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英語(English)

第6回大阪市立大学数学研究所-慶北国立大学GRG共催 微分幾何学ワークショップ



対称空間の部分多様体論と

有限次元および無限次元リー理論

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Date: February 1 (Fri)- February 3 (Sun), 2013
Place: Lecture Room 301, Department of Mathematics, Graduate School of Science, Osaka City University


Organizers: Yoshihiro Ohnita(OCU, Vice-director of OCAMI, Japan), Young Jin Suh (KNU, Leader of KNUGRG, Korea),
Naoyuki Koike (Tokyo University of Science, Japan), Takashi Sakai (Tokyo Metropolitan University & OCAMI, Japan)

Sponsors Osaka City University Advanced Mathematical Institute (OCAMI).
Department of Mathematics, Kyungpook National University (KNU), Grassmann Research Group (GRG)

Invited speakers Professor Claudio Gorodski (University of Sao Paulo, Brasil)
Professor Laura Geatti (University of Rome II Tor Vergata, Italy)
Professor Young Jin Suh (Kyoungpook National University, Korea)
Professor Byung Hak Kim (Kyung Hee University, Korea)
Professor Tohru Morimoto (Emeritus Professor of Nara Women University, Japan)
Professor Naoyuki Koike (Tokyo University of Science, Japan)
Professor Hiroyuki Tasaki (University of Tsukuba, Japan)
Professor Osamu Ikawa (Kyoto Institute of Technology, Japan)
Professor Makiko Tanaka (Tokyo University of Science, Japan)
Professor Hiroshi Tamaru (Hiroshima University, Japan)
Professor Takashi Sakai (Tokyo Metropolitan University & OCAMI, Japan)
Doctor Hyunjin Lee (Kyoungpook National University, Korea)
Doctor Imsoon Jeong (Kyoungpook National University, Korea)
Doctor Misa Ohashi (Meijo University & OCAMI, Japan)
Mr. Chang Hwa Woo(Kyoungpook National University, Korea)
Miss Eunmi Pak (Kyoungpook National University, Korea)
Mrs. Mi Jeong Kim(Kyoungpook National University, Korea)
Mr. Takahiro Hashinaga(Hiroshima University, Japan)
Mr. Akira Kubo(Hiroshima University, Japan)

       
Title and Abstract of TalksPDF
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 Kac-Moody 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 Cauchy-Riemann 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 two-plane 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 two-plane Grassmannians $\GBt$ which consists of all complex two dimensional linear subspaces in ${\Bbb C}^{m+2}$, respectively. And we give non-existence 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 two-plane Grassmannians"
Abstract: In this paper we consider a new notion of Reeb parallel shape operator for real hypersurface $M$ in complex two-plane Grassmannian $\GBt$. When $M$ has the Reeb parallel shape operator and non-vanishing 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 two-plane Grassmannians with g-Tanaka-Webster 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 Tanaka-Webster (in short, g-Tanaka-Webster) connection for real hypersurfaces in complex two-plane Grassmannians $\GBt$. In this talk, we consider a new notion of recurrent hypersurfaces in $\GBt$ for g-Tanaka-Webster connection and give a non-existence theorem for a Hopf hypersurface in $\GBt$ with g-Tanaka-Webster recurrent shape operator.
Imsoon Jeong, Eunmi Pak and Young Jin Suh
Title: "Parallelism of various generalized Tanaka-Webster invariant shape operators for real hypersurfaces in complex two-plane Grassmannians"
Abstract: In this talk, we introduce new notions of Lie invariant shape operator for a real hypersurface in complex two-plane Grassmannian $\GBt$, and give classifications for Hopf hypersurfaces in $\GBt$ with Lie invariant shape operators in the generalized Tanaka-Webster connection.   
Tohru Morimoto (Emeritus Professor of Nara Women University, Japan)
Title: A Klein-Cartan 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 semi-simple.    
Naoyuki Koike (Tokyo University of Science, Japan)
Title: Certain kind of isoparametric submanifolds in symmetric spaces of non-compact type and Hermann actions
Abstract: In this talk, we state that full irreducible curvature-adapted isoparametric real analytic submanifolds of codimension greater than one in a symmetric space $G/K$ of non-compact 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 anti-Kaehler space through an anti-Kaehler 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 R-spaces 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 left-invariant metrics on a Lie group
Abstract: We introduce the space of left-invariant metrics on a Lie group up to isometry and scaling, the moduli space. This moduli space derives a Milnor-type theorem, a generalization of the Milnor frame for three-dimensional unimodular Lie groups. Our Milnor-type theorem is useful to examine the existence and the nonexistence of a distinguished left-invariant 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 pseudo-Riemannian 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 Calabi-Yau 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) Calabi-Yau 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 Jost-Hinrich 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: Low-dimensional solvsolitons and the minimality of the corresponding submanifolds
Abstract: The notion of the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups is defined through the study on the space of left-invariant Riemannian metrics on a Lie group. Our interest is to characterize a geometric property of left-invariant Riemannian metrics in terms of the corresponding submanifolds. In this talk, we introduce the relationship between the existence of solvsolitons on low-dimensional 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.

