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　京都大学数理解析研究所の共同研究事業の一つとして， 下記の通り研究集会を開催いたします．
　《エリー・カルタンと２１世紀》というテーマで 「Ｄ１くらいが聞いて，勉強になってかつ元気が出る話」を いくつか依頼しておりますので， 院生の方も奮って御参加ください．

January 24

13:30 -- 14:30 KENMOTSU Katsuei (Tohoku University)

E. Cartan in the Bonnet Problem : E. Cartan and the Twenty-First Century

Abstract: A generic surface in Euclidean three space is determined uniquely by its metric and the second fundamental form, but there are special surfaces where this is not the case. Classification of the surfaces is the Bonnet Problem. After reviewing the E. Cartan's contribution to this problem, the recent progress is summarized with emphasis on the theory of integrable systems.

14:45 -- 15:45 HASEGAWA Kazuyuki (Science University of Tokyo)

The fundamental theorems for affine immersions into hyperquadrics
Abstract: In Riemannian geometry, the fundamental theorems for submanifolds, that is, existence and congruence theorems, are important and useful. The fundamental theorems for affine immersions into the affine spaces have been also investigated. We prove the fundamental theorems for affine immersions into hyperquadrics (including affine spaces) with arbitrary codimension. Our fundamental theorems are generalization of those for isometric immersions into space forms. The fundamental theorems for equiaffine immersions into hyperquadrics with arbitrary codimension are also obtained.

16:00 -- 17:00 NAGANO Tadashi

Geometric Theory of Symmetric Spaces, Prelude (1) : E. Cartan and the Twenty-First Century

Abstract: A symmetric space M is defined with a point symmetry assigned to each point of M along with a homomorphism of M into another symmetric space (as an object of a category). Then M has a unique affine connection which is left invariant under the point symmetries. Thus the automorphism group G is a Lie group. If M is connected, then G is transitive on M and a homomorphism into another is nothing but a totally geodesic map. Locally, a symmetric space is characterized, in terms of the connection, by the vanishing of the torsion and of the covariant derivative of the curvature R.
Basically, we owe all these facts and further developments to E. Cartan (1869-1951). He made significant contributions to the theories of Lie groups, differential systems and geometry, which were closely related to each other in his works.
Symmetric spaces have been studied in various areas of mathematics. From a geometric point of view, one can define the root system R(M) of M with the Jacobi equations along geodesics, for example. Hence R(M) is a convenient expression of R, as one will see. (PS file)

January 25

09:30 -- 10:30 NAGANO Tadashi

Geometric Theory of Symmetric Spaces, Prelude (2) : E. Cartan and the Twenty-First Century

Abstract: The geometric theory of symmetric spaces in these lectures differs from the conventional one in that M can be a disconnected space such as a finite group. Another example is an arbitrary finite set S whose point symmetry at every point is the identity map of S. S is called trivial. The supremum of the cardinalities of the trivial subspaces in a given M is called the 2-number of M. For the existence of a monomorphism of M into another one, N, it is necessary that the 2-number of M does not exceed that of N, obviously. The 2-number equals the dimension of the cohomology group of M in a certain case (of the R-spaces).
In this lecture, certain basic concepts will be defined with the fixed point set of the point symmetry at a given point. They will be used to discuss the structures and properties of symmetric spaces together with relationships between them. (PS file)

10:45 -- 11:45 YAMADA Kotaro (Kyushu University)

Total curvature of CMC-1 surfaces in H^3
Abstract: Several properties of total absolute curvature of complete CMC-1 (constant mean curvature one) surfaces in hyperbolic 3-space will be introduced. In particular, classification of complete CMC-1 surfaces in H^3 with total absolute curvature not grater than 4 pi, and related topics are treated. (PS file)

13:30 -- 14:30 AIYAMA Reiko (University of Tsukuba)

A construction of Lagrangian surfaces in the complex 2-space

Abstract: I will give a construction of Lagrangian surfaces in the complex 2-space through rotational surfaces in the Euclidean 3-space. This construction is based on the following two facts:
(1) Every Lagrangian conformal immersion from a Riemann surface into the complex 2-space can be given by certain algebraic combination of a solution of a linear system of first order differential equations.
(2) For a surface in the Euclidean 3-space, there exists a spinor representation of the Gauss map satisfying the similar equations.

14:45 -- 15:45 SASAHARA Tooru (Hokkaido University)

On Chen invariant of CR-submanifolds

Abstract : Bang-Yen Chen has introduced new type of Riemannian curvature invariants and obtained sharp inequalities involving these invariants and the square mean curvature for arbitrary submanifolds in real and complex space forms.
We study CR-submanifolds of maximal CR dimension satisfying the equality case of Chen's inequalities. In particular, we are able to establish the explicit representation of such submanifolds in complex hyperbolic space under the condition that the shape operator with respect to the distinguished vector field has constant principal curvatures. Also, we classify three-dimensional normal CR-submanifold satisfying the equality in complex projective space and complex Euclidean space.

16:00 -- 17:00 MIYAOKA Reiko (Sophia University)

Isoparametric hypersurfaces, old and new (1) : E. Cartan and the Twenty-First Century

Abstract: E. Cartan has made a remarkable step in the study of isoparametric hypersurface theory, originated from primitive works in geometric optics. This was the first systematic research of the subject, and many basic facts have been discovered. We review these first, and then mention to recent results and related topics.

January 26

09:30 -- 10:30 MIYAOKA Reiko (Sophia University)

Isoparametric hypersurfaces, old and new (2) : E. Cartan and the Twenty-First Century

10:45 -- 11:45 YAMAGUCHI Keizo (Hokkaido University)

Geometry of Differential Systems (1) : E. Cartan and the Twenty-First Century
Abstract: In this talk, I'd like to give an overview of the development on the Geometry of Differential Systems. Especially I will talk about what we inherit from E.Cartan and N.Tanaka. In the first talk, I will start with an elementary subject of the local classification of regular differential system on a space of dimension less than five, which is the introductory part of the notorious "five variables" paper of E.Cartan.

13:30 -- 14:30 YAMAGUCHI Keizo (Hokkaido University)

Geometry of Differential Systems (2) : E. Cartan and the Twenty-First Century

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