Integrability conditions for a certain class of nonlinear evolution equations and Kähler geometry (Q1311942)

The classification of simply connected and geodesically complete pseudo- Kählerian (indefinite) manifolds of constant holomorphic sectional curvature is given. These are well known spaces \(P^ n_ s(\mathbb{C})\), \(H^ n_ s(\mathbb{C})\), \(\mathbb{C}^ n_ s\) in the notation of the book of \textit{J. A. Wolf} [Spaces of constant curvature (Russian) (Nauka 1982; Zbl 0518.53040), Ch. 12]. The main result of the paper consists in a characterization of the above-mentioned manifolds in terms of integrability conditions (by inverse scattering transformation) for Hamiltonian nonlinear evolution equations of the form \(z^ a_ t = \{H,z^ a\}\), associated with pseudo-Kählerian manifolds \((M,h)\), where \((z^ a(x,t)): \mathbb{R}^ 2\to M\) is given in local complex coordinates \((z^ a)\), \(H = \int dx\sum h_{a\overline{b}}(z(x))z^ a_ x(x)\overline{z}^ b_ x(x)\), \(h\) is the metric, \(h = \sum h_{a\overline{b}}dz^ ad\overline{z}^ b\), and the Poisson brackets are generated by the Kählerian form of the metric \(h\). It is shown that the obtained integrable equation for \(P^ n(\mathbb{C})\) is a generalized ferromagnet (in the sense of Fordy and Kulish) and a recurrent formula for its local conservation laws is calculated.