We study isometric immersions of surfaces of constant curvature into the homogeneous spaces H 2 × R and S 2 × R. In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into H 2 ×R and a unique one into S 2 × R when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into H 2 × R or S 2 × R.MSC: 53C42, 53C40.
Abstract. We give a conformal representation for improper affine spheres which is used to solve the Cauchy problem for the Hessian one equation. With this representation, we characterize the geodesics of an improper affine sphere, study its symmetries and classify the helicoidal ones. Finally, we obtain the complete classification of the isolated singularities of the Hessian one MongeAmpère equation.
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