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Curvature in noncommutative geometry

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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.

Dedicated to Alain Connes with admiration, affection, and much appreciation

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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. Department of Mathematics, Computational Foundry, Swansea University, Swansea, UK

    Farzad Fathizadeh

  2. Department of Mathematics, University of Western Ontario, London, ON, Canada

    Masoud Khalkhali

Authors
  1. Farzad Fathizadeh
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  2. Masoud Khalkhali
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Correspondence to Masoud Khalkhali .

Editor information

Editors and Affiliations

  1. Department of Physics Institute for Mathematics, Astrophysics and Particle Physics, Nijmegen, The Netherlands

    Ali Chamseddine

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA

    Caterina Consani

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Nigel Higson

  4. Department of Mathematics, The University of Western Ontario, London, ON, Canada

    Masoud Khalkhali

  5. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Henri Moscovici

  6. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Guoliang Yu

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Fathizadeh, F., Khalkhali, M. (2019). Curvature in noncommutative geometry. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_6

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