JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (12): 115-119.doi: 10.6040/j.issn.1671-9352.0.2018.604

Previous Articles    

On the fundamental group of complete noncompact Riemannian manifolds

CHEN Ai-yun1, XUE Qiong1*, XIAO Xiao-feng2   

  1. 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

PDF (PC)

386

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.

CLC Number: 

  • O186
[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.
[1] CHEN Ai-yun, XUE Qiong, CHEN Huan-huan, XIAO Xiao-feng. Complete noncompact Riemannian manifold with asymptotically nonnegative Ricci curvature [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 1-6.
[2] HE Guo-qing, ZHANG Liang, LIU Hai-rong. Chen-Ricci inequalities for submanifolds of generalized Sasakian space forms with a semi-symmetric metric connection [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 56-63.
[3] WEN Hai-yan, LIU Jian-cheng. Space-like submanifolds with constant scalar curvature in the pseudo-Riemannian space forms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 89-96.
[4] DENG Yi-hua. Estimates for the Hessian of f-harmonic functions and their applications to splitting theorem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 83-86.
[5] LIU Xu-dong, PAN Xu-lin, ZHANG Liang. Inequalities for Casorati curvatures of submanifolds in a Riemannian manifold of quasi-constant curvature with a semi-symmetric metric connection [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(2): 55-59.
[6] HE Chao, LI Ying, SONG Wei-dong. On the 2-harmonic timelike submanifolds in locally symmetric pseudo-riemannian manifolds [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(10): 54-58.
[7] SU Man, ZHANG Liang. Two results on Chens inequalities for spacelike submanifolds of a pseudo-Riemannian space form [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(10): 59-64.
[8] LI Ying, SONG Wei-dong. On pseudo-umbilical timelike submanifold in a de Sitter space [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(10): 64-67.
[9] PAN Xu-lin, ZHANG Pan, ZHANG Liang. Inequalities for Casorati curvatures of submanifolds in a riemannian manifold of quasi-constant curvature [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(09): 84-87.
[10] LIU Min, SONG Wei-dong. Totally real minimal submanifolds in quasi-complex projective space [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(12): 71-75.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd