the 20th international workshop on hermitian symmetric spaces and submanifoldsrircm.knu.ac.kr ›...

The 20th International Workshop

on Hermitian Symmetric Spaces and Submanifolds& The 12th RIRCM-OCAMI Joint Differential Geometry Workshop

July 26 (Tuesday), 2016

11:30~12:30 Chair Professor Juan de Dios Perez

11:30~12:00 Recent results about Jacobi operator for real hypersurfaces in complex two plane GrassmanniansEunmi Pak* (Kyungil University & RIRCM, Korea) and

Young Jin Suh (Kyungpook National University & RIRCM)

12:00~12:30 The parallelism for real hypersurfaces in complex Grassmannians of rank twoHyunjin Lee* (Research Institute of Real and Complex Manifolds of KNU (RIRCM), Korea) and

Young Jin Suh (Kyungpook National University & RIRCM, Korea)

12:30~14:00 Lunch Time

14:00~15:00 Chair Professor Byung Hak Kim

14:00~14:30 Hypersurfaces in complex hyperbolic two-plane Grassmannians with GTW Reeb Lie derivative structure Jacobi operator

Gyu Jong Kim* (Kyungpook National University, Korea) and Young Jin Suh (Kyungpook National University & RIRCM, Korea)

14:30~15:00 Comparison on classification results of real hypersurfaces in compact and noncompact complex two-plane Grassmannians

Young Jin Suh (Kyungpook National University & RIRCM, Korea) and Changhwa Woo* (Kyungpook National University, Korea)

15:00~15:10 Break Time

15:10~16:10 Chair Professor Alfonso Romero

15:10~15:40 Characterizations of a Clifford hypersurface in a unit sphereKeomkyo Seo (Sookmyung Women’s University, Korea)

15:40~16:10 On the pointwise slant submanifolds Kwang-Soon Park (University of Seoul, Korea)


16:30~18:00 Chair Professor Young Jin Suh

16:30~17:00 CR geometry on contact manifoldsJong Taek Cho (Chonnam National University, Korea)

17:00~17:30 Quaternionic projective spaces and 7-sphereJae-Hyouk Lee (Ewha Womans University, Korea)

17:30~18:00 Gromov-Witten invariants on the products of almost contact manifolds Yong Seung Cho (Ewha Womans University, Korea)



July 27 (Wednesday), 2016

09:00~12:40 Chair Professor Yoshihiro Ohnita

09:00~10:00 Integral Geometry along flat subvarieties in symmetric spaces of compact type (I)Eric Grinberg (University of Massachusetts Boston, USA)

10:10~11:10 Hamiltonian non-displaceability of the Gauss images of isoparametric hypersurfaces (I)Reiko Miyaoka (Tohoku University, Japan)


11:20~12:20 LVMB manifolds and transverse Kahler foliations Jin Hong Kim (Chosun University, Korea)

12:20~12:40 Opening Address & Group Photo I


14:00~15:30 Chair Professor Eric Grinberg

14:00~14:45 On Floer homology of the Gauss images of isoparametric hypersurfacesYoshihiro Ohnita (OCAMI & Osaka City University, Japan)

14:45~15:30 On Bonnesen-style inequalities for mixed volumesJiazu Zhou (Southwest University, China)

15:30~17:00 Discussion Time

17:00~18:00Chair Professor Jiazu Zhou

17:00~18:00 Quantitative rigidity on manifolds of Ricci curvature bound below (I)Xiaochun Rong (Rutgers, The State University of New Jersey, USA)



July 28 (Thursday), 2016

09:00~11:10 Chair Professor Xiaochun Rong

09:00~10:00 Integral Geometry along flat subvarieties in symmetric spaces of compact type (II)Eric Grinberg (University of Massachusetts Boston, USA)

10:10~11:10 Hamiltonian non-displaceability of the Gauss images of isoparametric hypersurfaces (II)Reiko Miyaoka (Tohoku University, Japan)


11:20~12:40 Chair Professor Jiazu Zhou

11:20~12:20 The classical Calabi-Bernstein theorem: history and new approachesAlfonso Romero (University of Granada, Spain)

12:20~12:40 Celebration Address & Group Photo II


[Session 1]

14:00~15:30 Chair Professor Keom Kyo Seo

14:00~14:30 Sequences of maximal antipodal sets of oriented real Grassmann manifolds IIHiroyuki Tasaki (University of Tsukuba, Japan)

14:30~15:00 Maximal antipodal subgroups of the automorphism groups of compact Lie algebrasMakiko Tanaka* (Tokyo University of Science, Japan) and

Hiroyuki Tasaki (University of Tsukuba, Japan)15:00~15:30 Biharmonic homogeneous submanifolds in compact symmetric spaces

Shinji Ohno (Osaka City University Advanced Mathematical Institute (OCAMI), Japan), Takashi Sakai* (Tokyo Metropolitan University, Japan) and

Hajime Urakawa (Tohoku University, Japan)


17:00~18:00 Chair Professor Reiko Miyaoka

17:00~18:00 Quantitative space from rigidity under lower Ricci curvature bound (II)Xiaochun Rong (Rutgers, The State University of New Jersey, USA)

[Session 2]

14:00~15:30 Chair Professor Byung Hak Kim

14:00~14:30 Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact typeNaoyuki Koike (Tokyo University of Science, Japan)

14:30~15:00 Twistor lifts and factorization for conformal maps of a surfaceKazuyuki Hasegawa* (Kanazawa University, Japan) and

Katsuhiro Moriya (University of Tsukuba, Japan) 15:00~15:30 A duality between compact symmetric triads and semisimple pseudo-Riemannian symmetric pairs with

applications to geometry of Hermann type actionsKurando Baba (Tokyo University of Science, Japan)



July 29 (Friday), 2016


09:00~10:00 Integral Geometry along flat subvarieties in symmetric spaces of compact type (III)Eric Grinberg (University of Massachusetts Boston, USA)

10:15~11:15 Hamiltonian non-displaceability of the Gauss images of isoparametric hypersurfaces (III)Reiko Miyaoka (Tohoku University, Japan)



