Geometric structures on submanifolds in space forms and differential equations related to their structures.

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(1) Suyama obtained remarkable results in the research on generic conformally flat hypersurfaces. The subject is an open problem since the work by E. Cartan. Suyama and Hertrich-Jeromin (Bath Univ.) constructed all conformally flat hypersurfaces with cyclic Guichard net and gave a complete classification of them by conformal equivalence. This work gives new development in the research on higher dimensional submanifolds from the integrable system view-points.(2) Shiohama studied the relation between radial curvature and topology of warped product manifolds. He gave Alexandrov-Toponogov comparison theorem for various geodesic triangles, and applied it for a classification of warped product manifolds.(3) Kurose studied the method of constructing improper affine hyperspheres and gave a new representation formula and a characterization of the class of improper affine hypersurfaces obtained by this formula.(4) Kurose and Fujioka gave a representation as a Hamiltonian system to the motions of curves in a complex hyperbola associated with the integrable equations of the Burgers hierarchy.(5) Kawakubo studied Kirchhoff elastic rods in three-dimensional space forms, and obtained the explicit formulas for them. Also, he proved that the energy functional for Kirchhoff elastic rods satisfies the Palais-Smale condition. The result gives an effective method of studying the stability of closed Kirchhoff elastic rods.(6) Matsuura studied discretizations of curves and surfaces in connection with the theory of discrete integrable systems. He made a geometric interpretation of the discrete KdV equation, so that he successfully proposed a non-autonomous version of it.