Revisiting Liebmann’s theorem in higher codimension

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.