11:30~12:30 Spacelike surfaces in GRW spacetimes whose mean curvature function satisfies a nonlinear inequalityAlfonso Romero (University of Granada, Spain)


[Session 1]

14:00~15:30 Chair Professor Jong Taek Cho

14:00~14:30 Almost complex structures on four-dimensional neutral manifoldsYasuo Matsushita (Osaka City University Advanced Mathematical Institute (OCAMI), Japan)

14:30~15:00 Weyl, projective and conformal semi-symmetric complex hypersurfaces in the semi-Kaehler space forms

Young Suk Choi (Kyungpook National University, Korea)

15:00~15:30 On 3-dimensional real hypersurfaces in complex space formsMayuko Kon (Shinshu University, Japan)


17:00~18:00 Chair Professor Alfonso Romero

17:00~18:00 Inequalities for algebraic Casorati curvatures and their applicationsMukut Mani Tripathi (Banaras Hindu University, India)

[Session 2]

14:00~15:30 Chair Professor Jin Hong Kim

14:00~14:30 The Schwarz lemma for super-conformal mapsKatsuhiro Moriya (University of Tsukuba, Japan)

14:30~15:00 Transversally complex submanifolds of a quaternion projective spaceKazumi Tsukada (Ochanomizu University, Japan)

15:00~15:30 A construction of weakly reflective submanifolds in compact symmetric spacesShinji Ohno (Tokyo Metropolitan University & OCAMI, Japan)



July 30 (Saturday), 2016

10:00~12:20 Chair Professor Jiazu Zhou

10:00~11:00 Derivatives on real hypersurfaces in nonflat complex space formsJuan de Dios Perez (University of Granada, Spain)

11:20~12:20 Constant mean curvature spacelike hypersurfaces in spacetimes with symmetriesAlfonso Romero (University of Granada, Spain)



14:00~14:45 Improved Chen-Ricci inequality for algebraic curvature tensors and its applicationsMukut Mani Tripathi (Banaras Hindu University, India)



15:00~15:30 On totally geodesic surfaces in Misa Ohashi (Nagoya Institute of Technology, Japan)

15:30~16:00 Some topics related to Clifford algebras and the octonionsHideya Hashimoto (Meijo University, Japan)


17:00~17:10 Closing Remarks

The 20th International Workshop on Hermitian Symmetric Spaces and Submanifolds

Abstracts

July 26, 2016(Tuesday)

The 20th International Workshopon Hermitian Symmetric Spaces and Submanifolds

& The 12th RIRCM-OCAMI Joint Differential Geometry Workshop

Kyungpook National University, Daegu, Korea

July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-1 Eunmi Pak* E-mail : [email protected](Kyungil University & RIRCM, Gyeongsan-si, Gyeongsangbuk-do 38428, & Daegu 41566,

Korea)Young Jin Suh E-mail : [email protected]

(Department of Mathematics, Kyungpook National University, Daegu 41566, Korea)

Recent results about Jacobi operator for real hypersurfaces in complex two plane Grassman-nians

In relation to Jacobi operator, we consider a new notion for real hypersurfaces incomplex two-plane Grassmannians G2(Cm+2) and show results about real hyper-surfaces in G2(Cm+2) with structure Jacobi operator and normal Jacobi operatorunder some condition.

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.

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July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-2 Hyunjin Lee* E-mail : [email protected](Research Institute of Real and Compelx Manifolds, Kyungpook National University, Daegu

41566, India)Young Jin Suh E-mail : [email protected]

(Department of Mathematics and RIRCM, Kyungpook National University, Daegu 41566,Korea)

The parallelism for real hypersurfaces in complex Grassmannians of rank two

In this talk we study the cyclic parallel hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m/S(U2Um) = G∗2(Cm+2) which have a remarkable geo-metric structure as Hermitian symmetric spaces of rank 2. At first we prove that ifthe Reeb vector field belongs to the distribution Q⊥ = span{ξν | JνN = −ξν , ν =1, 2, 3}, then a cyclic parallel hypersurface M in G∗2(Cm+2) must be Reeb parallelby virtue of the equation of Codazzi. Using this relation, we classify all cyclic (orReeb, resp.) parallel hypersurfaces in G∗2(Cm+2) with geodesic Reeb flow.

References

[1] B.Y. Chen, G.D. Ludden and S. Montiel, Real submanifolds of a Kaehler manifold, AlgebrasGroups Geom. 1 (1984), 176-212.

[2] H. Lee, Y.S. Choi and C. Woo, Hopf hypersurfaces in complex two-plane Grassmannians

with Reeb parallel shape operator, Bull. Malays. Math. Sci. Soc. 38 (2015), 617-634.[3] H. Lee and Y.J. Suh, Real hypersurfaces of type (B) in complex two-plane Grassmannians

related to the Reeb vector, Bull. Korean Math. Soc. 47 (2010), no. 3, 551-561.

[4] Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Jpn. 28 (1976),529-540.

[5] J.D. Perez and Y.J. Suh, The Ricci tensor of real hypersurfaces in complex two-plane Grass-mannians, J. Korean Math. 44 (2007), 211-235.

[6] Y.J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grass-

mannians, Adv. Appl. Math., 50 (2013), 645659.

[7] Y.J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reebvector field, Adv. Appl. Math. 55 (2014), 131-145.

[8] Y.J. Suh, Real hypersurfaces in compelx quadric with Reeb parallel shape operator, Internat.J. Math. 25 (2014), no. 6, 1450059.

[9] Y.J. Suh, Real hypersurfaces in the compelx quadric with parallel Ricci tensor, Adv.

Math. 281 (2015), 886-905.

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July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-3 Gyu Jong Kim* 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)

Hypersurfaces in complex hyperbolic two-plane Grassmannians with GTW Reeb Lie deriva-tive structure Jacobi operator

In this talk, we consider the complex hyperbolic two-plane GrassmanniansG∗2(Cm+2)

as the abient space. Using previous results of Berndt and Suh, we give a existencetheorem of a real hypersurface in complex hyperbolic two-plane Grassmanniansrelated to a certain condition of the structure Jacobi operator.

