完备非紧黎曼流形的基本群

doi:10.6040/j.issn.1671-9352.0.2018.604

《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (12): 115-119.doi: 10.6040/j.issn.1671-9352.0.2018.604

• • 上一篇

完备非紧黎曼流形的基本群

陈爱云¹,薛琼^1*,肖小峰²

1.武汉理工大学理学院, 湖北武汉 430070;2.武汉纺织大学机械工程与自动化学院, 湖北武汉 430073

发布日期:2019-12-11
作者简介:陈爱云(1993— ),女,硕士研究生,研究方向为微分几何及应用. E-mail:1113603226@qq.com*通信作者简介:薛琼(1980— ),女,博士,副教授,研究方向为微分几何及应用. E-mail:rabbit_801005@163.com
基金资助:
国家自然科学基金资助项目(61573012);中央高校基本科研业务费专项资金资助项目(2017IA006)

On the fundamental group of complete noncompact Riemannian manifolds

CHEN Ai-yun¹, XUE Qiong^1*, XIAO Xiao-feng²

1. School of Science, Wuhan University of Technology, Wuhan 430070, Hubei, China; 2. School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430073, Hubei, China

Published:2019-12-11

摘要/Abstract

摘要： 研究了一类完备非紧的n维黎曼流形,Ricci曲率满足Ric_M≥-(n-1)k(k>0),利用点到极小测地圈中点的距离的一致估计,证明了此流形在满足小的直径线性增长条件下,其基本群是有限生成的。

关键词: 黎曼流形, Ricci曲率, 小的直径线性增长, 基本群

Abstract: We study the topology of complete noncompact Riemannian manifolds with Ricci curvature satisfies RicM≥-(n-1)k(k>0). By using the uniform estimates for the distance from a point to halfway point of minimal geodesics, we prove that a manifold with linear diameter growth has a finitely generated fundamental group.

中图分类号:

O186

陈爱云,薛琼,肖小峰. 完备非紧黎曼流形的基本群[J]. 《山东大学学报(理学版)》, 2019, 54(12): 115-119.

CHEN Ai-yun, XUE Qiong, XIAO Xiao-feng. On the fundamental group of complete noncompact Riemannian manifolds[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(12): 115-119.

参考文献

[1] MILNOR J. A note on curvature and fundamental group[J]. Journal of Differential Geometry, 1968, 2(1):1-7.
[2] LI P. Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature[J]. Annals of Mathematics, 1986, 124(1):1-21.
[3] ANDERSON M. On the topology of complete manifolds of nonnegative Ricci curvature[J]. Topology, 1990, 29(1):41-55.
[4] ANDERSON M, RODRIGUEZ L. Minimal surfaces and 3-manifolds of nonnegative Ricci curvature[J]. Mathematische Annalen, 1989, 284(3):461-476.
[5] CHEEGER J, GROMOLL D. On the structure of complete manifolds of nonnegative Ricci curvature[J]. Annals of Mathematics, 1972, 96(3): 413-443.
[6] SORMANI C. Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups[J]. Journal of Differential Geometry, 2000, 54(3):547-559.
[7] 徐森林, 邓勤涛. 具有非负Ricci曲率的开流形的基本群[J]. 数学学报, 2006,49(2):353-356. XU Senlin, DENG Qintao. The fundamental group of open manifolds with nonnegative Ricci curvature[J]. Acta Mathenatica, 2006, 49(2):353-356.
[8] ABRESCH U, GROMOLL D. On complete manifolds with nonnegative Ricci curvature[J]. Journal of the American Mathematical Society, 1990, 3(2):355-374.
[9] 张运涛, 徐栩. 曲率二次衰减的完备流形的基本群[J]. 数学学报, 2007, 50(5):1093-1098. ZHANG Yuntao, XU Xu. On the fundamental groups of complete manifolds with lower quadratic curvature decay[J]. Acta Mathenatica, 2007, 50(5):1093-1098.
[10] 金亚东, 朱鹏. 曲率二次衰减的开流形的基本群[J]. 江苏理工学院学报, 2014,20(6):9-12. JIN Yadong, ZHU Peng. The fundamental group of open manifolds with 2-curvature decay[J]. Journal of Jiangsu Institute of Technology, 2014, 20(6):9-12.
[11] SHEN Zhongmin. Complete manifolds with nonnegative Ricci curvature and large volume growth[J]. Inventiones Mathematicae, 1996, 125(3): 393-404.
[12] XU Senlin, YANG Fangyun, WANG Zuoqin. Open manifolds with nonnegative Ricci curvature and large volume growth[J]. Northeastern Mathematical Journal, 2003, 19(2):155-160.
[13] SORMANI C, WEI G. Hausdorff convergence and universal covers[J]. Transactions of the American Mathematical Society, 2001, 353(9):3585-3602.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed

完备非紧黎曼流形的基本群

On the fundamental group of complete noncompact Riemannian manifolds

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

多维度评价

本文评价

推荐阅读 0

[1]	陈爱云,薛琼,陈欢欢,肖小峰. 具有渐近非负Ricci曲率完备非紧的黎曼流形[J]. 山东大学学报（理学版）, 2018, 53(4): 1-6.
[2]	何超,李影,宋卫东. 局部对称伪黎曼流形中的2-调和类时子流形[J]. 山东大学学报（理学版）, 2016, 51(10): 54-58.