Statistical submanifolds from a viewpoint of the Euler inequality
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Statistical submanifolds from a viewpoint of the Euler inequality Naoto Satoh1 · Hitoshi Furuhata1 · Izumi Hasegawa2 · Toshiyuki Nakane4 · Yukihiko Okuyama4 · Kimitake Sato4 · Mohammad Hasan Shahid3 · Aliya Naaz Siddiqui3 Received: 1 October 2019 / Revised: 15 April 2020 / Accepted: 21 August 2020 © Springer Nature Singapore Pte Ltd. 2020
Abstract We generalize the Euler inequality for statistical submanifolds. Several basic examples of doubly autoparallel statistical submanifolds in warped product spaces are described, for which the equality holds at each point. Besides, doubly totally-umbilical submanifolds are also illustrated. Keywords Statistical manifolds · Warped product · The Euler inequality · Doubly autoparallel submanifolds · Doubly totally-umbilical submanifolds Mathematics Subject Classification 53B25 · 53C15 · 53B35
1 Introduction The statistical submanifold theory originates in Information geometry and is now deepening through pure differential geometry. We begin by taking another look at the following elementary fact. For a surface in the Euclidean 3-space, the mean curvature squared is not less than the Gaussian curvature at each point. This property is often called the Euler inequality. A totally umbilical surface, namely, a part of a plane or a part of a round sphere, is characterized as the surface such that the equality holds at each point. This simple and beautiful theorem has been generalized to various cases (e.g. [7,8] as the Chen inequality, [9,13,15] and [22] as the Wintgen inequality). In this paper, we give the counterpart in the statistical submanifold theory. A statistical manifold is a manifold endowed with a pair of torsion-free affine connection and Riemannian metric satisfying the Codazzi equation. It can be considered as a generalization of a Hessian manifold, which is in the research area Jean–Louis Koszul strongly influenced. The geometry of statistical manifolds is developing as deforma-
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Naoto Satoh [email protected]
Extended author information available on the last page of the article
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Information Geometry
tion of Riemannian geometry, related to Information geometry, Hessian geometry, the differential geometry of affine hypersurfaces and many other fields [1,18]. The geometric study of the statistical submanifold theory began in the early 2010s. Several authors have worked on generalizations of the Euler inequality for statistical submanifolds ([2,3,16,17], and see also more recent similar researches [4–6,19] and references therein). However, it is hard to find researches about the statistical submanifolds such that the equality holds, though such submanifolds must play important roles because they should be the objects like a plane or a sphere in each setting. In this paper, we will illustrate several such examples. In Sect. 2, we review the statistical manifold theory briefly. We give several examples of statistical manifolds. One of them has the hyperbolic space (Hm , g) of constant negative curvature −κ as an underlying Riemannian manifold (Exam
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