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July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-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)

Comparison on classification results of real hypersurfaces in compact and noncompact com-plex two-plane Grassmannians

In this talk, we introduce a new commuting condition between the structure (resp.,normal) Jacobi operator and symmetric (1,1)-type tensor field T , that is, RξϕT =TRξϕ (resp., RNϕT = TRNϕ), where T = A or T = S for Hopf hypersurfacesin complex (hyperbolic) two-plane Grassmannians. By using simultaneous diag-onalzation for commuting symmetric operators, we give a complete classificationof real hypersurfaces in complex (hyperbolic) two-plane Grassmannians with com-muting condition respectively.

References

[1] J. Berndt and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatsh.

Math. 127 (1999), 1–14.[2] J. Berndt and Y. J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two,

Internat. J. Math., World Sci. Publ., 23 (2012), 1250103(35 pages).[3] J. Berndt H. Lee and Y. J. Suh, Contact hypersurfaces in noncompact complex Grassman-

nians of rank two, Internat. J. Math., World Sci. Publ., 24 no. 11 (2013), 13500(11 page).[4] H. Lee, Y.J. Suh and C. Woo, Real hypersurfaces with commuting Jacobi operators in complex

two-plane Grassmannians, Houston J. Math. 40 (2014), no. 3, 751–766.[5] E. Pak, Y. J. Suh and C. Woo, Restricted Ricci conditions for real hypersurfaces in complex

two-plane Grassmannians, Houston J. Math. 41 (2015) no. 3 767-783.[6] Y.J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb

vector field, Adv. Appl. Math. 55 (2014), 131–145.[7] Y.J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commut-

ing Ricci tensor, Inter. J. Math. World Sci. Publ., 26 (2015), 1550008 (26 pages).

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July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-5 Keomkyo Seo E-mail : [email protected](Department of Mathematics, Sookmyung Women’s University, Seoul 04310, Korea )

Characterizations of a Clifford hypersurface in a unit sphere

The Clifford hypersurface is one of the simplest compact hypersurfaces in a unitsphere. In this talk, we give two different types of characterizations of Cliffordhypersurfaces among constant m-th order mean curvature hypersurfaces with twodistinct principal curvatures. One is obtained by assuming embeddedness and bycomparing two distinct principal curvatures. The proof uses the maximum principleto the two-point function, which was used in the proof of Lawson conjecture byBrendle [1]. The other is given by obtaining a sharp curvature integral inequalityfor hypersurfaces in a unit sphere with constant m-th order mean curvature andwith two distinct principal curvatures, which generalizes Simons integral inequality[4]. This talk is based on joint works [2, 3] with Sung-Hong Min.

References

[1] S. Brendle, Embedded minimal tori in S3 and the Lawson conjecture, Acta Math. 211 (2013),no. 2, 177-190.

[2] S.H. Min, K. Seo, Characterizations of a Clifford hypersurface in a unit sphere via Simons’

integral inequalities, to appear in Monatsh. Math.[3] S.H. Min, K. Seo, A characterization of Clifford hypersurfaces among embedded constant

mean curvature hypersurfaces in a unit sphere, to appear in Math. Res. Lett.[4] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105.

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July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-6 Kwang-Soon Park E-mail : [email protected](Division of General Mathematics, University of Seoul, Seoul 02504, Republic of Korea)

On the pointwise slant submanifolds

In this talk, we consider several kinds of submanifolds in Riemannian manifolds,which are obtained by many authors. (i.e., slant submanifolds, pointwise slant sub-manifolds, semi-slant submanifolds, pointwise semi-slant submanifolds, pointwisealmost h-slant submanifolds, pointwise almost h-semi-slant submanifolds, etc.) Weinvestigate the properties of slant functions and semi-slant functions and also ob-tain the topological properties of proper pointwise slant submanifolds. We givean inequality for the squared norm of the second fundamental form in terms ofa warping function and a slant function for a warped product submanifold of aRiemannian manifold. Finally, we give some examples of such submanifolds.

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July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-7 Jong Taek Cho E-mail : [email protected](Department of Mathematics, Chonnam National University, Gwangju 500-757, Korea)

CR geometry on contact manifolds

For a contact manifold, we have two fundamental structures associated with thegiven contact form. One is a Riemannian structure and the other is a transversalalmost complex structure. In this talk, we study pseudo-Hermitian geometry, whichpreserves a transversal almost complex structure.

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July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-8 Jae-Hyouk Lee E-mail : [email protected](Department of Mathematics, Ewha Womans University, 52, Ewhayeodae-gil, Seodaemun-

gu, Seoul 03760, Republic of Korea)

Quaternionic projective spaces and 7-sphere

In this talk, we consider U(n+1) invariant polynomials on quaternionic projectivespaces HPn induced from an isoparametric function on sphere S4n+3 via Hopffibration, and discuss hypersurface geometry on the projective spaces HPn andrelated spheres along the polynomials. In particular, we study the hypersurfacegeometry of 7-sphere S7.

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July 26(Tue) - 30(Sat), 2016

July 26(Tue), 2016

1st-9 Yong Seung Cho E-mail : [email protected](Department of Mathematics, Ewha Womans University, Seoul 120-750, Korea)

Gromov-Witten invariants on the products of almost contact metric manifolds

We investigate Gromov-Witten invariants and quantum cohomologies on the prod-ucts of almost contact metric manifolds. In case the manifolds are cosymplectic,then the product has a Kaehelr structure. On the products we get some geomet-ric properties, Gromov-Witten invariants, and quantum cohomologies. We havesome relations between Gromov-Witten invariants, quantum cohomologies of theproducts and the ones of two cosymplectic manifolds,respectively.

References

[1] D.E. Blair and S.I.Goldberg, Topology of almost contact manifolds, J. Di er. Geom. 1 (1967),347-354.

