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).

We study the geometry of the tangent bundle equipped with a two-parameter family of metrics, deforming the Sasaki and Cheeger-Gromoll metrics. After deriving the expression for the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. We identify all metrics whose restriction to the fibres has positive sectional curvature. When the base manifold is a space form, we characterise metrics with non-negative sectional curvature and show that one can always find parameters ensuring positive scalar curvature. This extends to compact manifolds and, under some curvature conditions, to general manifolds.

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Abstract. A vector field σ on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised CheegerGromoll metrics on T M with respect to which σ is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vector fields that are harmonic. In particular, the harmonic Killing fields and conformal gradient fields are classified, a loop of non-congruent harmonic conformal fields on the hyperbolic plane constructed, and the 2-dimensional classification achieved for conformal fields. A classification is then given of all harmonic quadratic gradient fields on spheres.