SOLITON CURVATURES OF SURFACES AND SPACES

An intrinsic geometry of surfaces and three-dimensional Riemann spaces is discussed. In the geodesic coordinates the Gauss equation for two- dimensional Riemann spaces (surfaces) is reduced to the one-dimensiona l Schrodinger equation, where the Gaussian curvature plays a role of p otential. The use of this fact provides an infinite set of explicit ex pressions for curvature and metric of surface. A special case is gover ned by the KdV equation for the Gaussian curvature. Integrable dynamic s of curvature via the KdV equation, higher KdV equations, and 2+1-dim ensional integrable equations with breaking solitons is considered. Fo r a special class of three-dimensional Riemann spaces the relation bet ween metric and scalar curvature is given by the two-dimensional stati onary Schrodinger or perturbed string equations. This provides us an i nfinite family of Riemann spaces with explicit scalar curvature and me tric. Particular class of spaces and their integrable evolutions are d e scribed by the Nizhnik-Veselov-Novikov equation and its higher analo gs. Surfaces and three-dimensional Riemann spaces with large curvature and slow dependence on the variable are considered. They are associat ed with the Burgers and Kadomtsev-Petviashvili equations, respectively . (C) 1997 American Institute of Physics.