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Revisiting Liebmann’s theorem in higher codimension

Araújo Jogli G., Lima Henrique F. de

Bulletin of the Polish Academy of Sciences. Mathematics

2019

Vol. 67, no. 2

179--185

We deal with compact surfaces immersed with flat normal bundle and parallel normalized mean curvature vector field in a space form Qc2+p of constant sectional curvature c ϵ {−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.