[2] E. Calabi and B. Eckmann, A class of complex manifold which are not algebraic, Ann. ofMath. 58 (1953), 494-500.

[3] Y.S. Cho, Stabilization of 4-manifolds, Science China Mathematics 57 (2014), 1835-1840.[4] Y.S. Cho, Quantum type cohomologies on contact manifolds, Inter. J. of Geom. Methods in

Modern Phys. 10 (2013), no. 5, 1350012.

[5] Y.S. Cho and S.T. Hong, Dynamics of stringy congruences in the early universe, Phys. Rev.D 83 (2011), 104040.

[6] Y.S. Cho, Hurwitz number of triple rami ed covers, J. of Geom. and Phys. 56 (2008), no. 4,542-555.

[7] Y.S. Cho, Seiberg-Witten invariants on non-symplectic 4-manifolds, Osaka J. Math. 34(1997), 169-173.

[8] K. Fukaya and K. Ono, Arnold conjecture and Gramov-Witten invariant, Topology 38(1999), no. 5, 933-1048.

[9] D. Janssens and J. Vanhecke, Almost contact structures and curvature tensors, Kodai Math.J. 4 (1981), 1-27.

[10] M. Kontsevich and Y. Manin, Gromow-Witten classes, quantum cohomology and enumerategeometry, Comm. Math. Phys. 164 (1994), 525-562.

[11] D. McDuff and D.Salamon, J-holomorphic curves and quantum cohomology, Univ. Lect.Series, AMS, Provindence RI, 6 (1994).

[12] E. Witten, Topological sigma models, Comm. Math. Phys. 118 (1988), 411-449.

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Abstracts

July 27, 2016(Wednesday)




July 26(Tue) - 30(Sat), 2016

July 27(Wed), 2016

2nd-1 Eric Grinberg E-mail : [email protected](Department of Mathematics, University of Massachusetts Boston, 100 Morrissey Blvd.

Boston, MA 02125-3393, USA)

Integral geometry along flat subvarieties in symmetric spaces of compact type

We consider the Radon transform that integrates over flat totally geodesic sub-manifolds in a symmetric space of compact type. The salient theme is that eitherthe transform has a canonical kernel or else it is injective. We also explore variantsof this problem where ‘flat’ is replaced by ‘maximally curved’.

1




July 26(Tue) - 30(Sat), 2016

July 27(Wed), 2016

2nd-2 Reiko Miyaoka E-mail : [email protected](Institute of Liberal Arts and Sciences, Tohoku University, Sendai 9808576, Japan)

Hamiltonian non-displaceability of the Gauss images of isoparametric hypersurfaces (I)-(III)

I will give talks on(I) Introduction to the theory of isoparametric hypersurfaces(II) Introduction to symplectic geometry and Lagrangian submanifolds(III) Floer homology on intersections of Lagrangian submanifolds

Our results will be given by Ohnita’s talk.

References

[1] H. Ma and Y. Ohnita, On Lagrangian submanifolds in complex hyperquadrics and isopara-metric hypersurfaces in spheres, Math. Z. 261 (2009), 749-785.

[2] Hiroshi Iriyeh, Hui Ma, Reiko Miyaoka, and Yoshihiro Ohnita, Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces, preprint (2015).

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July 26(Tue) - 30(Sat), 2016

July 27(Wed), 2016

2nd-3 Jin Hong Kim E-mail : [email protected](Department of Mathematics Education, Chosun University, 309 Pilmun-daero, Dong-gu,

Gwangju, 501-759, Republic of Korea)

LVMB manifolds and transverse Kahler foliations

There is a well-known class of compact, complex, non-Kahlerian manifolds con-structed by Bosio, called the LVMB manifolds, which properly includes the Hopfmanifold, the Calabi-Eckmann manifold, and the LVM manifolds. As in the case ofLVM manifolds, these LVMB manifolds can admit a regular holomorphic foliationF . Moreover, Meersseman shows that if an LVMB manifold is actually an LVMmanifold, then the regular holomorphic foliation F is actually transverse Kahler.The aim of this talk is to address the converse question. Namely, we want to dis-cuss whether or not, when the holomorphic foliation F on an LVMB manifold Nis transverse Kahler, N is actually an LVM manifold.

1




July 26(Tue) - 30(Sat), 2016

July 27(Wed), 2016

2nd-4 Yoshihiro Ohnita E-mail : [email protected](Osaka City University Advanced Mathematical Institute & Department of Mathematics,

Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, JAPAN)

On Floer homology of the Gauss images of isoparametric hypersurfaces

The Gauss images of isoparametric hypersurfaces in the unit standard sphere pro-vide compact minimal (thus monotone) Lagrangian submanifoilds embedded incomplex hyperquadrics Qn(C). Recently we used the Floer homology and thelifted Floer homology for monotone Lagrangian submanifolds in order to studytheir Hamiltonian non-displaceability in the joint paper [1]. In this talk, I wouldlike to explain the spectral sequences for the Floer homology and the lifted Floer ho-mology of monotone Lagrangian submanifolds and their applications to the Gaussimages of isoparametric hypersurfaces, which are the main technical part in ourjoint work. Moreover I will suggest some related open problems

References

[1] H. Iriyeh, H. Ma, R. Miyaoka and Y. Ohnita, Hamiltonian non-displaceability of Gaussimages of isoparametric hypersurfaces, (submitted), arXiv: 1510.05057v1 [math.DG] 17Oct.2015.

1




July 26(Tue) - 30(Sat), 2016

July 27(Wed), 2016

2nd-5 Jiazu Zhou E-mail : [email protected](School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s

Republic of China)

On Bonnesen-style inequalities for mixed volumes

The classical Bonnesen-style inequality measures the deficit of a convex body anda standard ball in the Euclidean space Rn. The Alexandrov-Fenchel inequality formixed volumes is the natural extension of the classical isoperimetric inequality. Wemay interested in the homotheties of convex bodies.

