Title and Abstract of Talks：
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Claudio Gorodski
（University of Sao Paulo, Brasil) 
Title: Isoparametric submanifolds in Hilbert space


Abstract:
A proper Fredholm submanifold M in a separable Hilbert space
is called "isoparametric" if (a) its normal bundle is flat; and (b) the shape
operators along any parallel normal vector field are conjugate.
Standard (homogeneous) examples of such submanifolds
arise essentially as principal orbits of isotropy representations of
affine KacMoody symmetric spaces (Terng).
In fact, in codimension different from one Heintze and Liu
proved that M must be homogeneous, but little is known about the
structure of the group which acts transitively on it.
In this talk, we will explain our contribution to the conjecture that
in codimension different from one M must be one of the standard examples.
Namely, we introduce a "homogeneous structure" on M and
use it to prove a rigidity theorem asserting that M
is completely determined by the second fundamental form and its
covariant derivative at one point,
thereby making such submanifolds accessible to classification.
Joint work with Ernst Heintze (Augsburg).


Laura Geatti
（University of Rome II Tor Vergata, Italy) 
Title:
Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space


Abstract:
Let $G/K$ be a noncompact symmetric space and let $G^{\bf c}/K^{\bf c}$ be its Lie group complexification.
Then $G^{\bf c}/K^{\bf c}$ is a Stein manifold where the Lie group $G$ acts by holomorphic transformations.
Basic questions in the study of $G^{\bf c}/K^{\bf c}$ are the classification of invariant Stein subdomains
and the determination of the envelopes of holomorphy of arbitrary invariant subdomains. In general,
such questions have a complete answer only inside a dististinguished $G$invariant domain containing $G/K$,
namely the complex crown $\Xi\subset G^{\bf c}/K^{\bf c}$. We present some new results in this direction
which hold in the Hermitian case.
We also show how the CauchyRiemann structure of the $G$orbits plays a role in this context.


Byung Hak Kim
（Kyung Hee University, Korea) 
Title: "On conformal transformations and conformally flat spaces"

Abstract:
The conformal transformation on Riemannian manifolds is characterized by Riemannian metrics,
which does not change the angle between two vectors at a point.
In this talk, we are survey to the various conformal transformations and their properties.
Moreover we consider the conformal transformations between complete product Riemannian manifolds,
and conformally flatness in the warped product space or more general space.




Imsoon Jeong, Carlos J.G. Machado, Juan de Dios Perez and Young Jin Suh 
Title: "$\mathfrak D$parallelism of normal and structure Jacobi operators for hypersurfaces in complex twoplane Grassmannians"


Abstract:
In this talk, we introduce notions of normal Jacobi operator and
structure Jacobi operator related to the curvature tensor for
hypersurfaces in complex twoplane Grassmannians $\GBt$ which
consists of all complex two dimensional linear subspaces in ${\Bbb
C}^{m+2}$, respectively. And we give nonexistence theorems for Hopf
hypersurfaces in $\GBt$ with ${\mathfrak D}$parallel normal Jacobi operator and
${\mathfrak D}$parallel structure Jacobi operator, respectively.


Young Suk Choi, Hyunjin Lee and Changhwa Woo 
Title: "Classification of real hypersurfaces with Reeb parallel shape operator in Complex twoplane Grassmannians"


Abstract:
In this paper we consider a new notion of Reeb parallel shape
operator for real hypersurface $M$ in complex twoplane
Grassmannian $\GBt$. When $M$ has the Reeb parallel shape operator
and nonvanishing geodesic Reeb flow, it becomes a real hypersurface
of Type~$(A)$ with exactly four distinct constant principal
curvatures.


Mijung Kim, Hyunjin Lee and Young Jin Suh 
Title: "A Real hypersurface in complex twoplane Grassmannians with gTanakaWebster recurrent shape operator" 

Abstract:
It is known that submanifolds in Kaehler manifolds have many kinds of connections.
Among them, we introduce a new connection named generalized TanakaWebster (in short, gTanakaWebster)
connection for real hypersurfaces in complex twoplane Grassmannians $\GBt$.
In this talk, we consider a new notion of recurrent hypersurfaces in $\GBt$ for gTanakaWebster connection and
give a nonexistence theorem for a Hopf hypersurface in $\GBt$ with gTanakaWebster recurrent shape operator.


Imsoon Jeong, Eunmi Pak and Young Jin Suh 
Title: "Parallelism of various generalized TanakaWebster invariant shape operators for real hypersurfaces
in complex twoplane Grassmannians"


Abstract:
In this talk, we introduce new notions of Lie invariant shape operator for a real hypersurface in complex twoplane Grassmannian $\GBt$, and give
classifications for Hopf hypersurfaces in $\GBt$ with Lie invariant shape operators in the generalized TanakaWebster connection.


Tohru Morimoto
（Emeritus Professor of Nara Women University, Japan) 
Title:
A KleinCartan programme for differential equations and extrinsic geometries in flag manifolds


Abstract: In 1872 Klein declared the Erlangen programme to understand various geometries in a unified manner via transformation groups
as homogeneous spaces, then in 1920's Cartan invented the notion of espace g\'en\'eralis\'e (principal bundle with Cartan connection
in modern terminology) to treat still group theoretically not only the homogeneous spaces but also inhomogeneous spaces such as
Riemannian geometries, conformal or projective differential geometries. With modern approaches to general equivalence problems of
geometric structures we have now a general transparent view to intrinsic geometries. In this talk we propose a Klein Cartan programme
for differential equations in the framework of nilpotent geometry and analysis. In particular, we show a categorical correspondence
between integrable overdetermined systems of linear partial differential equations and submanifolds in flag manifolds.
We then have a general method to find the invariants of a submanifold in a flag manifold, based on an algebraic harmonic theory
and the moving frame method, in the case when the relevant Lie algebra is semisimple.


