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

Abstract

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 \({\mathcal{N}.}\)