[Show abstract][Hide abstract] ABSTRACT: Geometry of Fisher metric and geodesics on a space of probability measures defined on a compact manifold is discussed and is applied to geometry of a barycenter map associated with Busemann function on an Hadamard manifold X. We obtain an explicit formula of geodesic and then several theorems on geodesics, one of which asserts that any two probability measures can be joined by a unique geodesic. Using Fisher metric and thus obtained properties of geodesics, a fibre space structure of barycenter map and geodesical properties of each fibre are discussed. Moreover, an isometry problem on an Hadamard manifold X and its ideal boundary ∂X-for a given homeomorphism Φ of ∂X find an isometry of X whose ∂X-extension coincides with Φ-is investigated in terms of the barycenter map.
[Show abstract][Hide abstract] ABSTRACT: Using Busemann function of an Hadamard manifold X we define the barycenter map from the space 𝒫+(∂X, d θ) of probability measures having positive density on the ideal boundary ∂X to X. The space 𝒫+(∂X, d θ) admits geometrically a fiber space structure over X from Fisher information geometry. Following the arguments in [E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986) 23–48; G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de coubure strictement négative, Geom. Funct. Anal.5 (1995) 731–799; Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems16 (1996) 623–649], we exhibit that under certain geometrical hypotheses a homeomorphism Φ of the ideal boundary ∂X induces, by the aid of push-forward, an isometry of X whose extension is Φ.
International Journal of Mathematics 03/2015; 26(6):1541007. DOI:10.1142/S0129167X15410074 · 0.60 Impact Factor
[Show abstract][Hide abstract] 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; 35. DOI:10.1016/j.difgeo.2014.02.001 · 0.69 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let X be a Hadamard manifold. By applying the stable Jacobi tensor along horospherical foliations of X, a characterization of the real space form is obtained by means of the second fundamental form of the horospheres. The complex space form, the quaternionic space form and the other rank-one symmetric spaces of non-compact type are also similarly characterized. Geometrical characterization of horospheres are also given.
[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 08/2011; 29. DOI:10.1016/j.difgeo.2011.04.015 · 0.69 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: For Damek-Ricci 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 06/2010; 33(2010). DOI:10.3836/tjm/1279719582 · 0.22 Impact Factor
[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 Laplace-Beltrami operator ∆ = −− i i on the n-dimensional 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 n-dimensional Rie-mannian 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 bound-ary ∂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 [Sch-Y]. Thus, any Poisson kernel together with the standard vol-ume 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.
[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 non-compact type and, also discuss monotonicity of the homothety constant in time.
[Show abstract][Hide abstract] ABSTRACT: An isolation theorem of Weyl conformal tensor of positive Einstein manifolds is given, when its $L^{n/2}$-norm is small.
Proceedings of the Japan Academy Series A Mathematical Sciences 09/2002; 78(7). DOI:10.3792/pjaa.78.140 · 0.22 Impact Factor