摘要
研究了m维黎曼空间中的n维曲面在无穷小等距变分下保持平均曲率向量或平均曲率的充要条件。首先研究了δ算子的定义、运算规律等,接着计算了平均曲率向量及平均曲率在无穷小变分下的变差。最后得出,一般黎曼空间、常曲率空间和欧氏空间的子空间在无穷小等距变分下保持平均曲率向量或平均曲率的充要条件。
Studies the necessary and sufficient conditions of an n--dimensional curvedsurface in an m--dimensional Riemannian space, which will keep up the meancurvatures vector or mean curvatures unchanged under the infinitesmal isom-etric variation. Following the definition of a δ-operator and its operation rulestudied, the variations of mean curvafure vectros and mean curvatures are ca-lculated under the infinitesmal isometric variation. As a result, the necessaryand sufficient conditions for keeping up the mean curvature vectors or meancurvatures of a subspace unchanged in an ordinary Riemannian space, constantcurvature space or Euclidean space under the infinitesmal isometric variationare obtained.
关键词
无穷小
等距变换
平均曲率
infinitesmal isometric transformation
mean curvature vector
mean curvature
