Geometry of a class of non-symmetric harmonic manifolds

Franco Tricerri and Lieven Vanhecke

Abstract. We discuss the geometric and algebraic structure of the
non-symmetric examples of harmonic manifolds given by E. Damek and
F. Ricci.  In addition we focus on some aspects of their geometry
in relation with Osserman's conjecture and some open problems about
$k$-harmonicity.  This leads to a new conjecture about harmonic spaces
for which we provide partial answers.

Keywords.  Lie algebras of Heisenberg type and generalized Heisenberg
groups, solvable extensions, homogeneous Hadamard manifolds, harmonic and
$k$-harmonic spaces, Osserman's conjecture, Jacobi operator.

MS classification. 53C25, 53C30, 53C35.