Revisiting Liebmann’s theorem in higher codimension

Volume 67 / 2019

Jogli G. Araújo, Henrique F. de Lima Bulletin Polish Acad. Sci. Math. 67 (2019), 179-185 MSC: Primary 53C42; Secondary 53A10, 53C20. DOI: 10.4064/ba190514-30-5 Published online: 17 June 2019

Abstract

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.

Authors

  • Jogli G. AraújoDepartamento de Matemática
    Universidade Federal Rural de Pernambuco
    52.171-900 Recife, Pernambuco, Brazil
    e-mail
  • Henrique F. de LimaDepartamento de Matemática
    Universidade Federal de Campina Grande
    58.429-970 Campina Grande, Paraíba, Brazil
    e-mail

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