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The Hopf-Levi-Civita Data of Two-Dimensional Metrics

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Algorithms as a Basis of Modern Applied Mathematics

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 404))

Abstract

The aim of this chapter is to study three 2-dimensional geometries, namely Riemannian, Kähler and Hessian, in a unitary way by using three (local) Hermitian matrices. One of these matrices corresponds to the symmetric matrix of metric while the other two Hermitian matrices is provided by the Christoffel symbols. The secondary diagonal of these Hermitian matrices are generated by the Hopf invariant and its conjugate, where for this notion we adopt the definition of Jensen et al. (Surfaces in classical geometries. A treatment by moving frames. Springer, Cham, 2016 [13]). In the Riemannian case a special view is towards an expression of the Gaussian curvature in terms of these data while in Kähler and Hessian geometry we use the corresponding potential function and a new (again local) differential operator of first order, similar to \(\partial \).

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Authors and Affiliations

  1. Faculty of Mathematics, University Al. I. Cuza, 700506, Iasi, Romania

    Mircea Crasmareanu

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  1. Mircea Crasmareanu
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Correspondence to Mircea Crasmareanu .

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Editors and Affiliations

  1. Department of Mathematics and Physics, Faculty of Military Technology, University of Defence, Brno, Czech Republic

    Šárka Hošková-Mayerová

  2. Faculty of Mathematics and Computer Science, Ovidius University, Constanta, Romania

    Cristina Flaut

  3. Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”, Caserta, Italy

    Fabrizio Maturo

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Crasmareanu, M. (2021). The Hopf-Levi-Civita Data of Two-Dimensional Metrics. In: Hošková-Mayerová, Š., Flaut, C., Maturo, F. (eds) Algorithms as a Basis of Modern Applied Mathematics. Studies in Fuzziness and Soft Computing, vol 404. Springer, Cham. https://doi.org/10.1007/978-3-030-61334-1_4

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