1




July 26(Tue) - 30(Sat), 2016

July 27(Wed), 2016

2nd-6 Xiaochun Rong E-mail : [email protected](Mathematics Department, Capital Normal University, Beijing, P.R.C. & Mathematics

Department, Rutgers University New Brunswick, NJ 08903, USA)

Quantitative rigidity on manifolds of Ricci curvature bound below I

This is the first of two talks under the same tile. In Riemannian geometry, spaceforms (i.e., complete simply connected manifolds of constant sectional curvature)has been played an important role, serving as a model space and a source of ideas.In this talk, starting with two volume space form rigidity with lower Ricci curvaturebound, we will discuss recent work on quantitative versions of the two volume spaceform rigidity. This work is joint with Lina Chen and Shicheng Xu of Capital NormalUniversity.

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Abstracts

July 28, 2016(Thursday)




July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-1 Eric Grinberg E-mail : [email protected](Department of Mathematics, University of Massachusetts Boston, 100 Morrissey Blvd.




1




July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-2 Reiko Miyaoka E-mail : [email protected](Institute of Liberal Arts and Sciences, Tohoku University, Sendai 980-8576, Japan)




References



1




July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-3 Alfonso Romero E-mail : [email protected](Department of Geometry and Topology, University of Granada, 18071 Granada, Spain)

The classical Calabi-Bernstein theorem: history and new approaches

The classical Calabi-Bernstein theorem states that the only entire solutions of themaximal surface equation in 3-dimensional Lorentz-Minkowski spacetime are theaffine functions defining spacelike planes. This relevant result was firstly proved byCalabi in 1970 and later extended to any dimension by Cheng and Yau in 1976.With this result, a new class of elliptic problems of geometric and physical relevancestarted. In this talk, several proofs of historical interest are reviewed and othernew ones are explained, paying special attention to the different techniques used.

1




July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-S1-4 Hiroyuki Tasaki E-mail : [email protected](Division of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba,

Tsukuba, Ibaraki 305-8571, Japan)

Sequences of maximal antipodal sets of oriented real Grassmann manifolds II

Chen-Nagano [1] introduced the notion of antipodal sets of compact Riemanniansymmetric spaces. I showed a correspondece between maximal antipodal sets oforiented real Grassmann manifolds and certain families of subsets of finite sets andreduced the problem of the classifications of maximal antipodal sets of oriented realGrassmann manifolds to a combinatorial problem in [2]. Using this correspondenceI showed some sequences of maximal antipodal sets in [3]. In this talk I constructnew sequences of maximal antipodal sets from those obtained in [3].

References

[1] B.-Y. Chen and T. Nagano, A Riemannian geometric invariant and its applications to aproblem of Borel and Serre, Trans. Amer. Math. Soc. 308 (1988), 273-297.

[2] H. Tasaki, Antipodal sets in oriented real Grassmann manifolds, Internat. J. Math.24 no.8(2013), Article ID: 1350061, 28 pp.

[3] H. Tasaki, Sequences of maximal antipodal sets of oriented real Grassmann manifolds, inICM Satellite Conf. on “Real and Complex Submanifolds”, eds. Y. J. Suh et al., Springer

Proceedings in Mathematics and Statistics, 106 (2014), 515-524.

1




July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-S2-4 Naoyuki Koike E-mail : [email protected](Department of Mathematics, Faculty of Science, Tokyo University of Science, Tokyo

162-8601, Japan)

Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compacttype

First we state the evolutions of the radius function and its gradient along thevolume-preserving mean curvature flow starting from a tube (of nonconstant radius)over a compact closed domain of a reflective submanifold in a symmetric spaceunder certain condition for the radius function. Next, we prove that the tubenessis preserved along the flow in the case where the ambient space is a rank onesymmetric space of non-compact type (other than a (real) hyperbolic space) andthe reflective submanifold is an invariant submanifold and the radius function of theinitial tube is radial. Furthermore, in this case, we prove that the flow reaches tothe invariant submanifold or it exists in infinite time and converges to another tubeof constant mean curvature over the invariant submanifold in the C∞-topology ininfinite time.

References

[1] M. Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric sur-

faces, Comment. Math. Helv. 72 (1997) 52-66.[2] M. Athanassenas, Behaviour of singularities of the rotationally symmetric, volume-

preserving mean curvature flow, Calc. Var. 17 (2003) 1-16.[3] E. Cabezas-Rivas and V. Miquel, Volume-preserving mean curvature flow of revolution hy-

persurfaces in a rotationally symmetric space, Math. Z. 261 (2009) 489-510.[4] N. Koike, Volume-preserving mean curvature flow for tubes in rank one symmetric spaces

of non-compact type, submitted for publication.

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July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-S1-5 Makiko Sumi Tanaka* E-mail : tanaka [email protected](Department of Mathematics, Faculty of Science and Technology, Tokyo University of

Science, Noda, Chiba 278-8510, Japan)Hiroyuki Tasaki E-mail : [email protected]

(Division of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba,Tsukuba, Ibaraki 305-8571, Japan)

Maximal antipodal subgroups of the automorphism groups of compact Lie algebras

The automorphism group Aut(g) of a comapct Lie algebra g is a compact Liegroup. A finite subgroup A of a compact Lie group is an antipodal subgroup if A iscommutative and every element of A is involutive. We classify maximal antipodalsubgroups of Aut(g) when g=su(n), o(n), sp(n), g2. A maximal antipodal subgroupof Aut(g) gives us as many mutually commutative involutions of g as possible. Forthe classification we use our former results of the classification of maximal antipodalsubgroups of Lie groups G = U(n)/Zµ, SU(n)/Zµ, O(n)/{±1}, Sp(n)/{±1}, G2

in [1, 2]. They contain the classification of maximal antipodal subgroups of theadjoint group Ad(G) of G which is isomorphic to the identity component of Aut(g)if the Lie algebra of G is g. We also use canonical forms of elements in a compactLie group which is not connected.

References

[1] M. S. Tanaka and H. Tasaki, Maximal antipodal subgroups of compact Lie groups, in prepa-

ration.[2] M. S. Tanaka, H. Tasaki and O. Yasukura, Maximal antipodal subgroups of the compact Lie

group G2 of exceptional type, in preparation.

