The 19th International Workshop on Hermitian-Grassmannian ... 19th International Workshop on...

The 19th International Workshop on

Hermitian-Grassmannian Submanifolds and Its Applications

NIMS, Daejeon, KoreaOctober 26 (Mon) - 28 (Wed), 2015

Organized by NIMS & Research Institute of Real and Complex Manifolds Supported by NIMS & National Research Foundation

October 26 (Monday), 2015

09:00~09:15 Registration

09:15~11:15 Chairman Jiazu Zhou

09:15~10:15 Geometry of Lagrangian submanifolds in complex hyperquadricsYoshihiro Ohnita (Osaka City University & OCAMI, Japan)

10:15~11:15 From Euclidean geometry to manifold theory and some curvature properties in Riemannian geometry Uday Chand De (University of Calcutta, India)

11:15~11:35 Break Time

11:35~12:30 Chairman Yoshihiro Ohnita

11:35~12:15 Isoparametric foliation and a problem of BesseWenjiao Yan (Beijing Normal University, China & Tsukuba University, Japan)

12:15~12:30 Opening Address Young Jin Suh (Kyungpook National University, Korea)

Welcome Address President of NIMS

12:30~14:00 Lunch

Session 1

14:00~15:40 Chairman Makiko Sumi Tanaka

14:00~15:00 -Einstein real hypersurfaces in complex two-plane Grassmannians

Tee-How Loo (University of Malaya, Malaysia)15:00~15:40 On concircular vector fields in the fibred Riemannian space

Byung Hak Kim (Kyung Hee University, Korea)

15:40~16:00 Agreement Cooperation between RIRCM & OCAMI

Break Time

16:00~17:00 Chairman Yasuo Matsushita

16:00~17:00 Antipodal structure of the intersection of real flag manifolds in a complex flag manifold IITakashi Sakai (Tokyo Metropolitan University, Japan)

17:00~18:00 Chairman Young Jin Suh

17:00~18:00 On Bonnesen-style symmetric mixed inequalityJiazu Zhou (Southwest University, China)

Session 2

14:00~15:40 Chairman Nobuhiro Innami

14:00~15:00 Quantum and Floer type cohomologies on the almost contact metric manifolds with closed fundamental forms

Yong Seong Cho (Ewha Womans University, Korea)

15:00~15:40 Star-Ricci flat real hypersurfaces in complex space formsTatsuyoshi Hamada (Fukuoka University, Japan)


October 27 (Tuesday), 2015


09:00~10:00 Certain generalizations of Einstein manifolds with applications to relativity Uday Chand De (University of Calcutta, India)

10:00~11:00 Some new aspects of the Bernstein theorem IYuan Long Xin (Fudan University, China)


11:20~12:20 Chairman Yasuo Matsushita

11:20~12:20 Geometry of the Gauss images of isoparametric hypersurfacesYoshihiro Ohinta (Osaka City Universtiy & OCAMI, Japan)

12:20~12:30 Photo Time

12:30~14:00 Lunch

Session 1

14:00~16:40 Chairman Takashi Sakai

14:00~15:00 Symmetries on real hypersurfaces in Kahlerian manifoldsJong Taek Cho (Chonnam National University, Korea)

15:00~16:00 Real hypersurfaces in a 2-dimensional complex space form with transversal Killing tensor fieldsMayuko Kon (Shinshu University, Japan)

16:00~16:40 Classification of generalized Sasakian space formsAvik De (University Tunku Abdul Rahman, Malaysia)


17:00~18:00 Chairman Byung Hak Kim

17:00~18:00 Neutral geometry in 4-dimension and counter examples to Goldberg conjecture constructed on Walker 6-manifolds

Yasuo Matsushita (OCAMI, Japan)

Session 2

14:00~16:40 Chairman Uday Chand De

14:00~15:00 Geodesics in a Finsler torus of revolutionNobuhiro Innami (Niigata University, Japan)

