JournalscmhVol. 87, No. 3pp. 669–703

New results on noncompact harmonic manifolds

  • Gerhard Knieper

    Ruhr-Universität Bochum, Germany
New results on  noncompact harmonic manifolds cover
Download PDF

Abstract

The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z. I. Szabó [Sz] for harmonic manifolds with compact universal cover. E. Damek and F. Ricci [DR] provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient.

In this paper we study, using a notion of rank, the asymptotic geometry and the geodesic flow on simply connected nonflat and noncompact harmonic manifolds denoted by XX.

In the first part of the paper we show that the following assertions are equivalent. The volume growth is purely exponential, the rank of XX is one, the geodesic flow is Anosov with respect to the Sasaki metric, XX is Gromov hyperbolic. We also provide a characterization of manifolds of constant negative curvature by their minimal volume growth among those harmonic manifolds with fixed mean curvature of the horospheres.

In the second part of the paper we show that the geodesic flow is Anosov if XX is a nonflat harmonic manifold with no focal points. In the course of the proof we obtain that certain partially hyperbolic flows on arbitrary Riemannian manifolds without focal points are Anosov, which is of interest beyond harmonic manifolds.

Combining the results of this paper with the rigidity theorem’s of [BCG], [BFL], and [FL], we confirm the Lichnerowicz conjecture for all compact harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups.

Cite this article

Gerhard Knieper, New results on noncompact harmonic manifolds. Comment. Math. Helv. 87 (2012), no. 3, pp. 669–703

DOI 10.4171/CMH/265