We present statistical biharmonic maps, a new class of mappings between statistical manifolds naturally derived from a variation problem. We give the Euler-Lagrange equation of this problem and prove that improper affine hyperspheres induce examples of such maps.

A brief introduction of doubly minimal submanifolds of statistical manifolds is given. A complex submanifold of a holomorphic statistical manifold is doubly minimal. Similar properties are obtained in the case where the ambient space is a Sasakian statistical manifold.

We characterize centroaffine surfaces of cohomogeneity one such that each normal directrix lies on a plane containing the origin.

We define a kind of sectional curvature and delta-invariants for statistical manifolds. For statistical submanifolds the sum of the squared mean curvature and the squared dual mean curvature is bounded below by using the delta-invariant. This inequality can be considered as a generalization of the so-called Chen inequality for Riemannian submanifolds.

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.

We show that the statistical manifold of normal distributions is homogeneous. In particular, it admits a 2-dimensional solvable Lie group structure. In addition, we give a geometric characterization of the Amari-Chentsov alpha-connections on the Lie group.

We characterize centroaffine surfaces of cohomogeneity one which have vanishing Tchebychev vector fields. Moreover, we classify centroaffine minimal surfaces of cohomogeneity one which have centroaffine metrics of constant curvature.

We determine a nondegenerate centroaffine ruled surface such that the image of its center map lies on a plane containing the origin. Furthermore, we give a centroaffine minimal surface such that the image of its center map lies on a plane not containing the origin, and illustrate a one parameter family of centroaffine ruled surfaces having the same center map whose image lies on a circle on such a plane.

The statistical sectional curvature of the warped product of two statistical manifolds is studied. A notion of a Kenmotsu statistical manifold is introduced, which is locally obtained as the warped product of a holomorphic statistical manifold and a line.

A notion of Sasakian statistical structure is introduced. The condition for a real hypersurface in a holomorphic statistical manifold to admit such a structure is given.

We define the sectional curvature for a statistical structure, and study CR submanifolds in a holomorphic statistical manifold of constant holomorphic sectional curvature. We prove that this sectional curvature of such a space vanishes if it admits a totally-umbilical and dual-totally-umbilical generic submanifold. Furthermore, we show that a Lagrangian submanifold is of constant sectional curvature if the statistical shape operator and its dual operator commute. Similarly, we generalize several theorems in the classical CR submanifold theory.

Fujioka A., Furuhata H., and Sasaki T.;
Projective surfaces and pre-normalized Blaschke immersions of codimension two,
Int. Electron. J. Geom. 9(2016), 100--110.

We prove that any non-degenerate surface in the projective 3-space has a local lift as a minimal pre-normalized Blaschke immersion into the equicentroaffine 4-space. Furthermore, an indefinite surface in the projective 3-space has a local lift as a pre-normalized Blaschke immersion into the equicentroaffine 4-space satisfying the Einstein condition if and only if the surface is projectively applicable to an affine sphere.

Furuhata H., Hu N. and Vrancken, L.;
Statistical hypersurfaces in the space of Hessian curvature zero II,
J. Van der Veken, I. Van de Woestyne, L. Verstraelen, L. Vrancken (eds.), Pure and Applied Differential Geometry - PADGE 2012 In Memory of Franki Dillen, Shaker Verlag GmbH, Germany, ISBN-13: 978-3844023633, (2013), 136--142.
(HUSCAP)

We construct cylindrical statistical immersions between spaces of Hessian curvature zero.

Fujioka A., Furuhata H., and Sasaki T.;
Projective minimality for centroaffine minimal surfaces,
J. Geom. 105(2014), 87--102. (doi:10.1007/s00022-013-0196-9)

We give a classification of indefinite centroaffine minimal surfaces whose associated surfaces are minimal as surfaces in the projective 3-space.

Furuhata H.;
Statistical hypersurfaces in the space of Hessian curvature zero,
Differential Geom. Appl. 29(2011), S86--S90. (doi:10.1016/j.difgeo.2011.04.012)

A rigidity theorem for a statistical hypersurface of Hesse-Einstein type is given.

Furuhata H. and Kurose T.;
Hessian manifolds of nonpositive constant Hessian sectional curvature,
Tohoku Math. J. 65(2013), 31--42. (doi:10.2748/tmj/1365452623)

We classify the maximal Hessian manifolds of constant Hessaian sectional curvature non-positive.

Furuhata H.;
Hypersurfaces in statistical manifolds,
Differential Geom. Appl. 27(2009), 420--429. (doi:10.1016/j.difgeo.2008.10.019)

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.

Furuhata H.and Vrancken, L.;
The center map of an affine immersion,
Results Math. 49(2006), 201--217.

