# Differential Geometry Seminar (2010)

As a project of OCAMI, we shall promote the seminar on differential geometry in the wide sense of including the areas related to geometric analysis, topology, algebraic geometry, mathematical physics, integrable systems, information sciences etc.

Contact | Yoshihiro Ohnita Shin Kato Department of Mathematics Osaka City University Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, JAPAN |
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TEL | 06-6605-2617 (Ohnita) 06-6605-2616 (Kato) |

ohnita@sci.osaka-cu.ac.jp shinkato@sci.osaka-cu.ac.jp |

Speaker | Katsuhiro Moriya (University of Tsukuba) |
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Title | Vector-valued exact one forms on a surface |

Date | March 30 (Wed.) 2010, 16:20～17:50 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | A differential of a map from a surface to a Euclidean space is a vector-valued exact one form on a surface. In the case where a surface is simply connected, formulae to construct an exact one form, which is a differential of a map in the field of differential geometry. For example, the Weierstrass-Enneper formula for minimal surfaces and the Kenmotsu formula for surfaces of constant mean curvature. Indeed, these formulae do not construct exact one forms but closed one forms. Because the domain is simply connected, exact one forms are constructed. The necessary and sufficient condition for these local surfaces,to become global surfaces, is that all the periods of the closed one form become zero. In this talk, I will give a necessary and sufficient condition for a closed one form to become an exact one form, in the case where a surface has suitable differential automorphisms. I will explain that how this condition is applicable to theory of surfaces. |

Speaker | Takashi Sakai (Tokyo Metropolitan University & OCAMI visiting researcher) |
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Title | Cohomogeneity one special Lagrangian submanifolds in the cotangent bundle of the sphere |

Date | March 21 (Mon.) 2011, 13:00～ |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | This talk is based on my joint work with Kaname Hashimoto (doctoral student D3, Osaka City University). It is known that Stenzel first constructs cohomogeneity one complete Ricci-flat Kaehler metrics on the cotangent vector bundles over the n-dimensional standard spheres under the group action of SO(n+1). We classify cohomogeneity one special Lagrangian submanifolds in the cotangent vector bundle invariant under the group action of SO(p)SO(q) (p+q=n+1) and discuss the classification problem. Moreover we describe the asymptotic behavior and singularities of those obtained special Lagrangian submanifolds. |

Speaker | Atsushi FUJIOKA (Hitotsubashi University) |
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Title | Self-congruency of centroaffine minimal surfaces |

Date | Jan. 19 (Wed.) 2011, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Centroaffine minimal surfaces are defined as extremals for the area integral of the centroaffine metric and considered as a natural generalization of proper affine spheres. On the other hand, we can define the notion of the center map for affine surfaces, which is a generalization of the center for affine spheres. In this talk, I will introduce fundamental facts about these and talk about self-congruency of centroaffine minimal surfaces. |

Speaker | Tae HATTORI (Kanazawa University, Osaka City University) |
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Title | Functions with finite p-Dirichlet sums and quasimonomorphisms of infinite graphs |

Date | Dec. 2 (Thu.) 2010, 16:30～18:00 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | In this talk, we study some properties on the space of p-Dirichlet finite functions on an infinite graph, via quasimonomorphisms. Particularly, we focus on the variation of the function spaces as varying the indices p and the existence or vanishing of p-harmonic functions with finite p-Dirichlet sums. |

Speaker | Takahiro NODA （Nagoya University & OCAMI） |
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Title | Geometry of differential systems and Tanaka theory |

Date | Nov. 18 (Thu.) 2010, 13：30 ～ 15：30 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | In this talk, we will explain the theory of the construction of the normal Cartan connections (Tanaka connections) for geometric structures treated in Cartan geometry by Prof. Noboru Tanaka. |

Speaker | Yukiko FUKUKAWA ( Osaka City University ) |
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Title | The cohomology ring of the GKM graph of a flag manifold. |

Date | Nov. 10 (Wed.) 2010, 14：40～16：10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | If a closed smooth manifold with an action of a torus satisfies certain conditions, then its equivariant cohomology is determined by the fixed point sets of codimension one subtori of the torus. In addition, the labelled graph correspnding to the fixed point sets of the codimension one subtori and the cohomology of the labelled graph are defined. It is known that the equivariant cohomology of a flag manifold is isomorphic to the cohomology of the labelled graph associated with the flag manifold. In my talk, we determine the ring structure of the cohomology ring of the labelled graph without using this fact. |

Speaker | Professor Hajime Urakawa (Institute for International Education, Tohoku University) |
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Title | Generalized Chen's conjecture on biharmonic submanifolds, and biharmonic maps in two dimensions (after Y-L. Ou's works) |

Date | Oct. 14 (Thu.) 2010, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | The Chen's conjecture is that the only biharmonic submanifolds of the Euclidean space are minimal, and the generalized Chen's conjecture is that the same is true for biharmonic submanifolds of a Riemannian manifold of non-positive curvature. Recently, Y-L. Ou et al. announced that the generalized Chen's conjecture is false. They gave many counter examples in case of conformally flat negative curvature manifolds. This idea still works for constructing biharmonic maps between 2-dimensional Riemannian manifolds (Y-L. Ou et al.). In my talk, I will report these works of Y-L. Ou et al. with their proofs. I will also propose some ideas in order to solve the original Chen's conjecture in case of hypersurfaces of the Euclidean space. |

