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

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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.