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

TL;DR

This paper extends Ruh and Vilms' theorem, relating the Laplacian of the Gauss map to mean curvature, from Euclidean space to hypersurfaces in Weitzenb"ock geometry, which involves torsion.

Contribution

It generalizes the Ruh-Vilms theorem to hypersurfaces in Weitzenb"ock geometry, incorporating torsion into the classical framework.

Findings

Laplacian of the Gauss map relates to covariant derivative of mean curvature in Weitzenb"ock geometry

Gauss map is harmonic iff mean curvature is constant in this setting

Extension bridges Riemannian and Weitzenb"ock geometries

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.