ON THE REGULAR ISOMETRIC IMMERSION IN $ E^3$ OF UNBOUNDED DOMAINS OF NEGATIVE CURVATURE
Дмитрий Васильевич Туницкий · Mathematics of the USSR-Sbornik · 1989
A wide class of unbounded domains is considered in complete Riemannian manifolds of negative curvature that are homeomorphic to planes, and the possibility of immersing them regularly and isometrically in three-dimensional Euclidean space is investigated.Let a metric of the manifold under consideration be given on the parameter plane by a line element of the form , where . The set is considered, where 0$ SRC=http://ej.iop.org/images/0025-5734/62/1/A08/tex_sm_3230_img6.gif/> and is twice continuously differentiable. Let denote the corresponding domain on the manifold. Then the domain can be isometrically immersed in by means of a surface of class .This result is proved by constructing a smooth solution of a special form of the Gauss-Peterson-Codazzi system of equations in the domain .Figures: 2. Bibliography: 11 titles.