authors show how for a given spin\(^{\mathcal{C}}\) manifold, one can obtain a Riemannian manifold and conversely how for a given Riemannian manifold and a spin\(^{\mathcal{C}}\) structure one can obtain a spin\(^{\mathcal{C}}\) manifold.