Hipersuperfícies em Rp+q+2 de curvatura escalar nula invariantes por O(p+1) x O(q+1).

Abstract

This dissertation has as base Jocelino Sato and Vicente de Souza Neto's paper called Complete and Stable O(p + 1) x O(q + 1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space Rp+q+2, published on the Annals of Global Analysis and Geometry - 29 in 2006. The main result of this dissertation is the Classi_cation Theorem, which states: The O(p+1) x O(q+1)-Invariant Hypersurfaces in Rp+q+2, p; q > 1, with zero scalar curvature belong to one of the following classes: (1) Cones with a singularity at the orign of Rp+q+2; (2) Hypersurfaces having one orbit of singularity and asymptoting both of the cones Cα and Cβ; (3) Regular hypersurfaces asymptoting the cone Cα; (4) Regular hypersurfaces asymptoting the cone Cβ; (5) Regular hypersurfaces asymptoting both of the cones Cα and Cβ. It was reached by the studies of the ordinary differential equation on R2, involving the coordenate curves that generate these hypersurfaces. Such differential equation, in its turn, is associated with a vector field X : R22 → R2 on the plan. The study of the orbits space in this field is essential; after all, because of it, it was possible to translate the X orbits' behavior into information concerning the profile curves and, finally, reach the theorem.