Geometry of the flat tori in the 3-sphere and its higher dimensional generalization

在这项研究中，我们研究了平面环面在3-球面上的几何、亚纯映射和动力系统。本文的主要研究成果如下：(1)S球面中平面环面的等距变形研究。在本研究中，Y.Kitagawa研究了平面环在S^3球面中的等轴测变形。(2)亚纯映射的代数相关性研究。在这项研究中，Y.Aihara证明了占优亚纯映射的代数相关性从复m-空间上的解析有限覆盖空间X传播到射影代数流形的一些准则。此外，应用这些判据，他得到了亚纯映射的唯一性定理，并给出了两个从X到光滑椭圆曲线E的全纯映射是代数相关的条件。(3)具有拓扑稳定性的向量场的研究。在这项研究中，K.Sakai(与K.Moriyasu和N.Sumi)利用拓扑稳定性的概念给出了结构稳定的向量场的一个刻画。更确切地说，证明了所有拓扑稳定的C^1向量场的集合的C^1内部与满足公理A和强横截条件的所有向量场的集合重合。

In this research, we studied geometry of flat tori in the 3-sphere, meromorphic mappings and dynamical systems. The main results of this research are summarized as follows.(1) Studies on isometric deformations of flat tori in the S-sphere.In this research, Y. Kitagawa studied isometric deformations of flat tori isometrically immersed in the 3-sphere S^3 with constant mean curvature. As a result, he obtained a classification of the flat tori isometrically immersed in S^3 which admit no isometric deformation.(2) Studies on algebraic dependence of meromorphic mappings.In this research, Y. Aihara proved some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space X over the complex m-space into a projective algebraic manifold. Moreover, applying these criteria, he obtained unicity theorems for meromorphic mappings, and gave conditions under which two holomorphic mappings from X into a smooth elliptic curve E are algebraically related.(3) Studies on vector fields with topological stability.In this research, K. Sakai (with K. Moriyasu and N. Sumi) gave a characterization of the structurally stable vector fields by making use of the notion of topological stability. More precisely, it was proved that the C^1 interior of the set of all topologically stable C^1 vector fields coincides with the set of all vector fields satisfying Axiom A and the strong transversality condition.