关于拟常曲率空间中的伪脐子流形
发布时间：2019-05-27 20:25
【摘要】：本文主要研究了拟常曲率空间中具有平行平均曲率向量的紧致伪脐子流形.运用J.Simons研究常曲率空间中极小子流形的方法,估算了子流形的第二基本形式模长的平方的Laplacian,得到了这类子流形关于第二基本形式模长的平方及截面曲率和Ricci曲率的若干Pinching定理.本文一共分为4章.第1章介绍了国内外关于子流形的研究现状,特别是常曲率和拟常曲率空间中的子流形.第2章介绍了黎曼流形及其子流形的相关基础知识.第3章主要介绍了拟常曲率黎曼流形,并给出了拟常曲率空间中黎曼子流形的基本方程.第4章是本论文的中心内容,研究了拟常曲率空间中具有平行平均曲率向量的紧致伪脐子流形,给出了相关的引理及计算,得到了本论文的主要结果及其证明,推广了常曲率空间上的相关结论.
[Abstract]:In this paper, we mainly study compact pseudo-umbilical submanifolds with parallel mean curvature vectors in quasi-constant curvature spaces. By using the method of J.Simons to study the minimal submanifolds in constant curvature space, the Laplacian, of the square of the second basic form module length of the subfluidic is estimated. In this paper, some Pinching theorem of this kind of submanifolds with regard to the square and section curvature and Ricci curvature of the module length of the second basic form are obtained. This paper is divided into four chapters. In chapter 1, the research status of submanifolds at home and abroad is introduced, especially the submanifolds in constant curvature and quasi-constant curvature spaces. In chapter 2, the basic knowledge of Riemannian manifolds and their submanifolds is introduced. In chapter 3, we mainly introduce quasi-constant curvature Riemannian manifolds, and give the basic equations of Riemann submanifolds in quasi-constant curvature space. Chapter 4 is the central content of this paper. The compact pseudo-umbilical submanifolds with parallel mean curvature vector in quasi-constant curvature space are studied, and the related Lemma and calculation are given, and the main results and proof of this paper are obtained. The related results on constant curvature space are extended.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O186.1
本文编号：2486453
[Abstract]:In this paper, we mainly study compact pseudo-umbilical submanifolds with parallel mean curvature vectors in quasi-constant curvature spaces. By using the method of J.Simons to study the minimal submanifolds in constant curvature space, the Laplacian, of the square of the second basic form module length of the subfluidic is estimated. In this paper, some Pinching theorem of this kind of submanifolds with regard to the square and section curvature and Ricci curvature of the module length of the second basic form are obtained. This paper is divided into four chapters. In chapter 1, the research status of submanifolds at home and abroad is introduced, especially the submanifolds in constant curvature and quasi-constant curvature spaces. In chapter 2, the basic knowledge of Riemannian manifolds and their submanifolds is introduced. In chapter 3, we mainly introduce quasi-constant curvature Riemannian manifolds, and give the basic equations of Riemann submanifolds in quasi-constant curvature space. Chapter 4 is the central content of this paper. The compact pseudo-umbilical submanifolds with parallel mean curvature vector in quasi-constant curvature space are studied, and the related Lemma and calculation are given, and the main results and proof of this paper are obtained. The related results on constant curvature space are extended.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O186.1
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