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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.
Proceedings of The Twelfth International Workshop on Diff. Geom. 01/2008; 12:1-20.
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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.
International Journal of Pure and Applied Mathematics ————————————————————————– Volume. 01/2008; 46:347-353.
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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 29. · 0.65 Impact Factor
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ABSTRACT: An isolation theorem of Weyl conformal tensor of positive Einstein manifolds is given, when its $L^{n/2}$-norm is small.
