Soliton curvatures of surfaces and spaces (Q2785759)

The main goal of this paper is to describe exactly solvable cases for which metric and curvature are given by explicit formulas and to establish the interrelation between the intrinsic geometry of Riemannian spaces and multidimensional integrable equations. The author proposes the Gaussian curvature and metric in terms of multi-Bargmann potentials and the Riemann \(\theta\) functions parametrized by arbitrary functions on \(y\), constructs the surfaces, the Gaussian curvature of which is nothing but solitons of the KdV equation, and presents the integrable dynamics of surfaces of revolution and generic surfaces via the KdV and higher KdV equations.NEWLINENEWLINENEWLINEMoreover, several nonlinear PDEs associated with three-dimensional Riemannian spaces with particular diagonal metrics are also given. The \((2+1)\)-dimensional generalizations of the Liouville, sine-Gordon and Tsitseica-Dood-Bullough equation are among them.