Studies on Curvatures of Submanifolds

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In this research, using a method established by Y.Kitagawa, we studied flat tori in the unit 3-sphere SィイD13ィエD1, and obtained many interesting results. We summarize the results as follows.(1) Isometric deformations of flat tori in SィイD13ィエD1 with nonconstant mean curvature : In this research, we studied isometric deformations of an isometric immersion f of a flat torus M into SィイD13ィエD1, and proved that if the mean curvature of f is not constant, then the immersion f admits a nontrivial isometric deformation preserving the total mean curvature.(2) Isometric deformations of flat tori in SィイD13ィエD1 with constant mean curvature : It is easy to see that if f is an isometric immersion of a flat torus M into SィイD13ィエD1 with constant mean curvature, then f is a covering map onto a Clifford torus in SィイD13ィエD1. In this research, we classified the covering maps which admit no isometric deformation.(3) Periodicity of the asymptotic curves of n-dimensional flat tori isometrically immersed in SィイD12n-1ィエD1 : In 1988, it was shown that if M is a 2-dimensional flat torus isometrically immersed in SィイD13ィエD1, then every asymptotic curve of M is periodic. In this research, we studied periodicity of the asymptotic curves of n-dimensional flat tori isometrically immersed in SィイD12n-1ィエD1, and proved that there exists a 3-dimensional flat torus isometrically embedded in SィイD15ィエD1 all of whose asymptotic curves have no period.