Remarks on geodesics for multivariate normal models

MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」A complete description of the geodesic curves on the Riemann manifold of multivariate normal distributions equipped with the Fisher information metric has been accomplished by Eriksen in 1987, and later by Calvo and Oller in 1991 but in a different manner. The former describes geodesic curves in terms of an exponential map in somewhat mysterious way and the latter obtains a solution of the differential equation of a geodesic curve explicitly by solving much general system of differential equations. The method what Erikson had taken seems to have a group theoretic nature while it is still unclear. The purposes of this short note are to derive the explicit formula of the geodesic curve from the result obtained by Eriksen and to clarify why such exponential map may give geodesic curves for the one dimensional normal distribution case