Various geometric structures and Grassmann Geometry

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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.