15:00~16:00 Real toric manifolds, links, and integral cohomology groupsJin Hong Kim (Chosun University, Korea)

16:00~16:40 The Olivier Rey's inequality on the Heisenberg groupTakanari Saotome (Shibaura Institute of Technology, Japan)


October 28 (Wed), 2015

09:00~10:00 Chairman Yong Seung Cho

09:00~10:00 Some new aspects of the Bernstein theorem IIYuan Long Xin (Fudan University, China)


Session 1

10:15~11:25 Chairman Tatsuyoshi Hamada

10:15~10:55 Constant f-mean curvature surfaces in a smooth metric measure spaceJuncheol Pyo (Pusan National University, Korea)

10:55~11:25 Characterizations for real hypersurfaces in complex two-plane Grassmannians related to normal Jacobi operator

Imsoon Jeong (Chungbuk National University, Korea)


11:40~12:40 Chairman Tee-How Loo

11:40~12:10 On a Riemannian submanifold whose slice representation has no nonzero fixed pointYuichiro Taketomo (Hiroshima University, Japan)

12:10~12:40 Study on Jacobi operators in complex two-plane GrassmanniansEunmi Pak (Kyungpook National University, Korea)

12:40~14:00 Lunch


14:00~15:00 Neutral geometry in 4-dimension, Walker 4-manifolds and the Spinor approachYasuo Matsushita (OCAMI, Japan)

15:00~16:00 Isometries of extrinsic symmetric spacesMakiko Sumi Tanaka (Tokyo University of Science, Japan)

16:00~16:10 Closing RemarksByung Hak Kim (Kyung Hee University, Korea)

Session 2

10:15~11:25 Chairman Yuan Long Xin

10:15~10:55 On generalized Sasakain space-formsPradip Majhi (University of North Bengal, India)

10:55~11:25 Some hypersurfaces in GTW Reeb Lie derivative structure Jacobi operator in complex two-plane Grassmannians

Gyu Jong Kim (Kyungpook National University, Korea)


11:40~12:40 Chairman Mayuko Kon

11:40~12:10 Classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with Ricci tensorsChanghwa Woo (Kyungpook National University, Korea)

12:10~12:40 A generalized invariance of the structure Jacobi operator for real hypersurfaces in compact complex Grassmanians of rank 2

Hyunjin Lee (Kyungpook National University, Korea)

12:40~14:00 Lunch



ABSTRACT

Oct. 26, 2015(Monday)

The 19th International Workshop onHermitian-Grassmannian Submanifolds and Its Applications

& The 10th KNUHGRG-OCAMI Joint Differential Geometry Workshop

NIMS, Daejeon, Korea

October 26(Mon) - 28(Wed), 2015

October 26(Mon), 2015

1st-1 Yoshihiro Ohnita E-mail : [email protected](Osaka City University Advanced Mathematical Institute, & Department of Mathematics,

Osaka City University, 3-3-138 Sugimoto, Sumitoshi-ku, Osaka 558-8585, Japan)

Geometry of Lagrangian submanifolds in complex hyperquadrics

The complex hyperquadric Qn(C) is canonically identified with the real Grassmann

manifold Gr2(Rn+2) of all oriented 2-dimensional vector subspaces of Rn+2, whichis a compact Hermitian symmetric space of rank two (if n ≥ 2). In general it isan interesting problem to study submanifold geometry in symmetric space of rankgreater than one. In this talk we shall discuss geometry of Lagrangian subman-ifolds in complex hyperquadrics: We shall begin with elementary properties andexamples of such Lagrangian submanifolds and emphasis on the relationship withthe hypersurface geometry in the standard unit sphere such as the Gauss map con-struction of Lagrangian submanifolds. This talk is the first one of my two lectureson this area at this international workshop.

