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.