Ferapontov, E. V. Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type. (English. Russian original) Zbl 0742.58018 Funct. Anal. Appl. 25, No. 3, 195-204 (1991); translation from Funkts. Anal. Prilozh. 25, No. 3, 37-49 (1991). The author continues the investigations of the Poisson brackets \[ \{I,J\}=\int [\delta I/\delta u^ i(x)]A^{ij}[\delta J/\delta u^ j(x)]dx \] of the functionals \(I=\int f(u,u_ x,\dots)dx\) and \(j=\int g(u,u_ x,\dots)dx\) on infinite-dimensional phase space, where \(A^{ij}\) are operators of hydrodynamical type. The theory of these brackets was begun in the paper by B. A. Dubrovin and S. P. Novikov [Sov. Math., Dokl. 27, 665-669 (1983); translation from Dokl. Akad. Nauk SSSR 270, 781-785 (1983; Zbl 0553.35011)] and generalized onto more complicated operators \(A^{ij}\) in the papers by S. P. Tsarev [Sov. Math.,Dokl. 31, 488-491 (1985); translation from Dokl. Akad. Nauk SSSR 282, No. 3, 534-537 (1985; Zbl 0605.35075)] and by O. I. Mokhov and the author [Non-local Hamiltonian operators of hydrodynamic type related to metrics of constant curvature, Russ. Math. Surv. 45, No. 3, 218-219 (1990); translation from Usp. Mat. Nauk 45, No. 3(273), 191- 192 (1990)]. In the above papers it was shown that the theory has a geometric nature: certain objects \(g_{ij}\) and \(\Gamma^ k_{ij}\) were associated with \(A^{ij}\) such that \(g_{ij}\) formed a pseudo- Riemannian metric and \(\Gamma^ k_{ij}\) were the Christoffel symbols for the connection compatible with the metric and having zero or (for more complicated operators) constant curvature. Here the author considers a new more complicated case of the operators to which one more object, an affinor \(w^ i_ j\) is associated. It is shown that \(w^ i_ j\) plays the role of the Weingarten operator for the metric \(g_{ij}\) and all the objects satisfy the Gauss-Peterson-Codazzi relations. Some applications are discussed, a certain hypothesis is suggested. Reviewer: Yu.E.Gliklikh (Voronezh) Cited in 1 ReviewCited in 33 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 35Q35 PDEs in connection with fluid mechanics Keywords:Hamiltonian operators of hydrodynamical type Citations:Zbl 0553.35011; Zbl 0605.35075; Zbl 0712.35080 PDFBibTeX XMLCite \textit{E. V. Ferapontov}, Funct. Anal. Appl. 25, No. 3, 195--204 (1991; Zbl 0742.58018); translation from Funkts. Anal. Prilozh. 25, No. 3, 37--49 (1991) Full Text: DOI References: [1] B. A. Dubrovin and S. P. Novikov, ”Hamiltonian formalism for one-dimensional systems of hydrodynamic type and the method of Bogolyubov?Whitham averaging,” Dokl. Akad. Nauk SSSR,270, No. 4, 781-785 (1983). · Zbl 0553.35011 [2] S. P. Tsarev, ”Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type,” Dokl. Akad. Nauk SSSR,282, No. 3, 534-537 (1985). · Zbl 0605.35075 [3] B. A. Dubrovin and S. P. Novikov, ”Hydrodynamics of weakly deformed soliton grids. 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