Differential systems and submanifolds theory

给出了具有单多重度的六主曲率等参超曲面齐次性的一个新的简单证明。为此，我们研究了退化高斯映射球中的子流形，并构造了许多满足Ferus不等式中的等式的例子。所有等参超曲面的焦点子流形和所有具有退化高斯映射的等参超曲面都是严格子流形。已知朴素子流形的正规束的正则嵌入给出了一个特殊的拉格朗日子流形。我们证明了R^n上的一个适当完全朴素子流形与一个维数不大于该子流形半维数的CW复形具有相同的同伦类型。从体积最小化正常束的角度来看，这很有趣。作为一个在S^3中分割常平均曲率曲面高斯映射的反问题，我们得到了从一对调和映射中得到常平均曲率曲面到tos ^2的充分必要条件。Kaneyuki研究了广义共形变换群的轨道。内山研究了非线性常微分方程的奇异性。横山完成了扩展DS图的理论。Tamaru分类了非紧型对称空间的各向同性轨道。Aiyama给出了多种曲面的表示公式，特别是C^2中的Langarnian曲面。石川给出了空间曲线可展曲面的拓扑分类，并研究了接触流形的奇异性。Udagawa用椭圆函数描述了环面到格拉斯曼流形的调和映射。他还研究了厄米对称空间中的圆。Umehara研究了曲线的奇异性，最小曲面的端点，以及它们的高斯映射，研究了常曲率曲面的固有性质。Kimura构造了子流形上圆束和球束的许多齐次例子。

We gave a new simple proof of the homogeneity of isoparametric hypersurfaces with six principal curvatures with single multiplicity. Concerning this, we investigate submanifolds in the spheres with degenrate Gauss mapping, and constructed many examples satisfying the equality in the Ferus inequality. All focal submanifolds of isoparametric hypersurfaces and all isoparametric hypersurfaces with degenerate Gauss mapping are austere submanifolds. It is known that the canonical embedding of the normal bundle of a (cone of) austere submanifold gives a special Lagrangian submanifold. We have shown that a proper complete austere submanifolds in R^n have the same homotopy type as a CW complex of dimension not larger than half dimension of the submanifold. This is interesting from the view point of volume minimizing normal bundles. As an inverse problem of splitting Gauss maps of surfaces of constant mean curvature in S^3, we get necessary and sufficient conditions to get surfaces of constant mean curvature from a pair of harmonic maps intoS^2.Kaneyuki studied orbit of generalized conformal transformation groups. Uchiyama investigated singularities of a non-linear ordinary differential eqquations. Yokoyama completed the theory of extended DS diagram. Tamaru classified isotropy orbits of certain symmetric spaces of non-compact type. Aiyama gave many kind of reperesentation formulas for surfaces, especially Langarnian surfaces in C^2.Ishikawa gave a topological classification of developable surfaces of space curves and studied singularities of contact manifolds. Udagawa described harmonic maps from tori into Grassmannian manifolds by using elliptic functions. He also studied circles in Hermitian symmetric spaces. Umehara studied singularities of curves, ends of minimal surfaces and concerning with Gauss map of these, studied an intrinsic properties of surface with constant curvature 1.Kimura constructied many homogeneous examples of circle and sphere bundles over submanifolds.