Harmonic sections of Riemannian vector bundles, and metrics of Cheeger–Gromoll type

Abstract

We study harmonic sections of a Riemannian vector bundle E → M when E is equipped with a 2-parameter family of metrics h p , q which includes both the Sasaki and Cheeger–Gromoll metrics. For every k > 0 there exists a unique p such that the harmonic sections of the radius- k sphere subbundle are harmonic sections of E with respect to h p , q for all q. In both compact and non-compact cases, Bernstein regions of the ( p , q ) -plane are identified, where the only harmonic sections of E with respect to h p , q are parallel. Examples are constructed of vector fields which are harmonic sections of E = T M in the case where M is compact and has non-zero Euler characteristic.