# Differential Geometry Seminar (2011)

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 | Misa Ohashi（Meijo University & OCAMI） |
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Title | On structure of the $G_2$-moduli space of helix in ImO |

Date | January 31 (Tue.) 2012, 14:40～16:10 |

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

Abstract | Let $\gamma _0$ be a helix in purely imaginary octonions and $\Gamma _{\gamma _0}$ be all helices which are congruent to $\gamma _0$ under the action of oriented isometry of $\textbf{R}^7$. We shall consider the moduli of $\Gamma _{\gamma _0}$ under the action of $G_2$ which are subgroup of isometry group of $\textbf{R}^7$. |

Speaker | Toshihiro Nogi （OCAMI） |
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Title | On a non-extendibility of the Bers isomorphism |

Date | Jan. 25 (Wed.) 2012, 13:00～14:30 |

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

Abstract | Zhang showed that the Bers isomorphism can not be continuously extended to the closure of the Bers fiber space F(S), where S is a closed Riemann surface of genus g(>1). I try to generalize his result. Let A be the set of all points filling S. In this talk, I will consider a non-extendibility of the Bers isomorphism to a point not in A. |

Speaker | Nobutaka Boumuki （OCAMI） |
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Title | A non-existence theorem for Hessian metrics of positive constant Hessian sectional curvature |

Date | Dec. 21 (Wed.) 2011, 17：00～18：30 |

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

Abstract | This talk is based on a joint work with Prof. Tomonori Noda (Osaka Dental University). In this talk we assert the following non-existence theorem: THEOREM. There does not exist any simply connected Hessian manifold of positive constant Hessian sectional curvature in the case where its flat affine connection is complete. One can prove THEOREM by a rigidity theorem for holomorphic totally geodesic isometric immersions. We will explain the reason why. |

Speaker | Ryozo Yoshikawa（Hachiman Technical High School） |
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Title | Global Kropina spaces |

Date | Nov. 17 (Thu.) 2011, 14:40～16:10 |

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

Abstract | Kropina spaces are simple Finsler structures on manifolds that have been only locally studied until now. In the present lecture we will discuss the existence and non-existence of globally defined Kropina spaces, the conditions for these spaces to be of constant curvature as well as some geodesic behavior related problems. |

Speaker | Katrin Leschke （University of Leicester, UK） |
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Title | Dressing and Darboux transform |

Date | September 22 (Thu.) 2011, 15:30～17:00 |

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

Abstract | In the second part of the talk, we will discuss a generalisation of the classical Darboux transformation to a transformation on conformal maps. On the other hand, constant mean curvature surfaces, Hamiltonian stationary Lagrangians and Willmore surfaces are all given by a harmonicity condition which allows to define a dressing operation on the corresponding harmonic map. We will discuss how these two transformations are related. |

Speaker | Katrin Leschke （University of Leicester, UK） |
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Title | Introduction to Quaternionic Holomorphic Geometry |

Date | September 22 (Thu.) 2011, 13:30～15:00 |

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

Abstract | In this talk I will explain how a quaternionic generalization of complex algebraic geometry can be used to obtain results for conformal immersions of a Riemann surface into Euclidean space: conformal maps are exactly the quaternionic holomorphic maps of the new theory. |

Speaker | Giacomo De Leva (Kumamoto University & OCAMI) |
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Title | Weak convergence of laws of stochastic processes on non-compact manifolds |

Date | July 13 (Wed.) 2011, 16:00～17:30 |

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

Abstract | We prove that if a sequence of singular weighted manifolds $M_{n}$ is convergent in the sense of measured, pointed Gromov-Hausdorff convergence, then, under some conditions, the law of the stochastic process associated to the canonical Dirichlet form on the limit space can be approximated, in the sense of weak convergence of measures, by the laws of the stochastic processes obtained from the projections, by means of the $\epsilon_{n}$-isometries, of the discrete time processes associated to the canonical Dirichlet forms on $M_{n}$. |

Speaker | Hironao Kato (JSPS PD, OCAMI) |
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Title | Invariant flat projective structures on homogeneous spaces and prehomogeneous vector spaces II |

Date | May 18 (Wed.) 2011, 14:40～16:10 |

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

Abstract | Let G be a simply connected complex reductive linear algebraic group. Then I will give a correspondence between Zariski open orbits of prehomogeneous vector spaces (G, F, V) and invariant complex centered affine structures on homogeneous spaces G/H. Moreover I will explain that any prehomogeneous vector spaces induces an invariant flat complex projective structure. As an application of this correspondence, I will discuss the classification of complex Lie groups admitting irreducible left invariant flat complex projective structures. If we have enough time, centro affine hypersurface immersions also will be discussed as the application. |

Speaker | Hironao Kato (JSPS PD, OCAMI) |
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Title | Invariant flat projective structures on homogeneous spaces and prehomogeneous vector spaces I |

Date | May 11 (Wed.) 2011, 14:40～16:10 |

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

Abstract | In these two lectures I will present a correspondence between invariant flat complex projective structures on complex homogeneous spaces and prehomogeneous vector spaces over complex field, and also give its application to geometry. In the 1st lecture I will give a basics of this research. Particular emphasis will be on projective structures given by the linear connections and invariant flat linear connections on homogeneous spaces. Next I will explain that flat projective structures are also described by atlases and Cartan connections for subsequent discussions. |

Speaker | Tsuyoshi Houri (Osaka City University Advanced Mathematical Institute) |
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Title | On Hidden Symmetry of higher dimensional black holes |

Date | April 27 (Wed.) 2011, 14:40～16:10 |

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

Abstract | The most general known black hole solution of vacuum Einstein's equation in arbitrary dimension is called higher-dimensional Kerr-NUT-AdS space-times. It is known that such a class of black holes admit separation of variables in Hamilton-Jacobi equation for geodesics and its property is relevant for conformal Killing-Yano tensors. In this talk, I would like to talk about classification of space-times admitting a conformal Killing-Yano tensor. If the time permits, I will also talk about recent progress on a generalization of conformal Killing-Yano tensor. |