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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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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