Concentrated Lectures：
 Professor Juergen Berndt :
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 Title: Submanifolds in Symmetric Spaces

 Abstract:
A symmetric space is a Riemannian manifold whose curvature tensor is invariant
under parallel translation.
The theory of symmetric spaces has been initiated and developed
to a large extent by Elie Cartan.
It is closely related to the algebraic theory of semisimple Lie algebras.
In the lectures I will present some recent developments on submanifolds in
symmetric spaces.
Topics include: submanifolds with constant principal curvatures,
homogeneous submanifolds, hyperpolar foliations, cohomogeneity one actions,
normal holonomy, and branched fibrations from projective planes onto spheres.


Differential Geometry Special Seminar :


Juergen Berndt （University College Cork) : PDF file 
"Singular fibrations from projective planes and Severi varieties onto spheres" (50分)

Abstract:
There is an elementary but very striking result which asserts that the quotient of the
complex projective plane by complex conjugation is the 4dimensional sphere.
A few years ago Arnold and independently Atiyah and Witten proved that the quotient of the quaternionic
projective plane by a certain circle action is a 7dimensional sphere.
In the first part of our work we extend the above two results to the Cayley projective plane and provide
a unifying proof for all three projective planes.
Each projective plane over a normed real division algebra has a naturalcomplexification,
which is known as a Severi variety. In the second part of our work we extend the above results to
the Severi varieties. This is joint with with Sir Michael Atiyah (Edinburgh).


Andreas Kollross (University of Augsburg) : 
"On a class of generalized symmetric spaces" (50 minutes) 
Abstract:
The classical notion of symmetric spaces may be generalized by replacing
the group Z/2Z with an arbitrary finite group. In this talk, I will
speak about the case where this group is isomorphic to Z/2Z x Z/2Z.
Recently, Yuri Bahturin and Michel Goze have given a classification of
the Z/2Z x Z/2Zsymmetric spaces G/K where G is a simple classical Lie
group. I will present a classification for the case of exceptional
groups, complementing the results of Bahturin and Goze.
The results are equivalent to a classification of Z/2Z x Z/2Zgradings
on simple Lie algebras. In submanifold geometry, Z/2Z x Z/2Zsymmetric
spaces correspond to totally geodesic orbits of Hermann actions.


Andreas Kollross (University of Augsburg) 
"Lie group actions on symmetric spaces" (90 minutes) PPT file 
In this lecture, I will consider isometric Lie group actions on compact
Riemannian symmetric spaces. In particular, I am interested in
classification problems where one seeks to find all isometric actions on
a symmetric space such that the principal orbits fulfill a certain
geometric condition. If this condition implies that the group acting is
large in a certain sense, we may use a straightforward classification
strategy as follows. Starting with the maximal subgroups of the
isometry group, we recursively descend to smaller groups until we arrive
at groups which are too small to fulfill the condition. To illustrate
this method, I will present results e.g. on polar and low cohomogeneity
actions on compact symmetric spaces.


Naoyuki Koike（Tokyo U. of Science) : PDF file 
"Homogeneity theorem for a proper complex equifocal submanifold" (50 minutes)

Abstract : In this talk, I first explain the notion of a complex focal
radius of a submanifold in a symmetric space of noncompact type and,
in the case where the submanifold is real analytic, the complex focal
radii are the quantities indicating the positions of the focal points
of the complexified submanifold. Next I explain the notion of a proper
complex equifocal submanifold. Also I explain the notion of an infinite
dimensional proper antiKaehlerian isoparametric submanifold and complex
curvature distributions, complex principal curvatures and complex
curvature normals of the submanifold. Next I introduce the homogeneity
theorem for an infinite dimensional isoparametric submanifold by
HeintzeLiu and the homogeneity theorem for an equifocal submanifold in
a symmetric space of compact type by U. Christ. Next I state the
homogeneity theorem for a proper complex equifocal submanifold and its
proof. Finally I state the following future plan:
"I plan to obtain a submanifold geometrical characterization of a principal
orbit of a Hermann type action in terms of this homogeneity theorem".


Imsoon Jeong (NIMS), Young Jin Suh and Hyunjin Lee (KNU) : PDF file 
"Real hypersurfaces in complex twoplane Grassmannians
with anticommuting shape operator" (30 minutes) 
Abstract:
In this talk we give a nonexistence theorem for real hypersurfaces in
complex twoplane Grassmannians $G_2({\Bbb C}^{m+2})$ with anticommuting shape operator.


Imsoon Jeong (NIMS), Hee Jin Kim and Young Jin Suh (KNU) : PDF file 
"Real hypersurfaces in complex twoplane Grassmannians
with parallel normal Jacobi operator" (30 minutes) 
Abstract:
In this talk we give a nonexistence theorem for Hopf real
hypersurfaces in complex twoplane Grassmannians $G_2({\Bbb C}^{m+2})$
with parallel normal Jacobi operator ${\bar R}_N$.


