Harmonic homogeneous manifolds of nonpositive curvature

摘要

A Riemannian manifold is called harmonic if its volume density functionexpressed in polar coordinates centered at any point is radial. Flat andrank-one symmetric spaces are harmonic. The converse (the LichnerowiczConjecture) is true for manifolds of nonnegative scalar curvature and for someother classes of manifolds, but is not true in general: there exists a familyof homogeneous harmonic spaces, the Damek-Ricci spaces, containing noncompactrank-one symmetric spaces, as well as infinitely many nonsymmetric examples. Weprove that a harmonic homogeneous manifold of nonpositive curvature is eitherflat, or is isometric to a Damek-Ricci space.