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{(Japanese)

# "Submanifold Geometry and Lie Theoretic Methods"

 Date: 30 (Thu.) October - 3 (Mon.) November, 2008 Place: Graduate School of Science, Osaka City University

 Organizers: Hiroo NAITOH (Yamaguchi U.), Hiroshi TAMARU (Hiroshima U.), Naoyuki KOIKE (Tokyo U. of Science), Takashi SAKAI (OCAMI), Young Jin SUH (Kyungpook National U.), Yoshihiro OHNITAiOCU, headj

 Sponsors F Osaka City University Advanced Mathematical Institute (OCAMI). Kyungpook National University (KNU) Research Group on Real & Complex Grassmann Manifolds Math. Computation Divisin of The Post BK21.

 Invited speakers F Professor Juergen BerndtiUniversity College Cork, Ireland) Professor Andreas Kollross (University of Augsburg, Germany)

 Concentrated LecturesF Professor Juergen Berndt : PDF t@C 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 iUniversity College Cork) : PDF file
"Singular fibrations from projective planes and Severi varieties onto spheres" (50)
Abstract: There is an elementary but very striking result which asserts that the quotient of the complex projective plane by complex conjugation is the 4-dimensional 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 7-dimensional 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/2Z-symmetric 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/2Z-gradings on simple Lie algebras. In submanifold geometry, Z/2Z x Z/2Z-symmetric 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 KoikeiTokyo 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 non-compact 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 anti-Kaehlerian 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 Heintze-Liu 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 two-plane Grassmannians with anti-commuting shape operator" (30 minutes)
Abstract: In this talk we give a non-existence theorem for real hypersurfaces in complex two-plane Grassmannians $G_2({\Bbb C}^{m+2})$ with anti-commuting shape operator.
Imsoon Jeong (NIMS), Hee Jin Kim and Young Jin Suh (KNU) : PDF file
"Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator" (30 minutes)
Abstract: In this talk we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane 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 two-plane Grassmannians with ${\frak D}^{\bot}$-parallel Lie derivative" (30 minutes)
Abstract: In this talk we give some non-existence properties of real hypersurfaces in complex two-plane 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 KoniHokkaido UniversityOCAMI) : PDF file
"On a Hopf hypersurface of a complex space form" (30 minutes)
AbstractF 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 IriyehiTokyo Denki University) : DVI file
"On global tightness of real forms in Hermitian symmetric spaces" (30)
AbstractF 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 HondaiOchanomizu Women University):
"Conformally flat homogeneous Lorentzian manifolds" (30 minutes)
AbstractF We would like to classify conformally flat homogeneous semi-Riemannian manifolds. Conformally flat homogeneous Riemannian manifolds were classified by Hitoshi Takagi. They are all symmetric spaces. On the other hand, three-dimensional 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 KitagawaiOCAMI) : PDF file
"Geodesics on Sub-Riemannian manifolds" (30 minutes)
Abstract: A Sub-Riemannian 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 length-minimizing paths in Sub-Riemannian geomerty.
Kazuhiro ShibuyaiHokkaido 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 FujiiiHiroshima University, D3) : PDF file
"Homogeneous isoparametric hypersurfaces with four distinct principal curvatures and moment maps" (30 minutes)
AbstractF 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 KatoiHiroshima 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 non-existence 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 KimuraiOCAMI) : 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 Berndt-Tamaru'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 BabaiTokyo U. of Science, D3) : PDF file
"Austere submanifolds and s-representations of semisimple pseudo-Riemannian 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 pseudo-Riemannian manifold, which is a pseudo-Riemannian 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 s-representations of semisimple pseudo-Riemannian symmetric spaces, and demonstrate how to classify such orbits. The method is based on the theory of restricted root systems for semisimple symmetric spaces.

Program F

 10/30iThuj AM 10:00-12:00 Room 3040@ Koike (50min). Imsoon Jeong, Young Jin Suh and Hyunjin Leei30min). Imsoon Jeong, Hee Jin Kim and Young Jin Suhi30min). PM 1:30-5:30 Room 3040 Kollross (50min). Berndt (50min). Hae Young Yang, Hyung Jun Jin and Young Jin Suh (30min). Fujiii30min). Baba (30min). 10/31iFrij AM 10:00-12:30 Room 3040 Honda (30min). Kato (30min). Shibuya (30min). Kitagawa (30min). Iriyeh (30min). PM Discussion 11/1iSatj AM 10:00-10:30 Room 3040 Kon (30min). AM 10:40-12:10 Room 3040 Lecture by Berndt (1) PM Discussion 11/2 (Sun) AM 10:00-10:30 Room 3040 Kimura (30min). AM 10:40-12:10 Room 3040 Lecture by Berndt (2) PM 14:40-16:10 Room 3040 Kollross (90min). 11/3iMonj AM 10:40-12:10 Room 3040 Lecture by Berndt (3) PM Discussion

 Support : JSPS Grant-in-Aid for Scientific Research (A) No.17204006 "Research via the Infinite Dimensional Methods in Submanifold Theory" (Head researcher : Yoshihiro Ohnita)