On Spherical Image Maps Whose Jacobians do not Change Sign

摘要

The first part of the paper deals with mappings p = p(x) of an n-dimensional x region into n-dimensional p space with a Jacobian J(x) which does not change sign. A map p = p(z) is called a gradient map if p(x) is locally a gradient. One of the theorems proved, for example, states that a gradient map p = p(x) with a Jacobian which does not change sign is monotone in the sense that the boundary of the image set is contained in the image of the Boundary. Analogous results are valid for the spherical Image mapping of an n-dimensional hypersurfaces immersed in (n + l)-dimensional Euclidean space.nThe next part of the paper concerns applications of the results of the first part, for example, there are obtained an n-dimensional generalization of the Rado (von Neumann) theorem on saddle surfaces, a characterization of complete hypersur¬faces of constant zero curvature in Euclidean spaces as cylinders, and, for n = 2, properties of the x sets grad z(x) = const. if the Hessian determinant of z is identically zero.