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

Abstract

We study harmonic sections of a Riemannian vector bundle E → M whose total space is equipped with a 2-parameter family of metrics h p,q which includes both the Sasaki and Cheeger-Gromoll metrics. The restrictions of the h p,q to the total space of any sphere subbundle SE(k) of E (where k > 0 is the radius) are essentially the same for all (p, q), and it is shown that for every k there exists a unique p such that the harmonic sections of SE(k) are harmonic sections of E with respect to h p,q for all q. In both the 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 compact vector fields which are harmonic sections of E = T M in the case where M has non-zero Euler characteristic.1991 Mathematics Subject Classification. 53C43 (53C07, 53C24, 58E15, 58E20, 58G30).

Harmonic vector fields on space forms

“…All other harmonic conformal gradient fields σ have metric parameter q < 0. As remarked in [3], in the spherical case σ is q-Riemannian, with σ(M ) touching the boundary of the Riemannian ball bundle on the equator of σ. In the hyperbolic case the opposite occurs: if σ has no zeros then σ(M ) lies entirely outside the Riemannian ball bundle, and touches its boundary on the equator of σ.…”

“…Nevertheless, it is interesting to learn that metric non-uniqueness can occur for vector fields of non-constant length. Part of the problem when looking for harmonic vector fields is to identify metric parameters, and to this end the following result of [3] provides some guidance. 1/ √ p − 1, then q < 0, with q < 1 − p/2 if p 2.…”

“…One is that all vector fields considered can be constructed from conformal gradient fields. Such fields were considered as the prime example in [3], but for the sphere only, and it was shown that whilst none exist on the 2-sphere, for each dimension greater than two there exists a unique (up to congruence) harmonic representative, which is also metrically unique. In Section 2 we extend that classification to hyperbolic space.…”

“…In [3] it was proposed to address the rigidity problem for vector fields by embedding h in a 2-parameter family h p,q (p, q ∈ R) of generalised Cheeger-Gromoll metrics, within which h = h 0,0 , the Cheeger-Gromoll metric [5] appears as h 1,1 , and h 2,0 is the stereographic metric. The h p,q all belong to the infinite-dimensional family of g-natural metrics on T M [1,2], but are much more tightly controlled, being constructed from a spherically symmetric family of metrics on R n via the Kaluza-Klein procedure.…”

“…Then σ is said to be (p, q)-harmonic if σ is a harmonic section of T M with respect to h p,q (the metric g on M is fixed throughout). From [3] the corresponding Euler-Lagrange equations are:…”