Global Studies on Curvature and Geometric Structures of Riemannian Manifolds

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Head investigator, Itokawa in the papers "Maximal diameter theorem for manifolds with restricted radial curvature" and "Generalized Toponogov's theorem for manifolds with radial curvature bounded below" written jointly with Katsuhiro Shiohama of Saga University and Yoshiro Machigashira of Osaka University of Educations, investigated riemannian manifolds whose sectional curvature in radial directions from a fixed point is bounded from below by certain function. We have obtained some triangle comparison theorems generalizing that of Toponogov. Moreover, in case equality is attained in these comparisons, these manifolds exhibit very rigid geometrical structure containing some totally geodesic surfaces. We have also obtained some applications of these results including a new sphere theorem.Investigator Nishihara is continuing his study on the extendibility of certain functions on infinite dimensional locally pseudo-convex spaces. In "The extension of polynomials of integral type in locally … More convex spaces and its applications" and "The extension of entire functions of nuclear type on locally convex spaces", he has extended his own results on the extendibility to the total space of polynomials and holomorphic functions which are a priori only defined on locally convex spaces. In the paper "The extension of entire function of nuclear type", he has also succeeded in generalizing Hahn-Banach type theorems of Meise-Vogt and of Nishihara himself. Further extension will be discussed in a forthcoming paper "A Hahn-Banach extension theorem for entire functions of nuclear type". In another forthcoming paper "Pseudoconvex domains of infinite dimensional Grassmann manifolds", Nishihara has also proved a vanishing theorem for regular pseudoconvex domains in Banach spaces.Nishiyama who has substituted as an investigator for the year 2001 has written two papers "Pseudo-advection methods for the axisymmetric stationary Euler equations" and "Magnetohydrodynamic approach to the solvability of the three-dimensional stationary Euler equations". In these papers, Nishiyama studied the methods for obtaining stationary solutions to certain Euler type equations. In particular, he established the effectiveness of the method of using the Galerkin approximation on certain accompanying equations associated to the given equation in case the given equation has a rotationary symmetry around the axis and for general equations in dimension 3. Less