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英語(English)

The 9th OCAMI-KNUHGRG Joint Differential Geometry Workshop on



Submanifold Geometry and Lie Theory



Date: February 12 (Thu)- 14 (Sat), 2015
Place: Lecture Room E408 (Feb.12), F405 (Feb.13-14), Department of Mathematics, Graduate School of Science, Osaka City University


Organizers: Yoshihiro Ohnita(OCU, Director of OCAMI, Japan), Young Jin Suh (KNU, Leader of KNUHGRG, Korea),
Hyunjin Lee (SRC-GAIA, POSTECH, Korea), Kaneme Hashimoto (OCAMI, Japan)

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

Invited speakers Professor Kazumi Tsukada (Ochanomizu University, Japan)
Professor Yong Seung Cho(Ewha Women's University, Korea)
Professor Wayne Rossman (Kobe University, Japan)
Professor Mayuko Kon (Shinshu University, Japan)
Professor Young Jin Suh (Kyungpook National University, Korea)
Professor Juergen Berndt(King's College London, UK)
Professor Juan de Dios Perez(University of Granada, Spain)
Professor Takashi Sakai(Tokyo Metroplitan University, Japan)
Doctor Wenjiao Yan(Beijing Normal University, P.R. China & Tohoku University, Japan)
Doctor Hyunjin Lee (SRC-GAIA, POSTECH, Korea)
Doctor Imsoon Jeong (Kyungpook National University, Korea)
Doctor Kaname Hashimoto (OCAMI, Japan)
Doctor Takahiro Hashinaga (Hiroshima University, Japan)
Doctor Akira Kubo (Hiroshima University, Japan)
Doctor Saki Okuhara (OCAMI, Japan)
Doctor Tohru Kajigaya (Tohoku University & OCAMI, Japan)
Doctor Eunmi Pak (Kyungpook National University, Korea)
Doctor Volker Branding (University of Vienna, Austria)
Mr. Masashi Yasumoto (Kobe University, Japan)
Mr. Wolfgang Carl (TU-Graz University, Austria)
Mr. Changhwa Woo(Kyungpook National University, Korea)
Mr. Gyu Jong Kim(Kyungpook National University, Korea)
Mr. Yuta Ogata (Kobe University, Japan)
Mr. Keisuke Teramoto (Kobe University, Japan)
etc.

