Curvature in noncommutative geometry

Farzad Fathizadeh () and Masoud Khalkhali ()
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Farzad Fathizadeh: Swansea University, Department of Mathematics, Computational Foundry
Masoud Khalkhali: University of Western Ontario, Department of Mathematics

A chapter in Advances in Noncommutative Geometry, 2019, pp 321-420 from Springer

Abstract: Abstract Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past 10 years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.

Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-29597-4_6

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DOI: 10.1007/978-3-030-29597-4_6

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