Sobre diferenciais quadráticas e aplicações normais torcidas de superfícies em S2×R e H2×R.

Abstract

Ruh-Vilms’ theorem states that a hypersurface of the Euclidean space has constant mean curvature if and only if its Gauss map is harmonic. Bittencourt and Ripoll, in the article ”Gauss Map Harmonicity and Mean Curvature of the Hypersurface in a Homogeneous Manifold”, extended the result for homogeneous spaces. In this work, we consider the particular case of the Cartesian product of the two-dimensional sphere with the real line, for which it’s proved that the Gauss map defined by Bittencourt and Ripoll in the article mentioned above induces a quadratic differential that coincides with the Abresch-Rosenberg differential, and this differential is holomorphic for surfaces of constant mean curvature. Furthermore, it is proved that such application is the twisted normal map of the surface. This concept can be extended to the Cartesian product of two-dimensional hyperbolic space with the real line, for which the coincidence of both differential forms holds. Later, it extends the Ruh-Vilms theorem for the two ambient spaces mentioned above.