FURUHATA Hitoshi
Department of Mathematics
Hokkaido University
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Fujioka A. and Furuhata H.;
Centroaffine surfaces of cohomogeneity one with
planar curvature lines
,
Preprint.

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

Furuhata H., Hasegawa I. and Satoh N.;
Chen invariants and statistical submanifolds
,
Preprint.

We define a kind of sectional curvature and deltainvariants
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 deltainvariant.
This inequality can be considered as a generalization
of the socalled Chen inequality for Riemannian submanifolds.

Satoh N., Furuhata H., Hasegawa I., Nakane T., Okuyama Y., Sato K.,
Shahid, M.H. and Siddiqui, A.N.;
Statistical submanifolds from a viewpoint of the Euler inequality
,
Inf. Geom. 4(2021), 189213.
(doi:
10.1007/s41884020000324)

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 totallyumbilical submanifolds are also illustrated.

Furuhata H., Inoguchi J. and Kobayashi S.;
A characterization of the alphaconnections on the statistical manifold of normal distributions
,
Inf. Geom. 4(2021), 177188.
(doi:
10.1007/s4188402000037z)

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

Fujioka A. and Furuhata H.;
Centroaffine surfaces of cohomogeneity one
,
Bull. Braz. Math. Soc., NS. 50(2019), 291313.
(doi:
10.1007/s005740180120x)

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.

Fujioka A. and Furuhata H.;
The center map of a centroaffine ruled surface,
An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 64(2018), 343355.
(pdf)

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.

Furuhata H., Hasegawa I., Okuyama Y. and Sato K.;
Kenmotsu statistical manifolds and warped product,
J. Geom. 108(2017), 11751191.
(doi:
10.1007/s0002201704031)

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.

Furuhata H., Hasegawa I., Okuyama Y., Sato K. and Shahid, M.H.;
Sasakian statistical manifolds
,
J. Geom. Phys. 117(2017), 179186.
(doi:
10.1016/j.geomphys.2017.03.010)

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.

Furuhata H., and Hasegawa I.;
Submanifold theory in holomorphic statistical manifolds
,
S. Dragomir, M.H. Shahid, and F.R. AlSolamy (eds),
Geometry of CauchyRiemann Submanifolds,
Springer Singapore,
ISBN: 9789811009150, (2016),
179215.
(doi:
10.1007/9789811009167_7)

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 totallyumbilical and dualtotallyumbilical
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 prenormalized Blaschke immersions of codimension two,
Int. Electron. J. Geom. 9(2016), 100110.

We prove
that any nondegenerate surface in the projective 3space has a local lift
as a minimal prenormalized Blaschke immersion into the equicentroaffine 4space.
Furthermore, an indefinite surface in the projective 3space has a local lift
as a prenormalized Blaschke immersion into the equicentroaffine 4space
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, ISBN13: 9783844023633,
(2013), 136142.
(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), 87102.
(doi:10.1007/s0002201301969)

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

Furuhata H.;
Statistical hypersurfaces in the space of Hessian curvature zero,
Differential Geom. Appl. 29(2011), S86S90.
(doi:10.1016/j.difgeo.2011.04.012)

A rigidity theorem for a statistical hypersurface
of HesseEinstein type is given.

Furuhata H. and Kurose T.;
Hessian manifolds of nonpositive constant Hessian sectional
curvature,
Tohoku Math. J. 65(2013), 3142.
(doi:10.2748/tmj/1365452623)

We classify the maximal Hessian manifolds
of constant Hessaian sectional curvature nonpositive.

Furuhata H.;
Hypersurfaces in statistical manifolds,
Differential Geom. Appl. 27(2009), 420429.
(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), 201217.

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), 1719.

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.;
Selfdual centroaffine surfaces of codimension two
with constant affine mean curvature,
Bull. Belgian Math. Soc. 9(2002), 573587.

We explicitly determine
the minimal, selfdual centroaffine surfaces
in R^4 by giving a representation formula.
Moreover, we describe the selfdual 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
semiEuclidean spaces
with real codimension one,
which is a generalization of the RicciCurbastro 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 semiEuclidean 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 SemiEuclidean Spaces,
Tohoku Math. Publ. 1(1995), 1  70.

We prove a classification of isometric pluriharmonic immersions
of a Kaehler manifold into a semiEuclidean space,
which establishes a generalization of CalabiLawson'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), 487496.

A classification of
isometric minimal immersions of Kaehler manifolds
into Euclidean spaces is given,
which is a generalization
of CalabiLawson'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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