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

By M. Benyounes, Eric Loubeau, C. Wood. 2006

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We study harmonic sections of a Riemannian vector bundle E → M when E is equipped with a 2-parameter family of metrics hp,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 hp,q for all q. In both compact and noncompact cases, Bernstein regions of the (p, q)-plane are identified, where the only harmonic sections of E with respect to hp,q are parallel. Examples are constructed of vector fields which are harmonic sections of E = TM in the case where M is compact and has non-zero Euler characteristic.

Subfields: Applied Mathematics, Genetics, Geometry and Topology

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