Hae Young Yang, Hyung Jun Jin and Young Jin Suh (KNU) : PDF file 
"Real hypersurfaces in complex twoplane Grassmannians
with ${\frak D}^{\bot}$parallel Lie derivative" (30 minutes)

Abstract:
In this talk we give some nonexistence properties of real
hypersurfaces in complex twoplane Grassmannians $G_2({\Bbb C}^{m+2})$
in terms of {\it ${\frak D}^{\bot}$parallel Lie
derivatives}
for the structure tensor ${\phi}_i$, $i=1,2,3$, the shape operator
$A$, and the induced Riemannian metric tensor $g$ along the
distribution ${\frak D}^{\bot} =
\text{Span}\{{\xi}_1,{\xi}_2,{\xi}_3\}$.


Mayuko Kon（Hokkaido University＆OCAMI) : PDF file 
"On a Hopf hypersurface of a complex space form" (30 minutes)

Abstract：
A Hopf hypersurface is defined to be a real hypersurface
of a complex space form whose structure vector field is a
principal curvature vector field. We study some conditions
on the holomorphic distribution on real hypersurfaces which
contain the definition of a Hopf hypersurface. On the other
hand, we study real hypersurfaces whose structure vector field
is an eigenvector field of the Ricci operator.


Hiroshi Iriyeh（Tokyo Denki University) : DVI file 
"On global tightness of real forms in Hermitian symmetric spaces" (30分) 
Abstract：
We give an idea for proving global tightness of real forms in Hermitian
symmetric spaces. Our method is to combine Lagrangian intersection
theory and integral geometry. In this short talk, we show that it works
successfully for totally geodesic Lagrangian sphere in $S^2 \times S^2
\cong Q_2({\mathbb C})$.
This is a joint work with Takashi Sakai.


Kyoko Honda（Ochanomizu Women University): 
"Conformally flat homogeneous Lorentzian manifolds" (30 minutes) 
Abstract：
We would like to classify conformally flat homogeneous semiRiemannian
manifolds. Conformally flat homogeneous Riemannian manifolds were
classified by Hitoshi Takagi. They are all symmetric spaces.
On the other hand, threedimensional conformally flat
homogeneous Lorentzian manifolds were classified by Honda and Tsukada
and the examples which are not symmetric spaces were found.
In this talk, we report the results on the classification of
higher dimensional conformally flat homogeneous Lorentzian manifolds.


Yumiko Kitagawa（OCAMI) : PDF file 
"Geodesics on SubRiemannian manifolds" (30 minutes) 
Abstract:
A SubRiemannian manifold (M,D,g) is a differential
manifold M endowed with a subbundle D of the tangent
bundle TM and a Riemannian metric g on D. In this talk we
will treat the problem of lengthminimizing paths in
SubRiemannian geomerty.


Kazuhiro Shibuya（Hokkaido University, D3) : PDF file 
"A set of integral elements of higher order jet spaces" (30 minutes) 
Abstruct:
I consider the prolongation of jet spaces(multi independent variables).
Generally, it is not a manifold. But the prolongation of J^2(2,1) is
a manifold(in some sense, this is an only case), and a generalization of
so called "Monster Goursat Manifold"(one independent variable). In this
talk, I will introduce singularities of the prolongation of J^2(2,1) as a
sense of differential systems.


Shinobu Fujii（Hiroshima University, D3) : PDF file 
"Homogeneous isoparametric hypersurfaces with four distinct principal
curvatures and moment maps" (30 minutes) 
Abstract：
The isotropy representations of Hermitian symmetric spaces are
Hamiltonian actions.
In this talk, we consider the case of rank two, and we explain that
(weighted) square norms of their moment maps are isoparametric
functions, which define homogeneous isoparametric hypersurfaces with
four distinct principal curvatures in spheres.
We expect that isoparametric hypersurfaces with four distinct principal
curvatures in spheres are related to moment maps of group actions.


Hironao Kato（Hiroshima University, D1) : PDF file 
"Left invariant flat projective structures on low dimensional Lie
groups" (30 minutes)

Abstract:
A left invariant flat projective structure is a geometric structure,
which is derived through abstracting the properties about shapes of
geodesics from flat affine structures. We study the existence or
nonexistence of left invariant flat projective structures on Lie
groups. As a result we obtain any Lie group of dimension < 6 admits
a left invariant flat projective structure. We also classify Lie groups
of dimension <6 admitting left invariant flat affine structures.


Taro Kimura（OCAMI) : PDF file 
"Stability of certain reflective orbits of cohomogeneity one
actions on compact Riemannian symmetric spaces" (30 minutes) 
Abstract:
We know every totally geodesic singular orbit of cohomogeneity one
actions on simply connected irreducible Riemannian symmetric spaces
of compact type by virtue of BerndtTamaru's results. In this talk,
we consider their stability as minimal submanifolds. We determine the
stability of certain reflective orbits of cohomogeneity one actions on
compact Riemannian symmetric spaces.


Kurando Baba（Tokyo U. of Science, D3) : PDF file 
"Austere submanifolds and srepresentations of semisimple
pseudoRiemannian symmetric spaces" (30 minutes) 
Abstract:
The notion of an austere submanifold was introduced by Harvey
and Lawson. In this talk, we introduce the notion of an austere
submanifold in a pseudoRiemannian manifold, which is a
pseudoRiemannian submanifold where for each normal vector, the
spectrum of the complexification of its shape operator is invariant
under the multiplication by 1. We investigate austere semisimple orbits of
srepresentations of semisimple pseudoRiemannian symmetric spaces, and
demonstrate how to classify such orbits. The method is based on the
theory of restricted root systems for semisimple symmetric spaces.