 

ProgramPDF

2/1(Fri) AM 10:00-10:30 Lecture Room 301  Imsoon Jeong, Carlos J.G. Machado, Juan de Dios Perez and Young Jin Suh
AM 10:40-11:10 Lecture Room 301 Young Suk Choi, Hyunjin Lee and Changhwa Woo
AM 11:20-11:50 Lecture Room 301 Misa Ohashi slide of talk (in preparation)
PM 13:30-14:00 Lecture Room 301 Mijung Kim, Hyunjin Lee and Young Jin Suh
PM 14:10-14:40 Lecture Room 301 Imsoon Jeong, Eunmi Pak and Young Jin Suh
PM 15:00-15:30 Lecture Room 301 Akira Kubo slide of talk
PM 15:40-16:10 Lecture Room 301 Takahiro Hashinaga slide of talk
2/2(Sat) AM 9:45-10:45 Lecture Room 301 Claudio Gorodski slide of talk
AM 11:00-12:00 Lecture Room 301 Naoyuki Koike slide of talk
PM 14:00-15:00 Lecture Room 301 Laura Geatti
PM 15:30-16:20 Lecture Room 301 Byung Hak Kim
PM 16:30-17:20 Lecture Room 301 Tohru Morimoto
2/3 (Sun) AM 9:45-10:35 Lecture Room 301 Makiko Sumi Tanaka slide of talk
AM 10:45-11:35 Lecture Room 301 Osamu Ikawa slide of talk
PM 13:15-14:05 Lecture Room 301 Hiroyuki Tasaki slide of talk
PM 14:15-15:05 Lecture Room 301 Takashi Sakai slide of talk
PM 15:15-16:05 Lecture Room 301 Hiroshi Tamaru slide of talk



Suggestion to Speakers: At the lecture room there are enough blackboards, the computer projector and the visualizer. Please prepare your talk using them.


Link Osaka City University Advanced Mathematical Institute (OCAMI)
Department of Mathematics, Osaka City Univercity
The 1st OCAMI-KNUGRG Joint International Workshop on Differential Geometry and Related Fields (Oct. 30- Nov. 3, 2008)
The 2nd OCAMI-KNUGRG Joint Differential Geometry Workshop (Oct. 29- Nov. 1, 2009)
The 3rd KNUGRG-OCAMI Joint Differential Geometry Workshop (Nov. 4- Nov. 6, 2010)
The 4th KNUGRG-OCAMI Joint Differential Geometry Workshop (Nov. 2- Nov. 5, 2011)
The 5th KNUGRG-OCAMI Joint Differential Geometry Workshop (Oct. 31- Nov. 2, 2012)
Poster
Kansai Kenshu Center


Support : JSPS Grant-in-Aid for Scientific Research (S) No.23224002
"Proof of Homological Mirror Symmetry"
(Principal investigator: Kenji Fukaya)
JSPS Grant-in-Aid for Scientific Research (A) No.22244006
"Geometry of Curves, Surfaces and Hypersurfaces with Singularities"
(Principal investigator: Masaaki Umehara)
JSPS Grant-in-Aid for Scientific Research (C) No.24540090
"Research on Submanifold Geometry and Harmonic Map Theory in Symmetric Spaces"
(Principal investigator: Yoshihiro Ohnita)

Contact : (e-mail)
Yoshihiro Ohnita: ohnita (at) sci.osaka-cu.ac.jp
Osaka City University Advanced Mathematical Institute (OCAMI) & Department of Mathematics, Osaka City University
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585 JAPAN

Created by T. Noda (ODU) Last updated on 31/January/2013