LW-surfaces with higher codimension and Liebmann’s Theorem in the hyperbolic space

Abstract

We deal with complete surfaces immersed with flat normal bundle and parallel normalized mean curvature vector field in the hyperbolic space $${\mathbb {H}}^{2+p}$$ . Supposing that such a surface $$M^{2}$$ satisfies a linear Weingarten condition of the type $$K=aH+b$$ for some appropriate real constants a and b, where H and K denote the mean and Gaussian curvatures, respectively, we show that $$M^{2}$$ must be either totally umbilical or isometric to one of the following flat surfaces: $${\mathbb {S}}^{1}(r)\times {\mathbb {R}}$$ , $$\mathbb S^{1}(r)\times {\mathbb {S}}^1(\sqrt{{\tilde{r}}^2-r^2})$$ or $$\mathbb S^{1}(r)\times {\mathbb {H}}^1(-\sqrt{{\tilde{r}}^2+r^2})$$ . Furthermore, we obtain a version of the classical Liebmann’s Theorem (Liebmann in Math Phys Klasse 44–55, 1899) showing that the only compact (without boundary) surfaces having positive constant Gaussian curvature, immersed with flat normal bundle in the hyperbolic space $${\mathbb {H}}^{2+p}$$ , are the totally umbilical round spheres.