Algebraic geometry approach to constructing metrics of diagonal curvature, orthogonal nets and semi-Hamiltonian systems of hydrodynamic type

We develop generalizations of the well-known Krichever algebraic-geometric construction of orthogonal curvilinear coordinate systems in Euclidean spaces. In the theory of integrable systems of hydrodynamic type a fundamental role is played by special orthogonal coordinates in some nonflat spaces. The most important classes of such spaces are given by metrics of diagonal curvature and metrics of orthogonal nets. We propose a method for constructing such metrics from algebraic-geometric data related to a certain algebraic curve. In addition, we construct a class of semi-Hamiltonian diagonal systems of hydrodynamic type from the algebraic-geometric data. The talk is mainly based on the following joint papers with E.V.Glukhov:
 
[1] E.V.Glukhov and O.I.Mokhov. On algebraic-geometry methods for constructing submanifolds with flat normal bundle and holonomic net of curvature lines. Functional Analysis and Its Applications, 2020, 54:3, 169-178.
[2] E.V.Glukhov and O.I.Mokhov. Algebraic-geometry approach to construction of semi-Hamiltonian systems of hydrodynamic type. Izvestiya: Mathematics, 2023, 87:6.
 
This work was supported by the Russian Science Foundation under grant no. 21-11-00331, https://rscf.ru/en/project/21-11-00331/, and performed at Steklov Mathematical Institute of Russian Academy of Sciences.