       
Title and Abstract of TalksPDF
Kazumi Tsukada (Ochanomizu University, Japan)
Title: Totally complex submanifolds of a complex Grassmann manifold of 2-planes
Abstract: A complex Grassmann manifold ${\rm G}_2 (\mathbb{C}^{m+2})$ of all $2$-dimensional complex subspaces in $\mathbb{C}^{m+2}$ has two nice geometric structures - the K\"ahler structure and the quaternionic K\"ahler structure. We study totally complex submanifolds of ${\rm G}_2 (\mathbb{C}^{m+2})$ with respect to the quaternionic K\"ahler structure. We show that the projective cotangent bundle ${\rm P}(T^* \mathbb{C}P^{m+1})$ of a complex projective space $\mathbb{C}P^{m+1}$ is a twistor space of the quaternionic K\"ahler manifold ${\rm G}_2 (\mathbb{C}^{m+2})$. Applying the twistor theory, we construct maximal totally complex submanifolds of ${\rm G}_2 (\mathbb{C}^{m+2})$ from complex submanifolds of $\mathbb{C}P^{m+1}$. Then we obtain many interesting examples. In particular we classify maximal homogeneous totally complex submanifolds. We show the relationship between the geometry of complex submanifolds of $\mathbb{C}P^{m+1}$ and that of totally complex submanifolds of ${\rm G}_2 (\mathbb{C}^{m+2})$. PDF
Yong Seung Cho(Ewha Women's University, Korea)
Title: Gromov-Witten Invariants on Symplectic Fibrations
Abstract: We construct a symplectic fibration over almost contact metric manifolds with symplectic fibre. The total space of the fibration has an almost contact metricstructure with a group of symplectomorphisms of the fibre. If the fibre is simply connected, then the flux homomorphism is trivialand the total space has a coupled connection 2-form. We construct the coupled connection 2-form and investigate the form. When the connection of a fibration is flat, we study the properties of total space and base space. As a trivial case we consider the products of almost contact metric manifolds and symplectic manifolds. We study Gromov-Witten type invariants and quantum type cohomologies o f the products. As an example we consider the product of the 3-sphere and a Calabi-Yau 3-fold.
Wayne Rossman and Masashi Yasumoto (Kobe University, Japan)
Title: Singularities of discretized linear Weingarten surfaces
Abstract: In this talk we will explain singularities of discretized surfaces with special curvature conditions. In particular, singularities of discrete and semi-discrete maximal surfaces in Lorentz-Minkowski 3-space and singularities of discrete linear Weingarten surfaces of Bryant and Bianchi types will be discussed.
Mayuko Kon (Shinshu University, Japan)
Title: Shape operator of real hypersurfaces in a 2-dimensional complex space form
Abstract: We show that if a shape operator of a real hypersurface in a 2-dimensional complex space form satisies a transversal Killing equation, then it is locally conguent to a geodesic hypersphere. Moreover, we introduce some non-Hopf hypersurfaces which satisfies g(AX,Y)=ag(X,Y) for some function a and any vector field X and Y orthogonal to the structure vector field. This result is an example of non-Hopf real hypersurfaces given by Ivey-Ryan, 2011.
J\"urgen Berndt (King's College London, UK) and Young Jin Suh*(Kyungpook National University, Korea)
Title: Differential geometry of real hypersurfaces in Hermitian symmetric spaces with rank 2
Abstract: PDF
Juan De Dios Perez (University de Granada, Spain) and Changhwa Woo* (Kyungpook National University, Korea)
Title: Hopf hypersurfaces in complex two-plane Grassmannians with GTW connections
Abstract: In this talk, we will give some non-existence properties for Hopf real hypersurfaces in complex two-plane Grassmannians with certain geometric conditions. First, real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster recurrent shape operator $A$ will be talked in detail. Next, harmonic curvature with generalized Tanaka-Webster connection for Hopf hypersurfaces in complex two-plane Grassmannians and its related topics will be given. PDF
Hyunjin Lee (SRC-GAIA, POSTECH, Korea)
Title: A real hypersurface in complex two-plane Grassmannians with recurrent (1,1)-type tensor
Abstract: Kobayashi and Nomizu [Foundations of Differential Geometry, Vol. I] have introduced a notion of recurrent for (r,s) type tensor on Riemannian manifolds. By the definition, we see that the conception of recurrent naturally becomes a kind of generalized parallelism. Specifically, we consider this notion for (1,1) type tensor defined on a real hypersurface in complex two-plane Grassmannians. Among several (1,1) type tensors, let us consider a structure Jacobi operator given on the Riemannian curvature tensor in this talk. Actually, Jeong, Perez and Suh [Acta Math. Hungar. (2009)] verified that there does not exist any connected Hopf hypersurface in complex two-plane Grassmannians with parallel structure Jacobi operator. We consider more general notions, which are said to be Reeb or $\mathcal Q^{\bot}$-recurrent structure Jacobi operator. By using these two weaker conditions, we give some characterizations of Hopf hypersurfaces in complex two-plane Grassmannians. PDF
