Gauss map harmonicity and mean curvature of a hypersurface in a homogeneous manifold

Fidelis Bittencourt and Jaime Ripoll

Vol. 224 (2006), No. 1, 45–63

DOI: 10.2140/pjm.2006.224.45

Abstract

We define a Gauss map of an orientable hypersurface in a homogeneous manifold with an invariant Riemannian metric. Our main objective is to extend to this setting some results on the Gauss map of a constant mean curvature hypersurface of an Euclidean space, namely the Ruh–Vilm theorem relating the harmonicity of the Gauss map and the constancy of the mean curvature, and the Hoffman–Osserman–Schoen theorem characterizing the plane and the circular cylinder as the only complete constant mean curvature surfaces whose Gauss image is contained in a closed hemisphere of the sphere.

Keywords

Gauss map, harmonic map, mean curvature

Mathematical Subject Classification 2000

Primary: 53A10

Milestones

Received: 1 June 2004

Accepted: 14 December 2004

Published: 1 March 2006