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On two-dimensional manifolds with constant Gaussian curvature and their associated equations. (English) Zbl 1252.35129
Summary: The components for the frame field of a two-dimensional manifold with constant Gaussian curvature are determined for arbitrary nonzero curvature. The components of the frame fields are found from the structure equations and lead to specific nonlinear equations which pertain to surfaces with specific values of the Gaussian curvature. For negative curvature, the equation is of sine-Gordon type, and for positive curvature it is of sinh-Gordon type. The integrability and Bäcklund properties of these equations are then investigated by studying a differential ideal of two-forms which leads to the equations. As a consequence of studying the prolongation structure of each equation, a Lax pair and Bäcklund transformation are obtained.
MSC:
35G20General theory of nonlinear higher-order PDE
35Q53KdV-like (Korteweg-de Vries) equations
37J35Completely integrable systems, topological structure of phase space, integration methods
37K25Relations of infinite-dimensional systems with differential geometry
53A05Surfaces in Euclidean space
37K35Lie-Bäcklund and other transformations