Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants
type: Item
instance of: scholarly article
title: Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants
zbMATH Open document ID: 0847.53001
DOI: 10.1016/0926-2245(95)00011-R
published in: Differential Geometry and its Applications
publication date: 1996-09-30T00:00:00Z
review text: Systems of hydrodynamic type \[ u^i_t = v^i_j (u)u^i_x, \quad i,j = 1, \dots, n \tag{1} \] are considered. The system is Hamiltonian if the matrix \(v^i_j (u)\) coincides with the Hessian \[ v^i_j (u) = \varepsilon^i {\partial^2 h(u) \over \partial u_i \partial u_j}. \] The system is called weakly nonlinear if each eigenvalue \(\lambda^i (u)\) of the matrix \(v^i_j (u)\) is constant along the corresponding eigenfoliation. It is known that all diagonalizable Hamiltonian systems (i.e. the systems which can be transformed into the form with diagonal matrix \(v^i_j)\) are integrable. The author classifies the nondiagonalizable Hamiltonian systems. The classification is based on a close relationship between Hamiltonian systems of hydrodynamic type and hypersurfaces in a pseudo-euclidean space. Let \(n\) and \(r\) be the unit normal and the radius-vector of a hypersurface \(M^n\) in pseudo-euclidean space \(E^{n + 1}\). The equation (1) is rewritten in the form \(n_t = r_x\) with \(v^i_j = (w^{-1})^i_j\), where \(w^i_j (u)\) is the Weingarten operator of \(M^n\). The classification of the 3-dimensional Dupin surfaces (principal curvatures are constant along the corresponding curvature surfaces) in \(E^4\) [\textit{U. Pinkall}, Math. Ann. 270, 427-440 (1985; Zbl 0538.53004)] is used to prove the main result of the paper: Theorem. A \(3 \times 3\) nondiagonalizable Hamiltonian system is integrable if and only if it is weakly nonlinear. All \(3 \times 3\) weakly nonlinear nondiagonalizable Hamiltonian systems are equivalent under the reciprocal transformations.
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35L60
zbMATH DE Number: 778516
zbMATH Keywords: Dupin hypersurfaces
zbMATH Keywords: Lie sphere transformations
zbMATH Keywords: isoparametric hypersurfaces
zbMATH Keywords: Hamiltonian systems of hydrodynamic type
zbMATH Keywords: reciprocal transformations
Wikidata QID: Q115362568
author: E. V. Ferapontov
reviewed by: Alexander I. Bobenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 778516