Harmonic vector fields on space forms

Abstract

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.

Harmonic vector fields on pseudo-Riemannian manifolds

“…We note that for the Riemannian 2-sphere λ < 0 and ǫ = 1, so Theorem 7.2 precludes the existence of harmonic Killing fields, as already observed in [2]. Comparison of Lemma 7.1 and Theorem 7.2 shows that harmonic Killing fields on each of the remaining pseudo-Riemannian quadrics form a quadric in the 3-dimensional Lie algebra of Killing fields (although not necessarily of the same type as the underlying quadric or its anti-isometric counterpart).…”

“…As in [2], we refer to ζ as the spinnaker of σ. The following result is a direct generalisation of the Riemannian version used in [2].…”

“…It turns out [1] that the energy functional behaves no less rigidly when the Sasaki metric h 0,0 is replaced by h 1,1 or h 2,0 ; however, other members of C G permit greater flexibility. In [2], a harmonic vector field on the Riemannian manifold (M, g) was defined to be a harmonic section of T M with respect to the Riemannian metric g on M and some h p,q ∈ C G ; classifications of harmonic vector fields were then obtained for conformal gradient fields and Killing fields on non-flat Riemannian space forms. Typically (but not invariably) a harmonic vector field is metrically unique; that is, it picks a unique h p,q ∈ C G .…”

“…Despite this, and somewhat remarkably, the singularity in the energy functional is completely resolved at the level of the first variation: the Euler-Lagrange equations for harmonic sections with respect to any h p,q ∈ C G are in fact globally defined, and coincide (tensorially) with those in the Riemannian case (Section 3). This enables us (Section 5) to extend the classification of harmonic conformal gradient fields on Riemannian space forms obtained in [2] to hyperquadrics of pseudo-Euclidean space (Theorem 5.4), and then (Section 6) examine Killing fields on these spaces. In particular, we obtain a condition for a preharmonic Killing field to be harmonic (Theorem 6.5).…”

“…In particular, we obtain a condition for a preharmonic Killing field to be harmonic (Theorem 6.5). (The notion of preharmonicity was introduced in [2], and may be viewed as an integrability condition for harmonicity.) We show (Section 7) that all Killing fields on the 2-dimensional pseudo-Riemannian quadrics are preharmonic, and complete the classification of harmonic Killing fields in this case: up to pseudo-Riemannian congruence there is a unique harmonic Killing field on five of the six metrically distinct quadrics, the exception being the Riemannian 2-sphere, on which no Killing field is harmonic (Theorem 7.5).…”