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Curvature in noncommutative geometry

  • Farzad Fathizadeh
  • Masoud KhalkhaliEmail author
Chapter

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.

Notes

Acknowledgements

F.F. acknowledges support from the Marie Curie/ SER Cymru II Cofund Research Fellowship 663830-SU-008. M.K. would like to thank Azadeh Erfanian for providing the original picture used in page four, Saman Khalkhali for his support and care, and Asghar Ghorbanpour for many discussions on questions related to this article.

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Authors and Affiliations

  1. 1.Department of Mathematics, Computational FoundrySwansea UniversitySwanseaUK
  2. 2.Department of MathematicsUniversity of Western OntarioLondonCanada

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