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

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