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July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-S2-5 Kazuyuki Hasegawa* E-mail : [email protected](Faculty of teacher education, Institute of human and social sciences, Kanazawa university,

Kakuma-machi, Kanazawa, Ishikawa, Japan)Katsuhiro Moriya E-mail : [email protected]

(Division of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba,1-1-1 Tennodai, Tsukuba, Ibaraki, Japan)

Twistor lifts and factorization for conformal maps of a surface

The twistor theory serves an important role in the study of surfaces in four-dimensional Riemannian manifolds, in particular, minimal surfaces (see [1] and[3], for example). Quaternionic holomorphic geometry [2] is another useful theoryfor studying the surfaces in the special case. In this talk, we show explicitly a re-lation between the theory of twistor lifts and quaternionic holomorphic geometry.The use of twistor lifts has the advantage that it induces a factorization of thedifferential of a conformal map into a factor which describes intrinsic geometry ofa surface and other factors which describe extrinsic geometry of a surface. Moreprecisely, we show that the differential of a conformal map is factored by two mapsinto Sp(1) and a (1, 0)-form locally. We note that the (1, 0)-form gives the intrinsicinvariant of a conformal map. The maps into Sp(1) give the generalized Gauss mapof a surface. Using this factorization, we give a characterization for constrainedWillmore surfaces etc.

References

[1] R. L. Bryant, Conformal and minimal immersions of compact surfaces into the 4-sphere, J.

Differential Geom. 17, 1982, 3, 455–473.[2] F. E. Burstall, D. Ferus, K. Leschke, F. Pedit and U. Pinkall, Conformal geometry of surfaces

in S4 and quaternions, Lecture Notes in Mathematics 1772 (2002), Springer-Verlag, Berlin.

[3] T. Friedrich, On surfaces in four-spaces, Ann. Global Anal. Geom. 2 (1984), no. 3, 257-287.[4] K. Hasegawa and K. Moriya, Twistor lifts and factorization for conformal maps of a surface,

arXiv:1510.06923.

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July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-S1-6 Shinji Ohno E-mail : [email protected](Osaka City University Advanced Mathematical Institute (OCAMI), Osaka City University,

3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan)Takashi Sakai* E-mail : [email protected]

(Department of Mathematics and Information Sciences, Tokyo Metropolitan University,Minami-Osawa 1-1, Hachioji, Tokyo, 192-0397, Japan)Hajime Urakawa E-mail : [email protected]

(Global Learning Center, Tohoku University, Kawauchi 41, Sendai, 980-8576, Japan)

Biharmonic homogeneous submanifolds in compact symmetric spaces

In 1983, J. Eells and L. Lemaire extended the notion of harmonic map betweenRiemannian manifolds to that of biharmonic map, which is defined as a criticalpoint of the bienergy functional. G.Y. Jiang studied the first and second variationformulas of the bienergy functional and obtained the Euler-Lagrange equation,which is a fourth order PDE. In this talk, first we give a necessary and sufficientcondition for a submanifold whose tension field is parallel to be biharmonic. Fororbits of commutative Hermann actions in compact symmetric spaces, this con-dition can be described in terms of symmetric triads. By using this criterion, wedetermine all biharmonic hypersurfaces in irreducible symmetric spaces of compacttype which are regular orbits of commutative Hermann actions of cohomogeneityone. Moreover, we construct higher codimensional proper biharmonic submanifoldsin compact symmetric spaces and give some classification results. Here, we call abiharmononic map is proper if it is not harmonic.

References

[1] S. Ohno, T Sakai and H. Urakawa, Biharmonic homogeneous hypersurfaces in compact sym-metric spaces, Differential Geom. Appl. 43 (2015), 155–179.

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July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-S2-6 Kurando Baba E-mail : baba [email protected](Department of Mathematics, Faculty of Science and Technology, Tokyo University of

Science, 2641 Yamazaki, Noda, Chiba 278-8510, JAPAN)

A duality between compact symmetric triads and semisimple pseudo-Riemannian symmetricpairs with applications to geometry of Hermann type actions

In this talk, we introduce the notion of a duality between commutative compactsymmetric triads and semisimple pseudo-Riemannian symmetric pairs, which is ageneralization of the duality between compact/noncompact Riemannian symmetricpairs. As its application, we study the action of a symmetric subgroup of G on apseudo-Riemannian symmetric space G/H, which is called a Hermann type action.We also give an alternative proof for Berger’s classification of semisimple pseudo-Riemannian symmetric pairs from a viewpoint of compact symmetric triads. Thistalk is based on a joint work with Osamu Ikawa (Kyoto Institute of Technology)and Atsumu Sasaki (Tokai University).

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July 26(Tue) - 30(Sat), 2016

July 28(Thu), 2016

3rd-7 Xiaochun Rong E-mail : [email protected](Mathematics Department, Capital Normal University, Beijing, P.R.C. & Mathematics

Department, Rutgers University New Brunswick, NJ 08903, USA)

Quantitative space from rigidity under lower Ricci curvature bound II

This is the second of two talks under same title. In this talk, we will prove thequantitative volume space form rigidity conjecture under the additional condition:Ricci curvature is also bounded above. Using the upper bound on Ricci curvatureenable, we will employ techniques of Ricci flow to show the existence of nearbymetric of almost constant sectional curvature. This work is joint with Lina nChenand Shicheng Xu of Capital Normal University.

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Abstracts

July 29, 2016(Friday)




July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-1 Eric Grinberg E-mail : [email protected](Department of Mathematics, University of Massachusetts Boston, 100 Morrissey Blvd.




