Geometry of Fisher Information Metric and the Barycenter Map†

Entropy

Mitsuhiro Itoh¹ 

[1] Institute of Mathematics, University of Tsukuba, 1-1-1, Ten-noudai, Tsukuba, 305-8571, Japan

关键词: Fisher metric;    probability measure;    geodesic;    Busemann function;    barycenter;   

DOI  :  10.3390/e17041814

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland