論黎曼測度V_m在常曲率空間S_(m+1)中的變形

<正> 在歐氏空間E_(m+1)中的安裝及變形問題在近一世紀的幾何學者的工作中得到了解决,而所確定的V_m E_(m+1)一般是不能變形的。就是說,能够變形的只是狹窄的一類超曲面.運用了外微分形式的方法,很詳細地綜合了這些工作,並且完全地給出

ON THE DEFORMATION OF A RIEMANNIAN METRIC V_m IN A SPACE OF CONSTANT CURVATURE S_(m+1)

With the aid of the method of exterior differential forms N. N. Yanenko has recently given a complete classification of the deformation of m dimensional Riemannian metric ds~2 = gii du~i du~i (i,j=1,…,m) in Euclidean space E_(m+1). Here we propose to investigate the same problem in a space S_(m+1) of constant curvature k_(om+1). Introducing the definition of the k_o-rank of a metric, we obtain the following results:1. In general, a V_m S_(m+1) is indeformable and the only possible deformable metric must be of k_o-rank ≤ 2 (an extension of Beez's Theorem).2. When k_o-rank ≥4, the Peterson-Codazzi equations are consequences of Gauss equations (an extension of T. Y. Thomas' Theorem).A complete classification of deformable hypersur faces V_m S_(m+1) is given.