2004/07/02 by Yuri Nikolayevsky, Y. Nikolayevsky, Nikolayevsky, Y. · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C25 #msc:53C30
paper · pdf · doi:10.48550/arxiv.math/0407024
11 pages
arxiv created 2004/07/02 · arxiv updated 2009/12/01
A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifolds, but is not true in general: there exists a family of homogeneous harmonic spaces, the Damek-Ricci spaces, containing noncompact rank-one symmetric spaces, as well as infinitely many nonsymmetric examples. We prove that a harmonic homogeneous manifold of nonpositive curvature is either flat, or is isometric to a Damek-Ricci space.