Harmonic homogeneous manifolds of nonpositive curvature

TL;DR

This paper characterizes harmonic homogeneous manifolds with nonpositive curvature, proving they are either flat or Damek-Ricci spaces, thus extending understanding of the structure of such manifolds.

Contribution

It proves that harmonic homogeneous manifolds with nonpositive curvature are either flat or Damek-Ricci spaces, confirming a conjecture in this curvature setting.

Findings

Harmonic homogeneous manifolds of nonpositive curvature are either flat or Damek-Ricci spaces.

Damek-Ricci spaces include noncompact rank-one symmetric spaces and nonsymmetric examples.

The classification extends the understanding of harmonic manifolds in the nonpositive curvature case.

Abstract

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.