常曲率空间中的两类紧致子流形  

Two Types of Compact Submanifolds in the Constant Curvature Space

作　　者：官展聿 梁林 周鉴 梁馨月[3] GUAN Zhanyu;LIANG Lin;ZHOU Jian;LIANG Xinyue(School of Mathematics,Yunnan Normal University,Kunming,Yunnan Province,650500;School of Mathematics&Statistics,Chuxiong Normal University,Chuxiong,Yunnan Province,675000;School of Economics&Management,Yunnan Normal University,Kunming,Yunnan Province,650500)

机构地区：[1]云南师范大学数学学院,云南昆明650500 [2]楚雄师范学院数学与统计学院,云南楚雄675000 [3]云南师范大学经管学院,云南昆明650500

出　　处：《楚雄师范学院学报》2018年第3期13-17,共5页Journal of Chuxiong Normal University

基　　金：楚雄师范学院国家自然科学基金校级培育孵化项目

摘　　要：本文主要研究常曲率空间中的两类紧致等距浸入子流形,一类是紧致极小子流形,另一类是紧致非极小且具有平行平均曲率向量的子流形。对于前者,通过计算第二基本形式模长的平方的Laplace,使用极大值原理及常曲率的限制条件可得到它是全测地的;对于后者,在其沿平均曲率向量方向全脐的条件下,构造适当的张量并计算所构造张量模长的平方的Laplace,使用极大值原理及常曲率的限制条件可证得该子流形是全脐的。In this paper,we mainly study two types of compact submanifolds isometrically immersed in the constant curvature space,one is compact minimal submanifold,another is compact non-minimal submanifold with parallel mean curvature vector fields.For the former,we calculate the Laplacian of the square length of the second fundamental form,by using maximum principle and the restricted condition of the constant curvature,we can get that the submanifolds are totally geodesic;as for the latter,with the condition that it is totally umbilic along the mean curvature vector,we can construct the appropriate tensor and calculate the Laplacian of the square length of this tensor,by using maximum principle and the restricted condition of the constant curvature,we can get that the submanifolds is totally umbilic.

关 键 词：等距浸入 极小子流形 平均曲率向量 全测地的 全脐的 

分 类 号：O186.1[理学—数学]