The condition for the curvature of a statistical manifold 
to admit a kind of standard hypersurface is given 
as a first step of the statistical submanifold theory. 
A complex version of the notion of statistical structures 
is also introduced.

/pdf/hypersurfaces-in-statistical-manifolds-47wqoawofi.pdf

Hypersurfaces in Statistical Manifolds

Chapter 3 - Riemannian Submanifolds

We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss–Bonnet theorem using heat equation methods, to give a new proof of a result of Kuzʼmina and Labbi concerning the Euler–Lagrange equations of the Gauss–Bonnet integral, and to give a new derivation of the Euh–Park–Sekigawa identity.

Universal curvature identities

In this article, we introduce the notion of pseudo-Einstein real hypersurfaces in the complex quadric Qm=SOm+2/SO2SOm and give a complete classification of such hypersurfaces.

Pseudo-Einstein real hypersurfaces in the complex quadric

In this article, we derive Chen’s inequalities involving Chen’s δ-invariant δM, Riemannian invariant δ(m1,⋯,mk), Ricci curvature, Riemannian invariant Θk(2≤k≤m), the scalar curvature and the squared of the mean curvature for submanifolds of generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application of the obtain inequality, we first derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms.

/pdf/improved-chen-s-inequalities-for-submanifolds-of-generalized-2iz3ztwu.pdf

Improved Chen's Inequalities for Submanifolds of Generalized Sasakian-Space-Forms

We prove that if the sectional curvatures for plane sections containing the structure vector field of a real hypersurface in a complex space form are equal to the same constant at every point, then the real hypersurface is a Hopf hypersurface. We also find some applications of the main result.

https://www.sci.osaka-cu.ac.jp/~ohnita/2008/Kon.pdf

On a Hopf hypersurface of a complex space form

We consider Kahler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ ( 2 , 2 ) .

Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

/pdf/ricci-tensor-of-real-hypersurfaces-3slfkt7qq9.pdf

Ricci tensor of real hypersurfaces

Semi-parallel CR submanifolds in a complex space form

We study affine hypersurface immersionsf : M ! R 2n+1 , where M is an almost complex n-dimensional manifold. The main purpose is to give a condition for (M, J) to be a special Kahler manifold with respect to the Levi-Civita connection of an affine fundamental form.

/pdf/hypersurfaces-with-almost-complex-structures-in-the-real-2qemwfq45n.pdf

Mayuko Kon

Papers

On a Hopf hypersurface of a complex space form

Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

Ricci tensor of real hypersurfaces

Semi-parallel CR submanifolds in a complex space form

Hypersurfaces with almost complex structures in the real affine space