Eunmi Pak(Kyungpook National University, Korea)
Title: Study on structure Jacobi operator in complex two-plane Grassmannians
Abstract: We study classifying problems of real hypersurfaces in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$. In relation to structure Jacobi operator, we consider some recurrent condition. In this case, we prove a complete classification for a real hypersurface in $G_2({\mathbb C}^{m+2})$ satisfying such a condition. PDF
Gyujong Kim* and Young Jin Suh (Kyungpook National University, Korea)
Title: Real hypersurfaces in complex two-plane Grassmannians with GTW Lie derivative structure Jacobi operator
Abstract: In this talk, several kinds of structure Jacobi operator tensors are defined on a Real hypersurface M in complex two-plane Grassmannians $G_{2}(C^{m+2})$. Using Berndt and Suh's theory, we give some complete classifications of M in $G_{2}(C^{m+2})$ with these conditions about GTW Lie derivative structure Jacobi operator. PDF
Imsoon Jeong*(Kyungpook National University, Korea)
Title: Parallelism of normal Jacobi operator for real hypersurfaces in complex two-plane Grassmannians
Abstract: In this talk, we introduce a notion of normal Jacobi operator $\RN$ for hypersurfaces $M$ in a complex two-plane Grassmannians $G_2({\Bbb C}^{m+2})$ in such a way that $${\RN}X={\bar R}(X,N)N{\in}\text{End}\ (T_xM), \quad x{\in}M$$ for any tangent vector field $X$ on $M$, where $\bar R$ and $N$ respectively denote the Riemannian curvature tensor and a unit normal vector field of $M$ in $G_2({\Bbb C}^{m+2})$. The ambient space $G_2({\Bbb C}^{m+2})$ has a remarkable geometric structure. It was known that $G_2({\Bbb C}^{m+2})$ is the unique compact irreducible Riemannian symmetric space equipped with both a K\"{a}hler structure $J$ and a quaternionic K\"{a}hler structure ${\frak J}$. And the structure vecror field $\xi$, $\xi = -J N$, of a real hypersurface $M$ in $G_2({\Bbb C}^{m+2})$ is said to be a {\it Reeb vector field}. The almost contact structure vector fields $\{\xi_1 ,\xi_2 ,\xi_3 \}$ are defined by $\xi_i =-J_i N$, $i=1,2,3$, where $\{J_1, J_2, J_3\}$ denote a canonical local basis of quaternionic K\"ahler structure ${\frak J}$ on $G_2({\Bbb C}^{m+2})$. If the distributions $\frak D$ and ${\frak D}^{\bot}=\text{Span}\{{\xi}_1,{\xi}_2,{\xi}_3\}$ are invariant by the shape operator $A$ of $M$, that is, $g(A{\frak D}, {\frak D}^{\bot})=0 $, where $T_xM = {\frak D}{\oplus}{\frak D}^{\bot}$, $x{\in}M$, then we call $M$ is ${\frak D}^{\bot}$-invariant. The normal Jacobi operator $\RN$ is said to be {\it Reeb parallel} on $M$ if the covariant derivative of the normal Jacobi operator ${\bar R}_N$ along the direction of the Reeb vector $\xi$ identically vanishes, that is, ${\nabla}_{\xi}{\RN} = 0$ . \vskip 6pt Related to such a Reeb parallel normal Jacobi operator ${\RN}$, we give a complete classification of ${\frak D}^{\bot}$-invariant real hypersurfaces in complex two-plane Grassmannians $G_2({\Bbb C}^{m+2})$ with Reeb parallel normal Jacobi operator. PDF
Wenjiao Yan(Beijing Normal University, P.R. China & Tohoku University, Japan)
Title: Isoparametric foliation in spheres and Yau's conjecture on the first eigenvalue
Abstract: I will give a brief introduction of the isoparametric foliation in spheres and talk about our recent works on this subject related with Yau conjecture, which states that the first eigenvalue of every closed minimal embedded hypersurface in the unit sphere is its dimension.
Tohru Kajigaya(Tohoku University & OCAMI, Japan)
Title: Hamiltonian minimality of normal bundles over the isoparametric submanifolds
Abstract: A Hamiltonian minimal (shortly, H-minimal) Lagrangian submanifold in a Kahler manifold is a critical point of the volume functional under all compactly supported Hamiltonian deformations. This gives a nice extension of the notion of minimal submanifolds. In this talk, we focus on constructions of H-minimal Lagrangian submanifolds in the complex Euclidean space C^n. We show that any normal bundle of a principal orbit of the adjoint representation of a compact simple Lie group G in the Lie algebra g of G is an H-minimal Lagrangian submanifold in the tangent bundle Tg which is naturally regarded as C^n. Moreover, we specify these orbits with this property in the class of full irreducible isoparametric submanifolds in the Euclidean space.
Kaname Hashimoto (OCAMI, Japan)
Title: Special Lagrangian submanifolds invariant under the isotropy action of symmetric spaces of rank 2
Abstract: In this talk we will explain the construction of cohomogenity one special Lagrangian submanifolds in the cotangent bundle of the sphere in the tangent space of Riemannian symmetric spaces of rank 2.
Takahiro Hashinaga (Hiroshima University, Japan)
