[Paper Review] New results on noncompact harmonic manifolds

Gerhard Knieper|ArXiv.org|Oct 20, 2009

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper establishes that for simply connected, noncompact, nonflat harmonic manifolds, purely exponential volume growth, rank one, Gromov hyperbolicity, and Anosov geodesic flow are all equivalent. It further proves that the geodesic flow is Anosov for nonflat harmonic manifolds without focal points, confirming the Lichnerowicz conjecture for compact harmonic manifolds with such properties.

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. Szabó \cite{Sz} for harmonic manifolds with compact universal cover. E. Damek and F. Ricci \cite{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 $X$. In the first part of the paper we show that the following assertions are equivalent. The volume growth is purely exponential, the rank of $X$ is one, the geodesic flow is Anosov with respect to the Sasaki metric, $X$ is Gromov hyperbolic. In the second part of the paper we show that the geodesic flow is Anosov if $X$ 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 \cite{BCG}, \cite{BFL} and \cite{FL}, we confirm the Lichnerowicz conjecture for all compact harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups.

Motivation & Objective

• To resolve the Lichnerowicz conjecture for compact harmonic manifolds by identifying conditions under which they must be locally symmetric.
• To investigate the asymptotic geometry and geodesic flow on noncompact, simply connected, nonflat harmonic manifolds.
• To clarify the role of rank and horospherical mean curvature in determining volume growth behavior.
• To establish that harmonic manifolds without focal points have Anosov geodesic flows, extending beyond symmetric spaces.

Proposed method

• Introduces a generalized notion of rank for manifolds without conjugate points, extending the classical rank concept from nonpositively curved spaces.
• Uses Jacobi tensor analysis to relate the behavior of geodesic spread to curvature and volume growth, particularly through the evolution of symmetric tensors along geodesics.
• Applies the Sasaki metric to study the geodesic flow and proves that Anosovity is equivalent to rank one and purely exponential volume growth.
• Employs transversality and bounded curvature assumptions to show that geodesic flows on manifolds without focal points are partially hyperbolic, then proves full Anosovity via geometric arguments.
• Combines results with rigidity theorems from Besson-Courtois-Gallot and Benoist-Foulon-Labourie to deduce local symmetry for compact harmonic manifolds with Gromov hyperbolic fundamental groups or no focal points.
• Uses the Wronskian of stable and unstable Jacobi tensors to derive identities linking the derivative of the flow to curvature and geometry.

Experimental results

Research questions

• RQ1 Are purely exponential volume growth, rank one, Gromov hyperbolicity, and Anosov geodesic flow equivalent in noncompact, simply connected harmonic manifolds?
• RQ2 Does the absence of focal points in a harmonic manifold imply that its geodesic flow is Anosov?
• RQ3 Can the Lichnerowicz conjecture be confirmed for compact harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups?
• RQ4 What is the relationship between the mean curvature of horospheres and the volume growth rate in noncompact harmonic manifolds?
• RQ5 Is the rank of a harmonic manifold without focal points constant, and does it imply Anosov dynamics?

Key findings

• For noncompact, simply connected, nonflat harmonic manifolds, the following are equivalent: purely exponential volume growth, rank one, Gromov hyperbolicity, and Anosov geodesic flow.
• The geodesic flow is Anosov if the harmonic manifold has no focal points, extending the Anosov property beyond symmetric spaces.
• All harmonic manifolds without focal points have constant rank, and in odd dimensions, this rank is necessarily one.
• If a harmonic manifold without focal points admits a compact quotient, it must have constant negative curvature.
• The geodesic flow on any Riemannian manifold without focal points and constant rank is Anosov under bounded curvature and transversality conditions.
• Combining the results with prior rigidity theorems confirms the Lichnerowicz conjecture for all compact harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups.