Harmonic Hadamard Manifolds of Prescribed Ricci Curvature and Volume Entropy

A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of hypergeometric type is obtained with respect to volume density.Mathematics Subject Classification (2010). Primary 53C21; Secondary 43A90, 42B10.

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“…18),(9.19) we have for a fixed µ, since h = h(λ) |c(2λ)| 2 B(o;r) ϕ λ (r(x))ϕ µ (r(x))dv B(o;r)(10.14) …”

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confidence: 99%

Harmonic Hadamard Manifolds and Gauss Hypergeometric Differential Equations

Itoh

Satoh

2019

Publ. Res. Inst. Math. Sci.

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“…Ramachandran and Ranjan [17] classified certain non-compact harmonic spaces in terms of their density functions. Itoh et al [11] studied harmonic spaces in relation to prescribed Ricci curvature and volume entropy. Choe, Kim and Park [4] characterized certain harmonic spaces in terms of the radial eigen-functions of the Laplacian.…”

Section: Introductionmentioning

confidence: 99%

Harmonic radial vector fields on harmonic spaces

Gilkey

Park

2020

Preprint

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We characterize harmonic spaces in terms of the dimensions of various spaces of radial eigen-spaces of the Laplacian ∆ 0 on functions and the Laplacian ∆ 1 on 1-forms. We examine the nature of the singularity as the geodesic distance r tends to zero of radial eigen-functions and 1-forms. Via duality, our results give rise to corresponding results for radial vector fields. Many of our results extend to the context of spaces which are harmonic with respect to a single point.2010 Mathematics Subject Classification. 53C21.

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“…Here σ(r) is the mean curvature of the geodesic sphere Σ(x; r). See [10,14] for the precise definition of the hypergeometric type. A harmonic Hadamard manifold of a hypergeometric type is of purely exponential volume growth.…”

Section: Introduction and Main Theoremsmentioning

confidence: 99%

Hessian of Busemann functions and rank of Hadamard manifolds

Itoh,

Kim,

Park

et al. 2017

Preprint

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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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Harmonic Hadamard Manifolds of Prescribed Ricci Curvature and Volume Entropy

Cited by 4 publications

References 14 publications

Harmonic Hadamard Manifolds and Gauss Hypergeometric Differential Equations

Harmonic Hadamard Manifolds and Gauss Hypergeometric Differential Equations

Harmonic radial vector fields on harmonic spaces

Hessian of Busemann functions and rank of Hadamard manifolds

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