Revisiting Liebmann’s theorem in higher codimension

Jogli G. Araújo¹ ; Henrique F. de Lima²

¹ Departamento de Matemática Universidade Federal Rural de Pernambuco 52.171-900 Recife, Pernambuco, Brazil
² Departamento de Matemática Universidade Federal de Campina Grande 58.429-970 Campina Grande, Paraíba, Brazil

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 2, pp. 179-185.

Résumé

We deal with compact surfaces immersed with flat normal bundle and parallel normalized mean curvature vector field in a space form $\mathbb {Q}_c^{2+p}$ of constant sectional curvature $c\in \{-1,0,1\}$. Such a surface is called an LW-surface when it satisfies a linear Weingarten condition of the type $K=aH+b$ for some real constants $a$ and $b$, where $H$ and $K$ denote the mean and Gaussian curvatures, respectively. In this setting, we extend the classical rigidity theorem of Liebmann (1899) showing that a non-flat LW-surface with non-negative Gaussian curvature must be isometric to a totally umbilical round sphere.

Détail

DOI : 10.4064/ba190514-30-5

Keywords: compact surfaces immersed flat normal bundle parallel normalized mean curvature vector field space form mathbb constant sectional curvature surface called lw surface satisfies linear weingarten condition type real constants where denote mean gaussian curvatures respectively setting extend classical rigidity theorem liebmann nbsp showing non flat lw surface non negative gaussian curvature isometric totally umbilical round sphere