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Horospherical mean curvature functions and D'Atri spaces

Gerhard Knieper, JeongHyeong Park, Norbert Peyerimhoff

TL;DR

The paper introduces and analyzes manifolds without conjugate points that admit a continuous, reversible, geodesic-flow-invariant horospherical mean curvature function $h$. It develops a rank-one rigidity framework via Jacobi tensors, the Riccati equation, and stable/unstable foliations to show that, under mild curvature bounds, rank-one manifolds are asymptotically harmonic and, in the compact case, locally symmetric of negative curvature. It further extends these rigidity phenomena to rank-one D'Atri spaces without conjugate points, proving harmonicity when $h$ is continuous, and it establishes product-closure and partial 3D classifications. The results connect invariant horospherical data with classical harmonic and asymptotically harmonic manifolds, enriching the landscape of geometric rigidity for spaces without conjugate points and situating Druetta-type homogeneous results within a broader, non-homogeneous setting.

Abstract

We consider simply connected Riemannian manifolds without conjugate points for which the horospherical mean curvature function is continuous, reversible and invariant under the geodesic flow. We show under mild additional curvature tensor conditions that rank one manifolds in this family are automatically asymptotically harmonic. In particular, compact rank one manifolds of this kind must be locally symmetric spaces of negative curvature. Moreover, we show under the same conditions that rank one D'Atri spaces without conjugate points are harmonic. An earlier result of this type was proved by Druetta for certain homogeneous D'Atri spaces.

Horospherical mean curvature functions and D'Atri spaces

TL;DR

The paper introduces and analyzes manifolds without conjugate points that admit a continuous, reversible, geodesic-flow-invariant horospherical mean curvature function . It develops a rank-one rigidity framework via Jacobi tensors, the Riccati equation, and stable/unstable foliations to show that, under mild curvature bounds, rank-one manifolds are asymptotically harmonic and, in the compact case, locally symmetric of negative curvature. It further extends these rigidity phenomena to rank-one D'Atri spaces without conjugate points, proving harmonicity when is continuous, and it establishes product-closure and partial 3D classifications. The results connect invariant horospherical data with classical harmonic and asymptotically harmonic manifolds, enriching the landscape of geometric rigidity for spaces without conjugate points and situating Druetta-type homogeneous results within a broader, non-homogeneous setting.

Abstract

We consider simply connected Riemannian manifolds without conjugate points for which the horospherical mean curvature function is continuous, reversible and invariant under the geodesic flow. We show under mild additional curvature tensor conditions that rank one manifolds in this family are automatically asymptotically harmonic. In particular, compact rank one manifolds of this kind must be locally symmetric spaces of negative curvature. Moreover, we show under the same conditions that rank one D'Atri spaces without conjugate points are harmonic. An earlier result of this type was proved by Druetta for certain homogeneous D'Atri spaces.

Paper Structure

This paper contains 16 sections, 30 theorems, 141 equations.

Table of Contents

  1. Introduction and statement of results
  2. Results
  3. History, context and motivation
  4. Background, relevant concepts and notation
  5. Jacobi tensors, Riccati equation, Busemann functions and rank
  6. Riemannian products of manifolds with invariant horospherical mean curvature functions
  7. Rank one manifolds with invariant horospherical mean curvature functions
  8. Step 1: $\det D$ and $\mathop{\mathrm{trace}}\nolimits D$ are both reversible and invariant under the geodesic flow
  9. Step 2: $\det D$ and $\mathop{\mathrm{trace}}\nolimits D$ are both constant along weak stable and unstable manifolds of rank one vectors
  10. Step 3: Continuity of $(p,v) \mapsto - \mathop{\mathrm{grad}}\nolimits b_v(p)$ on the set $X \times SX_\alpha$
  11. Step 4: $\det D$ and $\mathop{\mathrm{trace}}\nolimits D$ are both globally constant
  12. $3$-dimensional manifolds with invariant horospherical mean curvature functions
  13. Rank one D'Atri spaces without conjugate points
  14. Proof of Theorem \ref{['thm:main2']}
  15. Homogeneous D'Atri spaces
  16. ...and 1 more sections

Key Result

Theorem 3

Let $(M_i,g_i)$, $i=1,2$, be two manifolds without conjugate points and $(M,g)$ be their Riemannian product. Then $M$ is a manifold with invariant horospherical mean curvature function if and only if both $M_1, M_2$ are manifolds with invariant horospherical mean curvature functions.

Theorems & Definitions (57)

  • Definition 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Theorem 7
  • Definition 8
  • Lemma 9
  • proof
  • ...and 47 more