A Ruh-Vilms theorem for hypersurfaces in Weitzenb\"ock geometry

Abstract

A well-known theorem by Ruh and Vilms states that the Laplacian of the Gauss map for a smooth immersion into Euclidean space is given by the covariant derivative of the mean curvature vector field. For hypersurfaces, this implies that the Gauss map is harmonic iff the mean curvature is constant. In this paper, we extend this result to hypersurfaces in Weitzenb\"ock geometry. While Riemannian geometry corresponds to the curved geometry without torsion, Weitzenb\"ock geometry is a flat geometry with torsion. They represent two opposite extremes of Riemann-Cartan geometry.