open access

https://hdl.handle.net/2115/60372

Statistical hypersurfaces in the space of Hessian curvature zero Ⅱ

英語

2013-12

Shaker Verlag GmbH

136-142

We construct cylindrical statistical immersions between spaces of Hessian curvature zero.The words “statistical submanifold” can be found in the paper [5] in 1989, which was written by Vos in the context of statistical inference or information geometry. Although the history of this geometry is not so short, it is hard to find classical differential geometric approaches for the study of statistical submanifolds. In this paper, we would like to continue to try it after [2], and give some of basic examples of statistical submanifolds apart from applications for statistics. In other words, we will study immersions between statistical manifolds preserving statistical structures, which are called statistical immersions, in particular, called statistical hypersurfaces if the codimension equals one. We take a space Nn in Definition 1.1, which can be considered as a basic model of a statistical manifold of dimension n. The space Nn has been known as a Hessian manifold of constant Hessian curvature zero. In [2], a statistical hypersurface of a Hessian manifold of constant Hessian curvature negative into the space Nn+1 is uniquely determined. Besides, there exist no statistical hypersurfaces of a Hessian manifold of constant Hessian curvature positive into the space Nn+1. On the other hand, we have plenty of statistical hypersurfaces of Nn into Nn+1. In this paper, we determine statistical diffeomorphisms of Nn onto itself, and statistical hypersurfaces of Nn into Nn+1 with vanishing statistical second fundamental form (Propositions 2.1 and 2.2). Moreover, we explicitly construct and determine statistical immersions of a domain of N2 into N3 of cylinder type (Theorem 3.1).

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