Publications (7)1.59 Total impact
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ABSTRACT: From geometrical study of horospheres we obtain, among asymptotically harmonic Hadamard manifolds, a rigidity theorem of the complex hyperbolic space CHmCHm with respect to volume entropy. We also characterize CHmCHm horospherically in terms of holomorphic curvature boundedness. Corresponding quaternionic analogues are obtained.Differential Geometry and its Applications 09/2014; · 0.48 Impact Factor 
Article: Horospheres and hyperbolic spaces
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ABSTRACT: Let X be a Hadamard manifold. The authors apply the stable Jacobi tensor along horospherical foliations of X, to give a characterization of the real space form by means of the second fundamental form of the horospheres. The complex space form, the quaternionic space form and the other rankone symmetric spaces of noncompact type are also similarly characterized. They also present some geometrical characterization of horospheres.Kyushu Journal of Mathematics 01/2013; 67(2). · 0.36 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Let (X,g) be an Hadamard manifold with ideal boundary ∂X. We can then define the map φ:X→P(∂X) associated with Poisson kernel on X, where P(∂X) is the space of probability measures on ∂X, together with the Fisher information metric G. We make geometrical investigation of homothetic property and minimality of this map with respect to the metrics g and G. The map φ is shown to be a minimal homothetic embedding for a rank one symmetric space of noncompact type as well as for a nonsymmetric Damek–Ricci space. The following is also obtained. If φ is assumed to be homothetic and minimal, then, (X,g) turns out to be an asymptotically harmonic, visibility manifold with the Poisson kernel being expressed in terms of the Busemann function.Differential Geometry and its Applications 01/2011; 29. · 0.48 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: For DamekRicci spaces $(X,g)$ we compute the exact form of the Busemann function which is needed to represent the Poisson kernel of $(X,g)$ in exponential form in terms of the Busemann function and the volume entropy. From this fact, we show that the Poisson kernel map $\varphi: (X,g) \rightarrow (\mathcal{P}(\partial X),G)$ is a homothetic embedding. Here $\mathcal{P}(\partial X)$ is the space of probability measures having positive density function on the ideal boundary $\partial X$ of $X$, and $G$ is the Fisher information metric on $\mathcal{P}(\partial X)$.Tokyo Journal of Mathematics 01/2010; 33(2010). · 0.26 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A heat kernel map is defined from a complete Riemannian manifold (X, h) into the space P(X) of probabilities on X whose density function is positive. By pulling back, by this heat kernel map, the Fisher information metric on it, we assert that this induced metric is homothetic to the original metric of X, provided (X, h) is a rank one symmetric space of noncompact type and, also discuss monotonicity of the homothety constant in time.International Journal of Pure and Applied Mathematics ————————————————————————– Volume. 01/2008; 46:347353.  [Show abstract] [Hide abstract]
ABSTRACT: Information geometry of Poisson kernels and heat kernel on an Hadamard manifold X which is harmonic is discussed in terms of the Fisher information metric. The Poisson kernel map and the heat kernel map, both, turn out to be a homothetic immersion from X into the space of probability measures. Certain geometric properties of Shannon's entropy for the Poisson kernels and the heat kernel are shown. It is verified that there exists a crucial relation associated with the heat kernel between the homothety constant and the Shannon's entropy. 0 Introduction and Main Theorems The classical Dirichlét problem for the LaplaceBeltrami operator ∆ = −− i i on the ndimensional unit ball in R n is solved by the integration as u(x) = ∂B n Φ(x, θ)ψ(θ)dθ by using the Poisson kernel. Now, let (X, h) be a connected, simply connected, complete ndimensional Riemannian manifold with curvature strictly bounded as −b 2 ≤ K X ≤ −a 2 < 0. We can consider in this case the Dirichlét problem with respect to the ideal boundary ∂X. (X, h) admits, then, Poisson kernels Φ(x, θ) such that any solution to the Dirichlét problem with boundary condition at infinity is written as the Poisson kernel integration, similarly as on the Euclidean unit ball. The detailed argument is referred to [SchY]. Thus, any Poisson kernel together with the standard volume form gives rise to a probability measure on the ideal boundary so that we can consider a map from X into P(∂X), the space of probability measures on ∂X, by assigning to any point x in X a probability measure ρ = Φ(x, θ)dθ on ∂X.Proceedings of The Twelfth International Workshop on Diff. Geom. 01/2008; 12:120.  [Show abstract] [Hide abstract]
ABSTRACT: The Fisher information is a Riemannian metric on the space P(Y) of all probability measures with density of a fixed measure space (Y,μ). The sectional curvature of this infinitedimensional Riemannian manifold is positive and constant [see Th. Friedrich, Math. Nachr. 153, 273–296 (1991; Zbl 0792.62003)]. Consider a Hadamard manifold X and its boundary Y=∂X. The Poisson kernel, related with the Dirichlet problem at infinity, defines an embedding of X into P(∂X). The authors announce some geometric properties of this map. In case that X is a rank one symmetric space of noncompact type or a nonsymmetric DamekRicci space, the map X→P(∂X) is homothetic and harmonic. Conversely, suppose that the map has the two properties for a fixed Hadamard manifold. Then the Poisson kernel can be represented by the Busemann function and X is a visible, asymptotically harmonic space. Moreover, if X admits a compact quotient, then X must be a rank one symmetric space of noncompact type.
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1  Citation  
1.59  Total Impact Points  
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Institutions

2011–2014

Tokyo Denki University
 School of Information Environment
Edo, Tōkyō, Japan