We study the center map of an equiaffine immersion which is introduced using the equiaffine support function. The center map is a constant map if and only if the hypersurface is an equiaffine sphere. We investigate those immersions for which the center map is affine congruent with the original hypersurface. In terms of centroaffine geometry, we show that such hypersurfaces provide examples of hypersurfaces with vanishing centroaffine Tchebychev operator. We also characterize them in equiaffine differential geometry using a curvature condition involving the covariant derivative of the shape operator. From both approaches, assuming the dimension is 2 and the surface is definite, a complete classification follows.

Furuhata H.;
Codazzi structures induced by minimal affine immersions,
Banach Center Publ. 57(2002), 17--19.

We give a necessary and sufficient condition for a statistical structure to be realized as a minimal affine hypersurface or a minimal centroaffine immersion of codimension two.

Furuhata H. and Kurose T.;
Self-dual centroaffine surfaces of codimension two with constant affine mean curvature,
Bull. Belgian Math. Soc. 9(2002), 573--587.

We explicitly determine the minimal, self-dual centroaffine surfaces in R^4 by giving a representation formula. Moreover, we describe the self-dual centroaffine surfaces with affine mean curvature -1.

Ichiyama T., Furuhata H. and Urakawa H.;
A conformal gauge invariant functional for Weyl structures and the first variation formula,
Tsukuba Math. J. 23(1999), 551 -- 564.

We consider a new conformal gauge invariant functional which is a natural curvature functional on the space of Weyl structures. We derive the first variation formula of its functional and characterize its critical points.

Furuhata H.;
Minimal centroaffine immersions of codimension two,
Bull. Belgian Math. Soc. 7(2000), 125 -- 134.

Minimal centroaffine immersions of codimension two are characterized as solutions of a certain variational problem. Moreover, we determine the moduli space of such immersions of R^2 into R^4 whose induced connection and affine fundamental form coincide with the ones of the Clifford torus respectively.

Furuhata H., Matsuzoe H. and Urakawa H;
Open problems in affine differential geometry and related topics,
Interdiscip. Inform. Sci. 4(1998), 125 -- 127.

Furuhata H. and Matsuzoe H.;
Holomorphic centroaffine immersions and the Lelieuvre correspondence,
Results Math. 33(1998), 294 -- 305.

The rigidity and intrinsic characterization of holomorphic centroaffine immersions are given. We also obtain a method to construct nondegenerate holomorphic affine hypersurfaces from centroaffine immersions and metrics satisfying some conditions.

Furuhata H.;
An intrinsic characterization of isometric pluriharmonic immersions with codimension one,
J. Geom. 65(1999), 111 -- 116.

We give an intrinsic characterization of isometric pluriharmonic immersions of Kaehler manifolds into semi-Euclidean spaces with real codimension one, which is a generalization of the Ricci-Curbastro theorem.

Furuhata H.;
A cylinder theorem for isometric pluriharmonic immersions,
Geom. Dedicata 66(1997), 303 -- 311.

We prove a cylinder theorem for isometric pluriharmonic immersions of complete Kaehler manifolds into semi-Euclidean spaces under an assumption concerning the index of relative nullity.

Furuhata H.;
Moduli space of isometric pluriharmonic immersions of Kaehler manifolds into indefinite Euclidean spaces,
Pacific J. Math. 176(1996), 1 -- 14.

We classify isometric pluriharmonic immersions of a Kaehler manifold into an indefinite Euclidean space. The moduli space of such immersions is explicitly constructed in terms of complex matrices. Some examples of these immersions are also given.

Furuhata H.;
Isometric Pluriharmonic Immersions of Kaehler Manifolds into Semi-Euclidean Spaces,
Tohoku Math. Publ. 1(1995), 1 -- 70.

We prove a classification of isometric pluriharmonic immersions of a Kaehler manifold into a semi-Euclidean space, which establishes a generalization of Calabi-Lawson's theory concerning minimal surfaces in Euclidean spaces. Then we study these immersions for complete Kaehler manifolds with low codimensions, and prove, in particular, a cylinder theorem and a Bernstein property. Moreover, we construct new examples of isometric pluriharmonic immersions.

Furuhata H.;
Construction and classification of isometric minimal immersions of Kaehler manifolds into Euclidean spaces,
Bull. London Math. Soc. 26(1994), 487--496.

A classification of isometric minimal immersions of Kaehler manifolds into Euclidean spaces is given, which is a generalization of Calabi-Lawson's theory concerning minimal surfaces. Moreover, we explicitly construct a nonholomorphic isometric minimal immersion of a complete Kaehler manifold, biholomorphic to C^2, into R^6.

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