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1st-2 Uday Chand De E-mail : [email protected](Department of Pure Mathematics, University of Calcutta, 35, B. C. road, Kolkata-700019,

West Bengal, India)

From Euclidean geometry to manifold theory and some basic properties of the curvaturetensors in Riemannian geometry

In the present talk we explain how the notion of manifolds come from Euclideangeometry. Next some basic properties of curvature tensors in Riemannian geometryhave been discussed. In particular, 2-dimensional and 3-dimensional Riemannianspace have been considered.

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1st-3 Wenjiao Yan E-mail : [email protected](School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

& Mathematical Institute, Graduate School of Sciences, Tohoku University, Sendai, 980-8578,Japan)

Isoparametric foliation and a problem of Besse

This talk gives a survey on the recent progress in the study of Willmore, Ein-stein and some Einstein-like properties of the focal submanifolds of isoparametrichypersurfaces in spheres with g = 4 and g = 6 distinct principal curvatures.

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1st-S1-4 Tee-How Loo* E-mail : [email protected](Institute of Mathematical Science, University of Malaya, 50603 Kuala Lumpur, Malaya)

Ruenn-Huah Lee E-mail : [email protected](Institute of Mathematical Science, University of Malaya, 50603 Kuala Lumpur, Malaya)

(η, ηa, θ)-Einstein real hypersurfaces in complex two-plane Grassmannians

In this paper, we introduce the notion of (η, ηa, θ)-Einstein real hypersurfacesin complex two-plane Grassmannians. We show that there does not exist any(η, ηa, θ)-Einstein real hypersurface in complex two-plane Grassmannians such thatξ is tangent to D. Some examples of (ηa, θ)-Einstein real hypersurfaces are given.

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1st-S2-4 Yong Seung Cho E-mail : [email protected](Division of Mathematical and Physical Sciences, Ewha Womans University, Seoul 120-750,

Korea)

Quantum and Floer type cohomologies on the almost contact metric manifolds with closedfundamental forms

To construct the quantum type cohomologies, we study pseudo-coholomorphiccurves, moduli spaces of the curves representing 2-dimensional homology classes,Gromov-Witten type invariants, and quantum type product on cohomology groups.For Floer type cohomologies on the manifolds, we study a symplectic type actionfuntional on the universal covering space of the space of contractible loops. Thecritical points of the funtional and the moduli space of gradient flow lines joiningcritical points induce a cochain complex and produce a Floer type cohomology. Weshow that the quantum type cohomology and the Floer type cohomology on themanifolds are isomorphic.

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1st-S1-5 Byung Hak Kim E-mail : [email protected](Department of Applied Mathematics, Kyung Hee University, Yong-In 17104, Korea)

On concircular vector fields in the fibred Riemannian space

The concircular transformation of Riemannian manifolds was introduced by K.Yanoin 1940, which is characterize by a conformal transformation preserving geodesiccircle. The notion of concircular vector fields was introduced by A.Falkow in 1939and developed the concircular geometry and many geometers obtained interest-ing results. Moreover a Ricci soliton on a Riemannian manifold is said to haveconcircular potential field if its potential field is a concircular vector field.

In this talk, we summarize of the recent results for the concircular geometry andconcircular vector field ,and willing to discuss about related problems in the fibredRiemannian space.

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1st-S2-5 Tatsuyoshi Hamada E-mail : [email protected](Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan &

Osaka City University Advanced Mathematical Institute (OCAMI), Osaka 558-8585, Japan)

Star-Ricci flat real hypersurfaces in complex space forms

The study of real hypersurfaces in complex space forms has been an active fieldof study over the past decade. Recently, Kaimakamis and Panagiotidou classifiedreal hypersurfaces with parallel star-Ricci tensor in complex projective plane P2(C)and complex hyperbolic plane H2(C). We can find the relations with their resultand flat star-Ricci tensor. In this paper, we classified real hypersurfaces of complexspace forms Pn(C) and Hn(C) with flat star-Ricci tensor.

