C. M. Wood

On the Energy of a Unit Vector Field

The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M → T1M, where the unit tangent bundle T1M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M=S5,S7,..., and an estimate on the index is obtained.

The energy of unit vector fields on the 3-sphere

The stability of the three-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of the three-dimensional Hopf vector field is compared and contrasted with that of the closely related Hopf map. Finally, it is shown that the Hopf vector fields are the unique global minima of the energy functional restricted to unit vector fields on the 3-sphere.

Harmonic sections of homogeneous fibre bundles

We show how the equations for harmonic maps into homogeneous spaces generalize to harmonic sections of homogeneous fibre bundles. Surprisingly, the generalization does not explicitly involve the curvature of the bundle. However, a number of special cases of the harmonic section equations (including the new condition of super-flatness) are studied in which the bundle curvature does appear. Some examples are given to illustrate these special cases in the non-flat environment. The bundle in question is the twistor bundle of an even-dimensional Riemannian manifold M whose sections are the almost-Hermitian structures of M.

A class of harmonic almost-product structures

Abstract The energy of Riemannian almost-product structure P is measured by forming the Dirichlet integral of the associated Gauss section γ, and P is decreed harmonic if γ criticalizes the energy functional when restricted to the submanifold of sections of the Grassman bundle. Euler-Lagrange equations are obtained, and geometrically transformed in the special case when P is totally geodesic. These are seen to generalize the Yang-Mills equations, and generalizations of the self-duality and anti-self-duality conditions are suggested. Several applications are then described. In particular, it is considered whether integrability of P is a necessary condition for γ to be harmonic.

HARMONIC CONTACT METRIC STRUCTURES AND SUBMERSIONS

We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby–Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre.