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Hiroyasu Satoh

Publications and source records attributed to Hiroyasu Satoh.

10 recordsLinked to original sources
arXiv2025

Infinitesimal Conformal Rigidity on Damek-Ricci Spaces

We show that every conformal vector field on a Damek-Ricci space is necessarily Killing, establishing a strong form of infinitesimal conformal rigidity. Although this rigidity phenomenon is classically known in the Einstein setting, our proof follows a completely different approach. We formulate the conformal Killing condition as an explicit system of partial differential equations on the solvable Lie group model and analyze it directly. This local and analytic method yields a constructive proof of rigidity without relying on global arguments or transformation groups.

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arXiv2025

A Constructive Approach to Infinitesimal Conformal Rigidity on Complex Hyperbolic Space

We prove that every conformal vector field on the complex hyperbolic space $\mathbb{C}H^n$ is Killing for all $n\ge 2$. Although this rigidity is classically known, our proof is entirely different in nature: it is local, analytic, and fully constructive. Our approach is local, analytic, and constructive: we view $\mathbb{C}H^2$ through its solvable Lie group model and express the conformal Killing equation as an explicit system of partial differential equations. By solving this system completely, we show that any conformal vector field must be determined by a Killing field. The analysis in complex dimension $2$ naturally extends to arbitrary $n$, yielding a unified and fully explicit proof of this rigidity phenomenon.

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arXiv2024

A note on the volume entropy of harmonic manifolds of hypergeometric type

Harmonic manifolds of hypergeometric type form a class of non-compact harmonic manifolds that includes rank one symmetric spaces of non-compact type and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of hypergeometric type to satisfy the Ricci curvature $\mathrm{Ric} = -(n-1)$, we show that the volume entropy of this manifold satisfies a certain inequality. Additionally, we show that manifolds yielding the upper bound of volume entropy are only real hyperbolic spaces with sectional curvature $-1$, while examples of Damek-Ricci spaces yielding the lower bound exist in only four cases.

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arXiv2022

Information geometry of the space of probability measures and barycenter maps

In this article, we present recent developments of information geometry, namely, geometry of the Fisher metric, dualistic structures and divergences on the space of probability measures, particularly the theory of geodesics of the Fisher metric. Moreover, we consider several facts concerning the barycenter of probability measures on the ideal boundary of a Hadamard manifold from a viewpoint of the information geometry.

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arXiv2020

Harmonic manifolds of hypergeometric type and spherical Fourier transform

The spherical Fourier transform on a harmonic Hadamard manifold $(X^n, g)$ of positive volume entropy is studied. If $(X^n, g)$ is of hypergeometric type, namely spherical functions of $X$ are represented by the Gauss hypergeometric functions, the inversion formula, the convolution rule together with the Plancherel theorem are shown by the representation of the spherical functions in terms of the Gauss hypergeometric functions. A geometric characterization of hypergeometric type is derived in terms of volume density of geodesic spheres. Geometric properties of $(X^n, g)$ are also discussed.

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arXiv2019

Almost Hermitian structures on tangent bundles

In this article, we consider the almost Hermitian structure on $TM$ induced by a pair of a metric and an affine connection on $M$. We find the conditions under which $TM$ admits almost K\"ahler structures, K\"ahler structures and Einstein metrics, respectively. Moreover, we give two examples of K\"ahler-Einstein structures on $TM$.

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arXiv2018

Harmonic Hadamard manifolds and Gauss hypergeometric differential equations

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.

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arXiv2017

Geometric mean of probability measures and geodesics of Fisher information metric

The space of all probability measures having positive density function on a connected compact smooth manifold $M$, denoted by $\mathcal{P}(M)$, carries the Fisher information metric $G$. We define the geometric mean of probability measures by the aid of which we investigate information geometry of $\mathcal{P}(M)$, equipped with $G$. We show that a geodesic segment joining arbitrary probability measures $\mu_1$ and $\mu_2$ is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of $\mathcal{P}(M)$ can be joined by a unique geodesic. Moreover, we prove that the function $\ell$ defined by $\ell(\mu_1, \mu_2):=2\arccos\int_M \sqrt{p_1\,p_2}\,d\lambda$, $\mu_i=p_i\,\lambda$, $i=1,2$ gives the Riemannian distance function on $\mathcal{P}(M)$. It is shown that geodesics are all minimal.

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arXiv2017

Hessian of Busemann functions and rank of Hadamard manifolds

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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