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1st-6 Osamu Ikawa E-mail : [email protected](Department of Mathematics and Physical Sciences, Faculty of Arts and Sciences, Kyoto

Institute of Technology, Sakyoku, Kyoto, 606-8585 Japan)

Hiroshi Iriyeh E-mail : [email protected](Mathematics and Informatics, College of Science, Ibaraki University, Bunkyo, Mito, 310-

8512 Japan)

Takayuki Okuda E-mail : [email protected](Department of Mathematics, Graduate School of Science, Hiroshima University,

Kagamiyama, Higashi-Hiroshima, 739-8526 Japan)

Takashi Sakai* E-mail : [email protected](Department of Mathematics and Information Sciences, Tokyo Metropolitan University,

Minami-Osawa, Hachioji, Tokyo, 192-0397 Japan)

Hiroyuki Tasaki E-mail : [email protected](Division of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba,

Tsukuba, Ibaraki, 305-8571 Japan)

Antipodal structure of the intersection of real flag manifolds in a complex flag manifold II

Tanaka and Tasaki studied the antipodal structure of the intersection of two realforms in Hermitian symmetric spaces of compact type. An orbit of the adjoint rep-resentation of a compact connected Lie group G admits a G-invariant Kahler struc-ture, and called a complex flag manifold. Furthermore, any simply-connected com-pact homogeneous Kahler manifold is a complex flag manifold. Using k-symmetricstructures, we can define (generalized) antipodal sets of a complex flag manifold.An orbit of the linear isotropy representation of the compact symmetric space G/Kis called a real flag manifold, and is embedded in a complex flag manifold as a realform. In this talk, we will give a necessary and sufficient condition for two real flagmanifolds, which are not necessarily congruent with each other, in a complex flagmanifold to intersect transversally in terms of symmetric triads. Moreover we willshow that the intersection is an orbit of a certain Weyl group and an antipodal set,if the intersection is discrete.

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1st-7 Hejun Wang E-mail : [email protected](School of Mathematics and Statistics, Southwest University, Chongqing 400715, China)

Pengfu Wang E-mail : [email protected](School of Mathematics and Statistics, Southwest University, Chongqing 400715, China)

Wenxue Xu E-mail : [email protected](School of Mathematics and Statistics, Southwest University, Chongqing 400715, China)

Jiazu Zhou* E-mail : [email protected](School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;

College of Mathematical Science, Kaili University, Kaili, Guizhou 556011, China; SoutheastGuizhou Vocational College of Technology for Nationalities, Kaili, Guizhou 556000, China)

On Bonnesen-style symmetric mixed inequality

The symmetric mixed isoperimetric deficit ∆2(K0,K1) of domains K0 and K1 inthe Euclidean plane R2 is defined in this paper. The symmetric mixed isoperimetricinequality and some new Bonnesen-style symmetric mixed inequalities are obtainedvia the known kinematic formulae of Poincare and Blaschke in integral geometry.

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ABSTRACT

Oct. 27, 2015(Tuesday)





October 27(Tue), 2015

2nd-1 Uday Chand De E-mail : [email protected](Department of Pure Mathematics, University of Calcutta, 35, B. C. road, Kolkata-700019,

West Bengal, India)

Certain generalizations of Einstein manifolds with applications to relativity

We study quasi-Einstein and generalized quasi-Einstein manifolds which are thenatural generalizations of Einstein manifolds. First we consider quasi-Einsteinmanifolds. We obtain some results regarding quasi-Einstein manifolds and givesome examples of such manifolds. We also study pseudo Ricci symmetric and spe-cial type of quasi-Einstein manifolds and give examples of such manifolds which jus-tify some theorems. Next, we study generalized quasi-Einstein manifolds. Finally,we give some applications of quasi-Einstein manifolds in the theory of relativity.