Naoyuki Koike
（Tokyo University of Science, Japan) 
Title:
Certain kind of isoparametric submanifolds in symmetric spaces of noncompact type and Hermann actions


Abstract: In this talk, we state that full irreducible curvatureadapted isoparametric real analytic submanifolds of
codimension greater than one in a symmetric space $G/K$ of noncompact type are principal orbits of Hermann actions on $G/K$
under certain condition.
In the proof, it is key to show the homogeneity of the lift of the complexification of the original submanifold to
an infinite dimensional antiKaehler space through an antiKaehler submersion.


Hiroyuki Tasaki
（University of Tsukuba, Japan) 
Title:
Antipodal sets in compact Riemannian symmetric spaces


Abstract:
There is a symmetry at each point in a Riemannian symmetric space.
An antipodal set is a subset where the restriction of the symmetry
at each point is the identity, which was introduced by Chen and
Nagano.
A set of two antipodal points (in a usual sense) in a sphere is
a typical example of antipodal sets.
A maximal antipodal set is a kind of frame of a compact Riemannian
symmetric space.
In this talk I mainly explain antipodal sets in symmetric
Rspaces and oriented real Grassmann manifolds.


Osamu Ikawa
（Kyoto Institute of Technology, Japan) 
Title:
The geometry of orbits of Hermann actions


Abstract: The isotropy actions of compact symmetric spaces are typical examples of Hermann actions.
Hermann actions have nice properties. For example these are hyperpolar actions and also variational complete actions.
We study the orbits of Hermann actions.
In order to do this, we introduce a notion of a symmetric triad, which is a generalization of an irreducible root system.


Hiroshi Tamaru（Hiroshima University, Japan)

Title: On the moduli space of leftinvariant metrics on a Lie group


Abstract:
We introduce the space of leftinvariant metrics on a Lie group
up to isometry and scaling, the moduli space.
This moduli space derives a Milnortype theorem,
a generalization of the Milnor frame for threedimensional unimodular Lie groups.
Our Milnortype theorem is useful to examine
the existence and the nonexistence of
a distinguished leftinvariant metric, such as Einstein and Ricci soliton.
In this talk, we explain the above mentioned framework of our study
by describing some explicit examples.
We will also mention a pseudoRiemannian version.


Takashi Sakai (Tokyo Metropolitan University & OCAMI, Japan) 
Title: Special Lagrangian submanifolds in the complex shpere and the complex cone


Abstract: In 1993 Stenzel constructed CalabiYau metrics on the cotangent bundles of compact rank one symmetric spaces.
As the limit of the Stenzel metric on the cotangent bundle of the sphere,
where we call it the complex shpere, we can obtain a (singular) CalabiYau metric on the complex cone.
In this talk, I would like to demonstrate two methods to construct special Lagrangian submanifolds
in the complex shpere and the complex cone.
One is the conormal bundle construction, and the other is the moment map technique.


Misa Ohashi
（Meijo University & OCAMI, Japan)

Title:
On $G_2$ moduli of curves in purely imaginary octonions


Abstract:
In this talk, first we explain the $G_2$congruence theorem
and give the interesting example of curves in purely imaginary octonions.
Usually, the $SO(7)$invariants of the curves in $\textbf{R}^7$
are obtained by a standard Frenet Serret formula.
Note that the $G_2$invariants are $SO(7)$invariants
but the converse is not true.
We explain this phenomenon.
Also we give the candidate of $G_2$ moduli
of curves in purely imaginary octonions.


Makiko Sumi Tanaka
（Tokyo University of Science, Japan)

Title: Isometries of Hermitian symmetric spaces


Abstract:
This is my joint work with JostHinrich Eschenburg and Peter Quast. We consider the following problem.
Let $P$ be a Riemannian symmetric space and $G$ its symmetry group, that is, the subgroup of the full isometry group
which is genetrated by all geodesic symmetries.
Let $\iota : p \to V$ be an isometric $G$equivariant embedding into some Euclidian space $V$.
Hence any $g \in G$ extends to a linear isometry of $V$. But what happens to those isometries of $P$ which
are not contained in $G$?
Do they also extend to linear isometries of the ambient space?
We obtained the affirmative answer to the problem when $P$ is a semisimple Hermitian symmetric space.
In this talk I will explain our result and refer to some related topics.


Mr. Takahiro Hashinaga（Hiroshima University, Japan) 
Title:
Lowdimensional solvsolitons and the minimality of the corresponding submanifolds


Abstract:
The notion of the corresponding submanifolds to leftinvariant
Riemannian metrics on Lie groups is defined through the study on the
space of leftinvariant Riemannian metrics on a Lie group.
Our interest is to characterize a geometric property of leftinvariant
Riemannian metrics in terms of the corresponding submanifolds.
In this talk, we introduce the relationship between the existence of
solvsolitons on lowdimensional solvable Lie groups and the minimality
of the corresponding submanifolds.


Mr. Akira Kubo（Hiroshima University, Japan) 
Title:
Homogeneous Ricci soliton hypersurfaces in the complex hyperbolic spaces


Abstract:
A Lie hypersurface in the complex hyperbolic space is a homogeneous real
hypersurface without focal submanifolds. In this talk, I will present
classification results of Ricci soliton Lie hypersurfaces in the complex
hyperbolic space. This is a joint work with Takahiro Hashinaga and
Hiroshi Tamaru.

