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 \(\partial X\)—for a given homeomorphism \(\Phi\) of \(\partial X\) find an isometry of \(X\) whose \(\partial X\)-extension coincides with \(\Phi\)—is investigated in terms of the barycenter map.
Highlights
The aim of this article is to deal with two subjects related with information geometry
One is Fisher metric G defined on a space P(M ) of probability measures having continuous positive density function over a connected, compact manifold M, and another one is barycenter map from P(∂X) to an Hadamard manifold X, where ∂X is the ideal boundary of X
With respect to these theorems we are interested in characterization of Damek-Ricci space from information geometry, especially from a viewpoint of Fisher metric G, since a Damek-Ricci space is a counterexample of Lichnerowicz conjecture of non-compact version ([12]) and its characterization is only given by Heber in [13] by Lie group theory argument
Summary
The aim of this article is to deal with two subjects related with information geometry. Remark that the equality C = Q/n is derived from asymptotical formula related with mean curvature of geodesic spheres and mean curvature of corresponding horospheres, level hypersurfaces of Busemann function ([11]) With respect to these theorems we are interested in characterization of Damek-Ricci space from information geometry, especially from a viewpoint of Fisher metric G, since a Damek-Ricci space is a counterexample of Lichnerowicz conjecture of non-compact version ([12]) and its characterization is only given by Heber in [13] by Lie group theory argument. By approaching from a viewpoint of the ideal boundary ∂X, we focus on barycenter of probability measures on ∂X with respect to Busemann function and shed a light on information geometry of barycenter map bar : P(∂X) → X
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