Hessian of Busemann functions and rank of Hadamard manifolds

2017/02/13 by Mitsuhiro Itoh, Sinwhi Kim, Itoh, Mitsuhiro +5 · 1 citation
Mathematics · Physics and Astronomy · #53C21 #58C40 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1702.03646

openalex publication_date 2017/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold (M,g) corresponding to eigenvalue zero is investigated with respect to rank of geodesics. On a harmonic Hadamard manifold which is of purely exponential volume growth, or of hypergeometric type it is shown that every Busemann function admits positive definite Hessian. A criterion for (M,g) fulfilling visibility axiom is presented in terms of positive definiteness of the Hessian of Busemann functions.

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• Geometry of Asymptotically harmonic manifolds with minimal horospheres

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