K-harmonic manifolds

Abstract

The present work is aimed toward the study of manifolds which admit k-harmonic metrics. These generalize the "classical" harmonic manifolds and in their definition, the k-th elementary symmetric polynomials of a certain endomorphism Φ of the fibres in the tangent bundle play a role similar to that of Ruse's invariant in classical harmonic spaces. We investigate some properties of k-harmonic manifolds analogous to those enjoyed by harmonic manifolds and obtain some results relating k-harmonic manifolds to harmonic ones. For instance we prove: (a) a k-harmonic manifold is necessarily Einstein, (b) a manifold is simply 1-harmonic iff it is simply n-harmonic. We also work out a general formulation of k-harmonic manifolds in terms of the Jacobi fields on the manifold. This enables us, in particular, to generalize the equations of Walker, and obtain in the case of symmetric spaces, a finite set of necessary conditions for k-harmonicity. As an application of this we are able to show that if a locally symmetric space is n-harmonic then it is k-harmonic for all k. Under the further assumption of compactness we prove that an irreducible k-harmonic manifold is necessarily a symmetric space of rank one. Consequently:(1) a compact simply connected riemannian symmetric manifold, k-harmonic for one k is k-harmonic for all k; and(2) by a theorem of Avez we can drop the assumption of symmetry in (l) but assume instead that the manifold is n-harmonic.