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A study of statistical submersions
摘要
In the sixties, A. Gray and B. O'Neill come with the notion of Riemannian submersions as a tool to study the geometry of a Riemannian manifold with an additional structure in terms of the fibers and the base space. Riemannian submersions have long been an effective tool to construct Riemannian manifolds with positive or nonnegative sectional curvature in Riemannian geometry and compare certain manifolds within differential geometry. In particular, many examples of Einstein manifolds can be constructed by using such submersions. It is very well known that Riemannian submersions have applications in physics, for example Kaluza-Klein theory, Yang-Mills theory, supergravity and superstring theories. In, Watson popularizes the knowledge of Riemannian submersions between almost Hermitian manifolds under the name of almost Hermitian submersions and many researchers discuss such submersions between various subclasses of almost Hermitian manifolds. Then, Sahin extends Riemannian submersions to many subclasses of almost contact metric manifolds under the title of contact Riemannian submersions in. Afterwards, B. Sahin comes with a self-contained exposition of recent developments in Riemannian submersions and maps. On the other hand, B. Nielsen and Jupp discuss the Riemannian submersion from the viewpoint of statistics. N. Abe and K. Hasegawa introduce the notion of statistical submersions between statistical manifolds by generalizing some basic results of B. O'Neill concerning Riemannian submersions and geodesics. Since then, the study of submersions became an active research subject, and many papers have been published by numerous of geometers (see). The purpose of this article is to provide a comprehensive survey on the study of recent developments in statistical submersions.