Harmonic Gauss maps

Gary R. Jensen and Marco Rigoli

Vol. 136 (1989), No. 2, 261–282

DOI: 10.2140/pjm.1989.136.261

Abstract

A construction is given whereby a Riemannian manifold induces a Riemannian metric on the total space of a large class of fibre bundles over it. Using this metric on the appropriate bundles, necessary and sufficient conditions are given for the Gauss map and the spherical Gauss map to be harmonic. A weak maximum principle is applied to the Gauss map of an isometric immersion into Euclidean space in order to prove a sufficient condition for when such an immersion with parallel mean curvature vector must be minimal.

Mathematical Subject Classification 2000

Primary: 53C42

Secondary: 58E20

Milestones

Received: 1 September 1987

Published: 1 February 1989