Title: On local isometric embeddings of low-dimensional Lie groups
Abstract: It is classically well known that any Riemannian manifold can be locally isometrically embedded into the Euclidean spaces of sufficiently large dimension. We are interested in determining the least dimension of Euclidean spaces into which the given Riemannian manifolds can be locally isometrically embedded. In this talk, we will classify left-invariant Riemannian metrics on three-dimensional Lie groups which can be locally isometrically embedded into the four-dimensional Euclidean space.
Akira Kubo (Hiroshima University, Japan)
Title: Totally geodesic submanifolds in some symmetric spaces
Abstract: Classification of totally geodesic surfaces in Riemannian symmetric spaces is one of the most fundamental problems in submanifold geometry. As a first step of our studies, we focus on the case of the symmetric space of type AI. In this talk, we prove that totally geodesic surfaces in such spaces correspond to certain nilpotent matrices, and as applications we give the explicit classifications in the cases of low rank.
Volker Branding (University of Vienna, Austria)
Title: The evolution equation for magnetic geodesics
Abstract: Magnetic geodesics describe the trajectory of a particle in a Riemannian manifold subjected to an external magnetic field. We use the heat flow method to deform a given curve and discuss in which cases we obtain a non-trivial magnetic geodesic. This is joint work with Florian Hanisch.
Wolfgang Carl (TU-Graz University, Austria)
Title: Calculus of variations on semidiscrete surfaces
Abstract: A semidiscrete surface is represented by a mapping into three-dimensional Euclidean space possessing one discrete and one continuous variable. It can be seen as a semidiscretization of a smooth surface, or as a partial limit case of a purely discrete surface, i.e., a mesh. In this talk we use variational principles to establish the following objects on semidiscrete surfaces: (i) a Laplace operator by variation of the Dirichlet energy functional, (ii) a mean curvature vector field by variation of the surface area, and (iii) a normal vector field by variation of the enclosed volume. Subsequently we analyze the properties of these objects, their connections with each other, and their convergence behavior. As examples of possible applications we discuss semidiscrete harmonic functions and semidiscrete surfaces of revolution with constant mean curvature.
Yuta Ogata(Kobe University, Japan)
Title:Criteria for singularities of spacelike constant mean curvature surfaces in Lorentzian spaceforms
Abstract: In this talk, we will explain singularities of spacelike constant mean curvature (CMC) surfaces in Lorentzian spaceforms. Unlike the case in Riemannian spaceforms, spacelike CMC surfaces in these spaceforms generally have singularities. After introducing the Lax representations for spacelike CMC surfaces, we will give criteria for their singularities. At the end of our talk, we will show some examples of spacelike CMC surfaces with singularities.
Keisuke Teramoto(Kobe University, Japan)
Title:Parallel surfaces of cuspidal edges
Abstract: We investigate parallel surfaces of cuspidal edges. We give a criterion for the parallel surfaces of cuspidal edges to have swallowtail singularities. Moreover, we also clarify relations between singularities of parallel surfaces and differential geometric properties of initial cuspidal edges.
Saki Okuhara (OCAMI, Japan)
Title: Geometry associated to the tt*-Toda equation
Abstract: Guest and Lin proved the existence of a smooth and global solution to the tt*-Toda equation. In this talk, I explain their result and possible applications to the construction of minimal submanifolds asuch as special Lagrangian cones.
Shinji Ohno and Takashi Sakai (Tokyo Metropolitan University, Japan)
Title: Area-minimizing cones over minimal embeddings of R-spaces
Abstract: Solutions of Plateau's problem may have singularities as integral currents. At an isolated conical singularity, the tangent cone is area-minimizing. Hence, in order to understand such singularities, we should study area-minimizing properties of minimal cones. In this talk, by constructing area-nonincreasing retractions, we discuss area-minimizing properties of some cones over minimal embeddings of R-spaces.
Yoshihiro Ohnita (Osaka City University & OCAMI, Japan)
Title: Lagrangian intersection theory of the Gauss images of isoparametric hypersurfaces (ioint work with Hiroshi Iriyeh, Hui Ma and Reiko Miyaoka)
Abstract: The Gauss images of isoparametric hypersurfaces in the standard unit sphere provide a nice class of compact minimal Lagrangian submanifolds embedded in the complex hypquadrics. In this talk I report new results on the Hamiltonian non-displaceability of the Gauss images of isoparametric hypersurfaces in the joint work with Hiroshi Iriyeh, Hui Ma and Reiko Miyaoka.

