Superfícies totalmente umbílicas em variedades homogêneas tridimensionais

Abstract

In this dissertation, we studied the complete classification of totally umbilical surfaces immersed in homogeneous 3-manifolds, as obtained in the article entitled "The classification of totally umbilical surfaces in homogeneous 3-manifolds" by Manzano and Soaum [Math. Z. 279 (2015) 557-576]. In the case of unimodular Lie groups it was shown that, except for the Euclidean space R3, unit sphere S3, soluble group Sol3 and the totally geodesics examples that appear in some special cases given by Tsukada [Kodai Math. J. 19 (3) (1996) 395-437], there are no fully umbilical surfaces. Moreover, there were obtained herein extensions of some results of Inoguchi and Van der Veken [Geom. Dedicata. 131 (2008) 159-172], Souam and Toubiana [Math. Helv. 84 (3) (2009) 673-704] and Tsukada [Kodai Math. J. 19 (3) (1996) 395-437], giving alternative proves for each case. In the case of non-unimodular Lie groups, the totally umbilical surfaces exist when the Lie group is isometric to the hyperbolic space H3 and to the product space H2 x R. Futhermore, in some special cases of non-unimodular Lie groups it was shown, up to ambient isometries, the existence of complete totally geodesics surfaces and complete totally umbilical one which are not totally geodesics.