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2nd-2 Yuan Long Xin E-mail : [email protected](Institute of Mathematics, Fudan University, Shanghai, Chian)

Some new aspects of the Bernstein theorem I

There are various generalizations of the well-known Bernstein theorem for minimalgraphs in R3 over the past decades. We will review some recent developmentsfor minimal parametric hypersurfaces in Euclidean space, higher codimensionalgeneralizations for Moser’s weak version of Bernstein’s theorem (so-called Lawson-Osserman problem), as well as the Bernstein type theorems for minimal hypersur-faces in general Riemannian manifolds with non-negative Ricci curvature.

1






2nd-3 Yoshihiro Ohnita E-mail : [email protected](Osaka City University Advanced Mathematical Institute, & Department of Mathematics,

Osaka City University, 3-3-138 Sugimoto, Sumitoshi-ku, Osaka 558-8585, Japan)

Geometry of the Gauss images of isoparametric hypersurfaces

Hypersurfaces with constant principal curvatures in the standard sphere are so-called isoparametric hypersurafaces. Through the Gauss map, the isoparametrichypersurafaces in the standard unit sphere provide a nice class of Lagrangian sub-manifolds in complex hyperquadrics. We know that the Gauss image, i.e. theimage of the Gauss map, of each isoparametric hypersuraface in the standard unitsphere is a compact minimal Lagrangian submanifold embedded in the complexhyperquadric. We studied geometric and topological properties of so obtained La-grangian submanifolds such as the minimal Maslov number, the classification prob-lem of homogeneous Lagrangian submanifolds and the Hamiltonian stability prob-lem. Moreover, we shall explain recent results on Hamiltonian non-displaceablityof the Gauss images of isoparametric hypersurafaces, which the Lagrangian Floertheory can be applied to. This lecture is based on my joint works with Hui Ma,Hiroshi Iriyeh and Reiko Miyaoka.

This talk is the second one of my two lectures at this international workshop.

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2nd-S1-4 Jong Taek Cho E-mail : [email protected](Department of Mathematics, Chonnam National University, Gwangju 500-757, Korea)

Symmetries on real hypersurfaces in Kahlerian manifolds

A real hypersurface of Kahlerian manifold admits the two fundamental structuresinduced from the Kahlerian structure of ambient space. One is Riemannian struc-ture and the other is CR structure. In this context, we may consider two naturalaffine connections on real hypersurfaces of Kahlerian manifold: the Levi-Civitaconnection and the generalized Tanaka-Webster connection. In this talk, we studyseveral symmetries using these two connections respectively.

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2nd-S2-4 Nobuhiro Innami E-mail : [email protected](Department of Mathematics, Niigata University, Niigata 950-2181, Japan)

Geodesics in a Finsler torus of revolution

Let (M,F ) denote a Finsler surface. Namely, M is a differentiable 2-manifoldwith fundamental function F : TM → R such that F is smooth and positive onTM r 0, F (x, tx) = tF (x, x), t > 0, for all x ∈ TxM and F (x, x) is strictlyconvex for x = 0. Here TM denotes the tangent bundle of M . The intrinsicdistance d on M is defined by d(p, q) := infLF (c) | c ∈ Ω(p, q), where Ω(p, q)denotes the set of all piecewise smooth curves from p to q and LF (c) is the lengthof a curve c : [0, 1] → M . An extremal of the variation problem of lengths of curvesc ∈ Ω(p, q) is called a geodesic in (M,F ). Let T (p, q) denote a minimizing geodesicsegment from p to q. If F is not absolutely homogeneous, then the distance d is notsymmetric and T (p, q) = T (q, p), in general. When M is orientable, an isometrywhich preserves the orientation of M is called a motion on (M,F ). We say thatφt : M → M , t ∈ (−∞,∞), is a one-parameter group of motions on (M,F ) if φt

satisfies F (φt(x), φt∗(x)) = F (x, x) for all x ∈ M , x ∈ TxM and all t ∈ (−∞,∞).

We study the global behavior of geodesics in a complete Finsler surface with one-parameter group of motions. In particular, we determine what types of geodesicsthere are in a Finsler 2-torus with revolution with non-symmetric distance.