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July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-2 Reiko Miyaoka E-mail : [email protected](Institute of Liberal Arts and Sciences, Tohoku University, Sendai 980-8576, Japan)




References



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July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-3 Alfonso Romero E-mail : [email protected](Department of Geometry and Topology, University of Granada, 18071 Granada, Spain)

Spacelike surfaces in GRW spacetimes whose mean curvature function satisfies a nonlinearinequality

A nonlinear inequality involving the mean curvature function of a spacelike sur-face and the restriction of the Hubble function in a 3-dimensional GeneralizedRobertson-Walker spacetime is considered and analyzed in detail. Under geomet-ric assumptions, such complete spacelike surfaces are classified. In the parametriccase, new Calabi-Bernstein type problems are then stated and solved as a conse-quence.

1




July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-S1-4 Yasuo Matsushita E-mail : [email protected](Osaka City University Advanced Mathematical Institute, Osaka , Japan)

Almost complex structures on four-dimensionsional neutral manifolds

The existence of a neutral metric (+ + −−) on a 4-manifold is equivalent to theexistence of a field of 2-planes, and also to the existence of two kinds of almost com-plex structures. Therefore, a neutral 4-manifold is necessarily an almost complexmanifold, and simultaneously an opposite almost complex manifold. Concerningthese almost complex structures, neutral 4-manifolds provides us various interest-ing problems, i.e., when they are integrable and when the corresponding Kahlerforms are symplectic, and moreover when they are Kahler. Nontrivial examplesof isotropic Kahler structures and a routine of producing non-flat neutral Kahler-Einstein 4-manifolds, and other interesting results are reported.

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July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-S2-4 Katsuhiro Moriya E-mail : [email protected](Division of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba,

1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan)

The Schwarz lemma for super-conformal maps

A super-conformal map from a Riemann surface to the Euclidean four-space is asurface with circular ellipse of curvature with respect to the induced metric. Thismap has properties similar to the holomorphic function on a Riemann surface. Inthis talk, the Schwarz lemma for super-conformal maps is explained.

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July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-S1-5 Young Suk Choi E-mail : [email protected](College of IT Engineering, Kyungpook National University, Daegu 702-701, Korea)

Weyl, projective and conformal semi-symmetric complex hypersurfaces in semi-Kaehler spaceforms

In this talk, we introduce the notion of Weyl semi-symmetric, projective semi-symmetric and conformal semi-symmetric curvature tensor defined on semi-Kaehlermanifolds. Moreover, we give a complete classification of complex hypersurfacesM in semi-Kaehler space forms Mn+1

s+t (c) with Weyl semi-symmetric, projectivesemi-symmetric or conformal semi-symmetric curvature tensor respectively andalso investigate a relation of these conditions.

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July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-S2-5 Kazumi Tsukada E-mail : [email protected](Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo

112-8610, Japan)

Transversally complex submanifolds of a quaternion projective space

We study a kind of complex submanifolds in a quaternion projective space HPn,which we call transversally complex submanifolds, from the viewpoint of quater-nionic differential geometry. There are several examples of transversally compleximmersions of Hermitian symmetric spaces. For a transversally complex immersionf : M → HPn, a key notion is a Gauss map associated with f , which is a mapS : M → End(Hn+1) with S2 = −id. Our theory is an attempt of a generalizationof the theory “ Conformal geometry of surfaces in S4 and quaternions” by Burstall,Ferus, Leschke, Pedit, and Pinkall [1].

References

[1] F.E. Burstall, D. Ferus, K. Leschke, F. Pedit, and U. Pinkall: Conformal geometry of surfaces

in S4 and quaternions, Springer Lecture Notes in Mathematics, 1772, Springer, Berlin, 2002.

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July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-S1-6 Mayuko Kon E-mail : mayuko [email protected](Faculty of Education, Shinshu University, 6-Ro, Nishinagano, Nagano 380-8544, Japan)

On 3-dimensional real hypersurfaces in complex space forms

In this talk, we classify real hypersurfaces of a 2- dimensional complex space formwhose Ricci operator S and the structure vector field ξ satisfy Sξ = βξ for somefunction β and g((∇XS)Y,Z) = g((∇Y S)X,Z), where X,Y and Z are perpendic-ular to ξ. And we consider some related conditions for the Ricci tensor S.

1




July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-S2-6 Shinji Ohno E-mail : [email protected](Osaka City University Advanced Mathematical Institute (OCAMI))

A construction of weakly reflective submanifolds in compact symmetric spaces.

A submanifold of a Riemannian manifold is called a weakly reflective submanifoldif for any point in the submanifold and each normal vector at the point, there existsa weakly reflection which is an isometry on the ambient Riemannian manifold. Thenotion of weakly reflective submanifold is a generalization of the notion of reflectivesubmanifold. Reflective submanifolds in irreducible compact symmetric spacesare classified. In contrast, examples of weakly reflective submanifolds in compactsymmetric spaces are not well known except for examples in hyperspheres. In thistalk we will give examples of weakly reflective submanifolds in compact symmetricspaces as orbits of Hermann actions which are generalizations of isotropy actions ofcompact symmetric spaces. To construct weakly reflective submanifolds in compactsymmetric spaces, we use symmetric triads which are generalizations of restrictedroot systems.

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July 26(Tue) - 30(Sat), 2016

July 29(Fri), 2016

4th-7 Mukut Mani Tripathi E-mail : [email protected](Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi

221005, India)

Inequalities for algebraic Casorati curvatures and their applications

In 1889, Casorati ([1],[2]) defined a curvature (now well known as the Casoraticurvature) which turns out to be the normalized sum of the squared principalcurvatures. For a Riemannian submanifold the Casorati curvature is defined to bethe normalized squared length of the second fundamental form [3].

The talk is organized as follows. An important Lemma for a convex optimiza-tion problem is presented. Given an n-dimensional Riemannian manifold (M, g), aRiemannian vector bundle (B, gB) over M , a B-valued symmetric (1, 2)-tensor fieldζ and a (curvature-like) tensor field T satisfying the algebraic Gauss equation, we

introduce the notion of different kind of algebraic Casorati curvatures δCT,B (n−1),

δ′CT,B (n− 1), δCT,B (n− 1), δCT,B (r;n− 1), δCT,B (r;n− 1), which in special casesof Riemannian submanifolds reduce to already known Casorati curvatures. Wepresent results expressing Casorati inequalities for algebraic Casorati curvatures.Equality cases are also discussed. After this, application parts begin. We obtain

basic inequalities for Casorati curvatures δ(n− 1), δ′(n− 1), δ(n− 1), δ(r;n− 1),

δ(r;n − 1) for Riemannian submanifolds. We further apply these results to ob-tain Casorati inequalities for Riemannian submanifolds of real space forms witha very short proof. We also present several results for Casorati ideal submani-folds/hypersurfaces of real space forms and Euclidean spaces. Finally, we presentsome problems for further studies.

