JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2018, Vol. 53 ›› Issue (4): 1-6.doi: 10.6040/j.issn.1671-9352.0.2017.480

    Next Articles

Complete noncompact Riemannian manifold with asymptotically nonnegative Ricci curvature

CHEN Ai-yun1, XUE Qiong1*, CHEN Huan-huan1, 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
  • Received:2017-09-22 Online:2018-04-20 Published:2018-04-13

PDF (PC)

588

Abstract: We study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. By using extensions of Excess function and Busemann function, it is proved that they have finite topological type under sectional curvature bounded from nonnegative below kp(r)≥- C/((1+r)α) and large volume growth, which extend a series known results.

Key words: asymptotically nonnegative Ricci curvature, Busemann function, finite topological type, Excess function, large volume growth

CLC Number: 

  • O186
[1] ABRESCH U. Lower curvature bounds, Toponogovs theorem, and bounded topology I[J]. Annales Scientifiques De L École Normale Supérieure, 1985, 18(4):651-670.
[2] ABRESCH U. Lower curvature bounds, Toponogovs theorem,and bounded topology Ⅱ[J]. Annales Scientifiques De L École Normale Supérieure, 1987, 20(3):475-502.
[3] ZHU Shunhui. A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications[J]. American Journal of Mathematics, 1994, 116(3):669-682.
[4] ABRESCH U, GROMOLL D. On complete manifolds with nonnegative Ricci curvature[J]. Journal of the American Mathematical Society, 1990, 3(2):355-374.
[5] MAHAMAN B. Open manifolds with asymptotically nonnegative curvature[J]. Illinois Journal of Mathematics, 2005, 49(3):705-717.
[6] HU Zisheng, XU Senlin. Complete manifolds with asymptotically nonnegative Ricci curvature and weak bounded geometry[J]. Archiv Der Mathematik, 2007, 88(5):455-467.
[7] SANTOS L, NEWTON L. Manifolds with asymptotically nonnegative minimal radial curvature[J]. Advances in Geometry, 2007, 7(3):331-355.
[8] YEGANEFAR N. Topological finiteness for asymptotically nonnegatively curved manifolds[J]. Mathematics, 2009, 108(1):109-134.
[9] ZHANG Yuntao. Open manifolds with asymptotically nonnegative Ricci curvature and large volume growth[J]. Proceedings of the American Mathematical Society, 2015, 143(18):4913-4923.
[10] MAHAMAN B. Topology of manifolds with asymptotically nonnegative Ricci curvature[J]. African Diaspora Journal of Mathematics, 2015, 18(2):11-17.
[11] SHEN Zhongmin. Complete manifolds with nonnegative Ricci curvature and large volume growth[J]. Inventiones Mathematicae, 1996, 125(3):393-404.
[12] MACHIGASHIRA Y. Complete open manifolds of non-negative radial curvature[J]. Pacific Journal of Mathematics, 1994, 165(1):153-160.
[13] CHEEGER J. Critical points of distance functions and applications to geometry[M]. Berlin:Springer, 1991:1-38.
[14] MAHAMAN B. A volume comparison theorem and number of ends for manifolds with asymptotically nonnegative Ricci curvature[J]. Revista Matemática Complutense, 2000, 13(2):399-409.
[15] SHA Jiping, SHEN Zhongmin.Complete manifolds with nonnegative Ricci curvature and quadratically nonnegatively curved infinity[J]. American Journal of Mathematics, 1997, 119(6):1399-1404.
[1] 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.
[2] 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.
[3] 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.
[4] 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.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[9] 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