Differential Equations on Manifolds and Their Singularities

首席调查员川上研究了以下内容。第一年：他和金泽大学的土家教授利用J.R. Munkres的方法证明了任意Cr，α流形（流形对）具有C∞平滑。他和村山博士（Shobi大学）等人研究了通过计算机网络提供数学教材的方法。第二年：他和土屋教授（金泽大学）研究了“Kurt Bryant and Lester F. caudill Jr., Inverse Problem 14 1429-1453（1998）”的概化。他们证明了在有限时间间隔内的数据唯一地决定了背表面的形状。他推测高斯-邦纳公式给出了在圆盘上获得正/负高斯曲率的度量变形存在的充分必要条件。他对这个猜想作了部分回答。调查员Kobayashi致力于研究通用地图与其鉴别物的奇怪关系。第一年的主要成果是；利用折叠映射的判别式描述了一般维上的“折叠成四”作用，找到了固定闭合4流形的映射与其判别式的无穷一对应关系，给出了闭流形稳定映射的判别式族。第二年的是；平面曲线的一种表征，它是一个封闭曲面在平面上的一般投影的临界值集；球束在球上的平面投影研究。

Head investigator Kawakami has studied the following. In the first year:He and Prof. Tsuchiya (Kanazawa Univ.) proved that any Cr,α manifold (manifold pair) has a C∞ smoothing by using a method of J.R. Munkres.He and Dr. Murayama (Shobi Univ.) and others studied about a means of giving teaching-materials of mathematics through a computer network.In the second year:He and Prof. Tsuchiya (Kanazawa Univ.) have studied a generalization of "Kurt Bryant and Lester F. caudill Jr., Inverse Problem 14 1429-1453 (1998)". They proved that the data in a finite time-interval uniquely determine the shape of the back surfice.He conjectured that the Gauss-Bonnet formula gives a necessary and sufficient condition for the existence of a metric deformation to obtain a positive/negative Gaussian curvature on a disk. He gave a partial answer of the conjecture.Investigator Kobayashi worked on studying the curious relation of generic maps to their discriminants. The main results in the first year are;a characterization of the 'folding into four' action in general dimensions by the discriminant of the folding map,finding of an infinite to one correspondence of maps of a fixed closed 4-manifolds to their discriminants,providing a family of discrimiants of stable maps of closed manifolds.Those in the second year are;a characterization of plane curves which are the critical value set of a generic projection of a closed surface into the plane;study of planar projections of sphere bundles over spheres.