Hipersuperfícies com curvatura média constante e hiperplanos

Abstract

In this work, we present results concerning hypersurfaces whose tangent geodesic does not intercept a nonempty special set. These results were obtained by Hilário Alencar and Kátia Frensel in a work which was published in the book Diferential Geometry - A Symposium in Honour of Manfredo do Carmo in 1991. Let us consider an isometric immersion x from M to Q where M denotes a diferentiable manifold of dimension n as well as Q stands for the (n + 1)-dimensional, space form of constant sectional curvature c. Initially, we extend for Q, c arbitrary, the notions of position vector and support function known in (n + 1)-dimensional Euclidean space and we present a geometric interpretation of such a support function in the Euclidean, Spherical and Hyperbolic space. We denote by W the set of points of Q for which does not cross any totally geodesic hypersurface tangent to the image of M by x. By using the support function of x, we characterize the immersions for which the set W is nonempty. We analyze separately the complete noncompact case as well as the compact case among hypersurfaces with constant mean curvature.