On Quadratic Differentials and Twisted Normal Maps of Surfaces in {mathbb{S}^{2} timesmathbb{R}} and {mathbb{H}^{2} timesmathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${mathcal{N}:M^{n}rightarrow{mathbb{S}}}$ on any hypersupersurface ${M^{n}looparrowright G/K}$ , where ${{mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${mathcal{Q}_{mathcal{N}}:=(mathcal{N}^{ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={mathbb{S}}^{2}times{mathbb{R}}}$ and compare ${mathcal{Q}_{mathcal{N}}}$ with the Abresch–Rosenberg differential ${mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${mathcal{Q}=mathcal{Q}_{mathcal{N}}}$ , after showing that ${mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{mathbb{H}}^{2}times{mathbb{R}}}$ and prove that ${mathcal{Q}=mathcal{Q}_{mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${mathcal{N}}$ and extend to surfaces in ${{mathbb{H}}^{2}times{mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${mathcal{N}.}$