On a nonlinear equation related to the geometry of the diffeomorphism group

David Dai-Wai Bao, Jacques Lafontaine and Tudor S. Ratiu

Vol. 158 (1993), No. 2, 223–242

DOI: 10.2140/pjm.1993.158.223

Abstract

Let M be a compact boundaryless Riemannian manifold. We derive the equations on M which characterize asymptotic vectors on Diff _vol(M). We classify those M’s whose volume-preserving diffeomorphism groups admit asymptotic vectors which are represented by harmonic vector fields on M. We then show that these harmonic solutions can be used to construct other (typically non-harmonic) solutions.

Mathematical Subject Classification 2000

Primary: 58B20

Secondary: 58B25, 58D05

Milestones

Received: 20 May 1991

Revised: 15 October 1991

Published: 1 April 1993