References

[1] H.Busemann, The Geometry of Geodesics, Academic Press, New York, 1955.[2] H.Busemann–F.Pedersen, Tori with one-parameter groups of motions, Math. Scan., 3, 209–

220 (1955).[3] N.Innami–T.Nagano–K.Shiohama, Geodesics in a Finsler surface with one-parameter group

of motions, Preprint (2014).

1






2nd-S1-5 Mayuko Kon E-mail : [email protected](Faculty of Education, Shinshu University, 6-Ro, Nishinagano, Nagano City 380-8544, Japan)

Real hypersurfaces in a 2-dimensional complex space form with transversal Killing tensorfields

Let M be a real hypersuface in a non-flat complex space form M2(c). The shapeoperator A is called transversal Killing tensor field if it satisfies (∇XA)X = 0for any vector field X ⊥ ξ, where ξ is a structure vector field. We prove thatA is transversal Killing tensor field if and only if M is a totally umbilical realhypersurface.

1






2nd-S2-5 Jin Hong Kim E-mail : [email protected](Department of Mathematics Education, Chosun University, 309 Pilmun-daero, Dong-gu,

Gwangju, 501-759, Republic of Korea)

Real toric manifolds, links, and integral cohomology groups

In view of various results and the nature of their constructions for real toric ob-jects, it is an intriguing question to answer how much amount of torsion can becontained in their integral cohomology groups. In this talk, I want to explain howto explicitly construct certain examples of real toric objects, called the real qua-sitoric manifolds, which shows the richness of their torsion. To do so, we first givesome necessary background for the notions such as links, quasitoric manifolds, andtheir real versions. We then describe some important steps for the construction,relatively in detail.

1






2nd-S1-6 Avik De E-mail : [email protected](Department of Mathematical and Actuarial Sciences, University Tunku Abdul Rahman,

Malaysia)

Classification of Generalized Sasakian space forms

In the present paper we review the advancement of the theory of generalizedSasakian spaceforms. We give a local characterization of generalized Sasakianspace forms of dimension greater than 5. We show that it is of constant sectionalcurvature, or an (α, β) trans-Sasakian manifold, or a canal hypersurface in a Eu-clidean space or Minkowski space, or a certain type of twisted product of R and aflat almost Hermitian manifold.

1






2nd-S2-6 Takanari Saotome E-mail : [email protected](College of Engineering, Shibaura Institute of Technology, 307, Fukasaku, Minuma, Saitama,

337-8570 Japan)

The Olivier Rey’s inequality on the Heisenberg group

We will study CR analogue of the Olivier Rey’s inequality on the Heisenberg group.In confromal setting this inequality is used to prove the existence of the solution tothe linearized Yamabe equation. This inequality shows that the energy functionalfor perturbed Yamabe equation is bounded below, if the perturbation is smallenough.

In this article, we identify the Heisenberg group and the standard sphere via theCayley transformation, and analyze the eigenvalues of the sub-Laplacian ∆b onS2n+1.

1






2nd-7 Yasuo Matsushita* E-mail : [email protected](Osaka City University Advanced Mathematical Institute (OCAMI), Osaka, Japan)

Peter R. Law E-mail : [email protected](4 Mack Place, Monroe, NY 10950, USA)

Noriaki Katayama E-mail : [email protected](Department of Industrial Systems Engineering, Osaka Prefecture University College of

Technology, Neyagawa 572-8572, Japan)

Neutral geometry in 4-dimension and counterexamples to Goldberg conjecture constructedon Walker 6-manifolds

Our talk covers two topics, the first being a survey of the existence problem for anindefinite metric of neutral signature (+2,−2) on a compact orientable 4-manidold,while the second presents a new counterexample to the Goldberg conjecture pro-vided by an indefinite metric of signature (+4,−2) on a compact six-dimensionalmanifold (a six-torus).