Speaker | Prof. Owen Dearricott (University of California Riverside, USA) |
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Title | Reflective submanifolds of affine symmetric spaces |

Date | Aug. 30 (Mon.) 2010, 15：00～16：30 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | The few manifolds of positive sectional curvature known arise as the orbit spaces of free isometric actions on Lie groups. We discuss a very different sort of metric of positive curvature that arises on 3-Sasakian 7-manifolds and a new manifold of positive sectional curvature that arises in this way. This example is deeply connected to curvature operators on self-dual Einstein orbifolds and conformal geometry in dimension 4. |

Speaker | Nobutaka BOUMUKI (Osaka City University Advanced Mathematical Institute) |
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Title | Reflective submanifolds of affine symmetric spaces II |

Date | Aug. 30 (Mon.) 2010, 13：30～14：30 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | In this talk I will determine reflective submanifolds of simple affine symmetric spaces G/H; and in addition, I will clarify relation between simple pseudo-Hermitian symmetric spaces G/R and simple para-Hermitian symmetric spaces G/U, in terms of Lagrangian reflective submanifolds (i.e., real forms) of G/R and para-complex reflective submanifolds of G/U. |

Speaker | Prof. Juergen Berndt (King's College London, UK) |
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Title | Hypersurfaces in symmetric spaces |

Date | Aug. 30 (Mon.) 2010, 10：40～12：10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | In this talk I will present a survey about some solved and unsolved problems related to hypersurfaces in symmetric spaces. Particular emphasis will be on homogeneous hypersurfaces, and hypersurfaces with constant principal curvatures or with some other geometric properties. Much of this is related to ongoing joint work with Jose Carlos Diaz Ramos, Young Jin Suh and Hiroshi Tamaru. |

Speaker | Toshiaki Omori ( Graduate School of Science, Tohoku University ) |
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Title | Compactification of moduli space of harmonic maps from surfaces |

Date | July 8 (Thu.) 2010, 16:20～17:50 <\br> July 9 (Fri.) 2010, 13:00～14:30 |

Place | Dept. of Mathematics, Sci. Bldg., 3053(July 8), 3040(July 9) |

Abstract | Japanese page only |

Speaker | Toshiaki Omori ( Graduate School of Science, Tohoku University ) |
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Title | Some existence theorems for harmonic maps via exponentially harmonic maps |

Date | July 7 (Wed.) 2010, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Exponentially harmonic maps are maps between Riemannian manifolds which extremize a functional with exponential growth. A smooth exponentially harmonic map is known to, unlike harmonic maps, always exist in a given homotopy class of continuous maps. In the present talk, I would like to mention that this remarkable fact enables us to re-establish some existence theorems for harmonic maps via exponentially harmonic maps. |

Speaker | Takanari SAOTOME (OCAMI) |
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Title | (Special Lecture) The J-Holomorphic Mapping over a Strongly Pseudo-convex Manifold |

Date | June 15 (Tue.) 2010, 15:00～19:00 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | In this talk, we will study the geometry of strongly pseudo-convex manifolds and $J$-holomorphic mappings between strongly pseudo-convex manifolds. I will do my talk with four parts.In the first two parts of my talk, we will review some basic topics in contact geometry and CR geometry, as an introduction. I will also discuss some known results which is related to CR geometry and the geometry of $J$-holomorphic mappings. In the talk of last two hours, I will introduce the parts of my present research of $J$-holomorphic mappings between strongly pseudo-convex manifolds. In particular, I investigate the removable singularity theorem and the sub-ellipticity of $J$-holomorphic mappings. |

Speaker | Hiroaki ISHIDA（ Dept. of Math., Osaka City University ） |
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Title | On real Bott manifolds which admit a symplectic form |

Date | May 26 (Wed.) 2010, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | The necessary and sufficient condition for a closed manifold to be a symplectic manifold is not known. In this talk, we will discuss about real Bott manifolds which admit a symplectic form. A real Bott manifold is the total space of an iterated $\mathbb{R}P1$-bundle over a point. We will give a complete characterization of real Bott manifolds which admit a symplectic form. |

Speaker | Kensuke ONDA（ Dept. of Math., Nagoya University ） |
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Title | Lorentzian Ricci Solitons and Cohomogeneity One Ricci-flat Metrics |

Date | May 12 (Wed.) 2010, 14:40～16:10 |

Place | Dept. of Mathematics, Sci. Bldg., 3040 |

Abstract | Einstein structures and the Ricci soliton structures are one of representative Riemann structures in the $C^{\infty}$ manifold. The Ricci soliton is one of the generalization of Einstein metrics, and it is the special solution of the Ricci flow equation. Recently many researchers was studying Ricci solitons. By this lecture, I introduce the left-invariant Lorenzian metrics with Ricci soliton structure on three-dimensional unimodular Lie groups. Furthermore, I introduce the study to constitute a Ricci-flat metric on the manifold which expanded the one dimension from Ricci soliton. |