References

[1] F. Casorati, Nuova definizione della curvatura delle superficie e suo confronto con quella di

Gauss (New definition of the curvature of the surface and its comparison with that of Gauss),Rend. Inst. Matem. Accad. Lomb. Series II 22 (1889), no. 8, 335-346.

[2] F. Casorati, Mesure de la courbure des surfaces suivant l’idee commune. Ses rapports avecles mesures de courbure gaussienne et moyenne, (Measuring the curvature of the surfaces

along the common idea. Its relations with Gaussian and mean curvature measurements) ActaMath. 14 (1890), no. 1, 95-110.

[3] S. Decu, S. Haesen, and L. Verstraelen, Optimal inequalities charaterising quasi-umbilical

submanifolds, J. Inequal. Pure Appl. Math. 9(3) (2008), Article ID 79, 07 pp.

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Abstracts

July 30, 2016(Saturday)




July 26(Tue) - 30(Sat), 2016

July 30(Sat), 2016

5th-1 Juan de Dios Perez E-mail : [email protected](Departamento de Geometrıa y Topologıa, Universidad de Granada, 18071-Granada, Spain)

Derivatives on real hypersurfaces of nonflat complex space forms

Let M be a real hypersurface of a nonflat complex space form, that is, eithera complex projective space or a complex hyperbolic space. On M we have theLevi-Civita connection abd for any nonnull real number k the corresponding gen-eralized Tanaka-Webster connection. Therefore on M we consider their associatedcovariant derivatives, the Lie derivative and, for any nonnull k, the so called Liederivative associated to the generalized Tanaka-Webster connection and introducesome classifications of real hypersurfaces in terms of the concidence of some pairsof such derivations when they are applied to the shape operator of the real hy-persurface, the structure Jacobi operator, the Ricci operator or the Riemanniancurvature tensor of the real hypersurface.

1




July 26(Tue) - 30(Sat), 2016

July 30(Sat), 2016

5th-2 Alfonso Romero E-mail : [email protected](Department of Geometry and Topology, University of Granada, 18071 Granada, Spain)

Constant mean curvature spacelike hypersurfaces in spacetimes with certain symmetries

The role of some symmetries of spacetime which naturally arise in General Rela-tivity is discussed. The importance of constant mean curvature (CMC) spacelikehypersurfaces in the study of the Einstein equation is recalled. In certain space-times with symmetry defined by a timelike conformal vector field or by a lightlikeparallel vector field, uniqueness theorems of complete CMC spacelike hypersurfacesare given. In some cases, uniqueness results of Calabi-Bernstein type are obtainedas applications.

1




July 26(Tue) - 30(Sat), 2016

July 30(Sat), 2016

5th-3 Mukut Mani Tripathi E-mail : [email protected](Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi

221005, India)

Improved Chen-Ricci inequality for algebraic curvature tensors and its applications

In 1999, B.-Y. Chen [1, Theorem 4] proved a basic inequality involving the Riccicurvature and the squared mean curvature of submanifolds in a real space form. In[2], the author presented Chen-Ricci inequality and improved Chen-Ricci inequalityfor curvature like tensors and applied the results to study Kaehlerian slant sub-manifolds of complex space forms and Legendrian submanifolds of Sasakian spaceforms. In this talk, the author will give a short survey of results and some problemsfor further studies.

References

[1] B.-Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with

arbitrary codimensions, Glasgow Math. J. 41 (1999), 33-41.

[2] M.M. Tripathi, Improved Chen-Ricci inequality for curvature-like tensors and its applications,Differential Geom. Appl. 29 (2011), no. 5, 685-698.

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July 26(Tue) - 30(Sat), 2016

July 30(Sat), 2016

5th-4 Misa Ohashi E-mail : [email protected](Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan)

On totally geodesic surfaces in G2/SO(4)

This is a joint work with H. Hashimoto, K. Mashimo and F. Nakata. We explainthe Quaternionic Kahler structure on G2/SO(4). Recently, Professor Mashimoobtained classification of totally geodesic surfaces in G2/SO(4). We give the ge-ometrical properties (for example totally complex or totally real) of these totallygeodesic surfaces. We give the relationship between the totally geodesic surfaces inG2/SO(4) and the 3-dimensional totally real submanifolds in 6-dimensional sphereS6 = G2/SU(3).

1




July 26(Tue) - 30(Sat), 2016

July 30(Sat), 2016

5th-5 Hideya Hashimoto E-mail : [email protected](Department of Mathematics, Meijo University, Nagoya 468-8502, Japan)

Some topics related to Clifford algebras and the octonions

This is a joint work with K. Mashimo, F. Nakata and M. Ohashi. Let C be theoctonions or Cayley algebra and let G2 be the group of automorphisms of C withrespect to the multiplication. There exist the set of homogeneous spaces relatedto G2. For example, G2/SU(3) is isomorphic to the unit sphere S6 in the purelyimaginary octonions ImC and G2/SO(4) can be considered as the Grassmann man-ifold of all associative 3-planes in ImC. Then there exists a twistor correspondencebetween the Nearly Kahler S6 and Quaternionic Kahler G2/SO(4). The purposeof this talk is to give the fibre bundle structures related to the homogeneous spacesof G2 and their geometrical structures. In particular, we want to describe the ge-ometrical structures on the homogeneous spaces by using the invariant differentialforms and moving frames of G2.

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the 20th international workshop on hermitian symmetric spaces and submanifoldsrircm.knu.ac.kr ›...

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