The geometry of k-harmonic manifolds

An n-dimensional Riemannian manifold is called k-harmonic for some integer k, 1 <= k <= n - 1, if the k-th elementary symmetric functions of the principal curvatures of small geodesic spheres are radial functions. We prove that k-harmonic manifolds are necessarily 2-stein and show that locally symmetric manifolds which are k-harmonic for one k, are k-harmonic for all k. We then establish some results relating the harmonic and k-harmonic conditions for the class of non-compact harmonic non-symmetric spaces constructed by Damek and Ricci. We also discuss other notions of k-harmonicity and the problem of their equivalence.