Various geometric structures and Grassmann Geometry

主要研究了六维球面的Grassmann几何。考虑到Cayley代数的乘法，6维球体具有几乎厄米特结构。特别地，Hashimoto给出了六维球面上的超极小J-全纯曲线的具体刻画和这种J-全纯曲线的Gauss曲率公式。Hashimoto，Taniguchi和Udagawa给出了包含在6维球面中的5维球面中所有类型(III)的几乎复数(J-全纯)2维环面的构造方法.Hashimoto和Sekigawa也给出了具有上齐性的6维球面上的J-全纯曲线的表示。Hashimoto和Mashimo研究了Kahler角为常数的6维球面上J-全纯曲线上的三维圆管。这一结果表明，在6维球面上存在许多三维全实CR子流形。

Mainly, we study the Grassmann geometry of 6-dimensional sphere. The 6-dimensional sphere has the almost Hermitian structure by taking account of the multiplication of the Cayley algebra. In particular, Hashimoto gives the concrete description of the super-minimal J-holomorphic curves in the 6-dimensional sphere and the formula of Gauss curvature of such J-holomorphic curves. The problem about the existence of the self intersection of the super-minimal J-holomorphic curves were left.Hashimoto, Taniguchi and Udagawa give the method of construction of all almost complex (J-holomorphic) 2-dimensional Tori of Type (III) in a 5-dimensional sphere which is contained in a 6-dimensional sphere, by taking account of the Theta functions. Also Hashimoto and Sekigawa give the representation of the J-holomorphic curves in the 6-dimensional sphere, which has the cohomogenity one.Hashimoto and Mashimo studied the 3-dimensional tubes over J-holomorphic curves in the 6-dimensional sphere, whose Kahler angle is constant. This result implies that there are many 3-dimensional totally real, CR-submanifolds in the 6-dimensional sphere.