 

Program (provisional) PDF

2/12(Thu) AM 10:00-10:30 Lecture Room E408 Hyunjin Lee
AM 10:35-11:05 Lecture Room E408 Eunmi Pak
AM 11:15-12:00 Lecture Room E408 Saki Okuhara
PM 13:30-14:00 Lecture Room E408 Imsoon Jeong
PM 14:05-14:35 Lecture Room E408 Gyujong Kim
PM 14:40-15:10 Lecture Room E408 Keisuke Teramoto
PM 15:30-16:00 Lecture Room E408 Yuta Ogata
PM 16:05-16:35 Lecture Room E408 Volker Branding
PM 16:40-17:10 Lecture Room E408 Wolfgang Carl
Party
2/13(Fri) AM 9:45-10:35 Lecture Room F405 Mayuko Kon
AM 10:45-11:35 Lecture Room F405 Young Jin Suh and Jurgen Berndt
AM 11:40-12:10 Lecture Room F405 Changwha Woo and Juan de Dios Perez
PM 13:30-14:20 Lecture Room F405 Kazumi Tsukada
PM 14:30-15:20 Lecture Room F405 Wenjiao Yan
PM 15:30-16:20 Lecture Room F405 Akira Kubo (Takahiro Hashinaga canceled his talk.)
PM 16:30-17:20 Lecture Room F405 Tohru Kajigaya
2/14 (Sat) AM 9:45-10:35 Lecture Room F405 Wayne Rossman and Masashi Yasumoto
AM 10:45-11:35 Lecture Room F405 Yong Seung Cho
PM 13:30-14:20 Lecture Room F405 Kaname Hashimoto
PM 14:30-15:20 Lecture Room F405 Takashi Sakai
PM 15:30-16:20 Lecture Room F405 Yoshihiro Ohnita



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


Link ICM 2014 Satellite Conference on Real and Complex Submanifolds
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)
The 6th OCAMI-KNUGRG Joint Differential Geometry Workshop (Feb. 1- Feb. 3, 2013)
The 7th KNUGRG-OCAMI Joint Differential Geometry Workshop (Sep. 30- Oct. 2, 2013)
The 8th OCAMI-KNUGRG Joint Differential Geometry Workshop (Apr. 14- Apr. 16, 2014)
Poster
Kansai Kenshu Center


Support : JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers
(Oct.2014-Mar.2017)
"Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI"
(Osaka City University - Kobe University - Waseda University, 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 13/January/2015