1



ABSTRACT

Oct. 28, 2015(Wednesday)





October 28(Wed), 2015

3rd-1 Yuan Long Xin E-mail : [email protected](Institute of Mathematics, Fudan University, Shanghai, Chian)

Some new aspects of the Bernstein theorem II

There are various generalizations of the well-known Bernstein theorem for minimalgraphs in R3 over the past decades. We will review some recent developmentsfor minimal parametric hypersurfaces in Euclidean space, higher codimensionalgeneralizations for Moser’s weak version of Bernstein’s theorem (so-called Lawson-Osserman problem), as well as the Bernstein type theorems for minimal hypersur-faces in general Riemannian manifolds with non-negative Ricci curvature.

1






3rd-S1-2 Juncheol Pyo E-mail : [email protected](Department of Mathematics, Pusan National University, Busan, 609-735 Korea)

Constant f-mean curvature surfaces in a smooth metric measure space

In this talk, we introduce surface theory in a smooth metric measure space andgive some rigidity results of constant f-mean curvature surfaces in a smooth metricmeasure space.

1






3rd-S2-2 Pradip Majhi E-mail : [email protected](Department of Mathematics, University of North Bengal, Raja Rammohunpur, Darjeeling,

Pin-734013, West Bengal, India)

On generalized Sasakain space-forms

The object of the present paper is to study some curvature properties of general-ized Sasakian space-forms. First we consider Ricci pseudosymmetric generalizedSasakian space-forms. We also study Ricci generalized pseudosymmetric and Weylsemisymmetric generalized Sasakian space-forms. Next, we consider ξ-conformallyflat, ϕ-conformally flat, ϕ-Weyl semisymmetric, ϕ-projectively semisymmetric gen-eralized Sasakian space-forms. As a consequence of the results we deduce someimportant corollaries. Finally, illustrative examples are given.

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3rd-S1-3 Imsoon Jeong* E-mail : [email protected](Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea)

Young Jin Suh E-mail : [email protected](Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea)

Characterizations for real hypersurfaces in complex two-plane Grassmannians related to nor-mal Jacobi operator

In this talk, we introduce a new notion of Reeb parallel normal Jacobi operator forhomogeneous real hypersurfaces in complex two-plane Grassmannians which has aremarkable geometric structure as a Hermitian symmetric space of rank 2. And wegive a complete classification for real hypersurfaces of Type (A) in complex two-plane Grassmannians G2(Cm+2), that is, a tube over a totally geodesic G2(Cm+1)in G2(Cm+2) with Reeb parallel normal Jacobi operator.

1






3rd-S2-3 Eunmi Pak E-mail : [email protected](Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea)

Gyu Jong Kim* E-mail : [email protected](Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea)


Some hypersurfaces in GTW Reeb Lie derivative structure Jacobi operator in complex two-plane Grassmannians

We considered a real hypersurfaceM in a complex two-plane GrassmannianG2(Cm+2)when the GTW Reeb Lie derivative of the structure Jacobi operator coincides withthe Reeb Lie derivative. And using the method of simultaneous diagonalization, wegive a complete classification for a real hypersurface in G2(Cm+2) satisfying sucha condition. In this case, we have proved that M is an open part of a tube arounda totally geodesic G2(Cm+1) in G2(Cm+2).

1






3rd-S1-4 Yuichiro Taketomi E-mail : [email protected](1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan)

On a Riemannian submanifold whose slice representation has no nonzero fixed point

In this talk, we introduce a new class of Riemannian submanifolds which we callarid submanifolds. We say a Riemannian submanifold is arid if no nonzero normalvectors are invariant under the full slice representation. Arid submanifolds are ageneralization of the notion of weakly reflective submanifolds introduced by Ikawa,Sakai and Tasaki. On the other hand, any arid submanifolds are minimal. Wealso introduce an application of the idea of arid submanifolds to the study of left-invariant metrics on a Lie group. We have obtained a sufficient condition for anarbitrary Riemannian Lie group to be a Ricci soliton.

1






3rd-S2-4 Young Jin Suh E-mail : [email protected](Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea)

Changhwa Woo* E-mail : [email protected](Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea)

Classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with Riccitensors

We have studied classifying problem of immersed submanifolds in Hermitian sym-metric spaces. Typically in this paper, we consider a new notion of Reeb parallelRicci tensor for homogeneous real hypersurfaces in complex hyperbolic two-planeGrassmannians which has a remarkable geometric structure as a Hermitian sym-metric space of rank 2. By using a common basis of simultaneously diagonalizablefinite dimensional operators, we give a complete classification for real hypersurfacesin complex hyperbolic two-plane Grassmannians with the given condition.

1






3rd-S1-5 Eunmi Pak* E-mail : [email protected](Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea)

Young Jin Suh E-mail : [email protected](Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea)

Study on Jacobi operators in complex two-plane Grassmannians

In relation to the generalized Tanaka-Webster connection, we consider a new no-tion of parallel Jacobi operator for real hypersurfaces in complex two-plane Grass-mannians G2(Cm+2) and show results about real hypersurfaces in G2(Cm+2) withgeneralized Tanaka-Webster parallel structure Jacobi operator and normal Jacobioperator.

References

[1] J. Berndt, Riemannian geometry of complex two-plane Grassmannian, Rend. Sem. Mat.

Univ. Politec. Torino 55 (1997), 19–83.[2] I. Jeong, H. J. Kim and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians

with parallel normal Jacobi operator, Publ. Math. Debrecen 76 (2010), 203–218.

[3] H. Lee, J. D. Perez and Y.J. Suh, On the structure Jacobi operator of a real hypersurface incomplex projective space, Monatsh. Math. 158 (2009), no. 2, 187–194.

1






3rd-S2-5 Hyunjin Lee* E-mail : [email protected](Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National Univer-

sity, Daegu 702-701, Korea)


A generalized invariance of the structure Jacobi operator for real hypersurfaces in compactcomplex Grassmannians of rank 2

In this talk, first we introduce a new invariance of the derivative Lξµ − ∇ξµ forthe structure Jacobi operator Rξ := R(·, ξ)ξ on real hypersurfaces in complex two-plane GrassmanniansG2(Cm+2), where Lξµ (∇ξµ , resp.) denotes the Lie (covariant,

resp.) derivative along the direction of ξµ ∈ Q⊥ and Q⊥ ⊂ TM a 3-dimensionaldistribution spanned by ξµ = −JµN (µ = 1, 2, 3). By using this notion, we give anew characterization of Hopf hypersurfaces in G2(Cm+2), m ≥ 3.

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3rd-6 Yasuo Matsushita* E-mail : [email protected](Osaka City University Advanced Mathematical Institute, Osaka, Japan)

Peter R. Law E-mail : [email protected](4 Mack Place, Monroe, NY 10950, USA)

Noriaki Katayama E-mail : [email protected](Department of Industrial Systems Engineering, Osaka Prefecture University College of

Technology, Neyagawa 572-8572, Japan)

Neutral geometry in 4-dimension, Walker 4-manifolds and spinor approach

This talk is a survey on neutral geometry in 4-dimension. Among pseudo-Riemannianmanifolds of various indefinite metric signatures, such a neutral metric in 4-dimensionexhibits many specific properties. These interesting aspects and significant struc-tures will be shown. Especially, Walker 4-manifolds and the spinor approach to theneutral 4-geometry will be also included.

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3rd-7 Makiko Sumi Tanaka E-mail : [email protected](Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510,

Japan)

Isometries of extrinsic symmetric spaces

We show that every isometry of an extrinsic symmetric space extends to an isometryof its ambient euclidean space. As a consequence, any isometry of a real form ofa Hermitian symemtric space extends to a holomorphic isometry of the ambientHermitian symmetric space. Moreover, every fixed point component of an isometryof a symmetric R-space is a symmetric R-space itself. This article is based on thejoint work with J. -H. Eschenburg and P. Quast.

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Date post:	22-Apr-2018
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