The Manifolds with Ricci Curvature Decay to Zero
Huashui Zhan
DOI: 10.4236/apm.2012.21008   PDF        3,657 Downloads   7,439 Views  


The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riemannian manifold M with Ricci curvature decay to zero, if there is a line in M, then the isometrically splitting M = R × N is true.

Share and Cite:

H. Zhan, "The Manifolds with Ricci Curvature Decay to Zero," Advances in Pure Mathematics, Vol. 2 No. 1, 2012, pp. 36-38. doi: 10.4236/apm.2012.21008.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. Cheeger and D. Gromoll, “The Splitting Theorem for Manifolds of Nonnegative Ricci Curvature,” Journal of Differential Geometry, Vol. 6, 1971, pp. 119-128.
[2] H. Wu and W. H. Chen, “The Selections of Riemannian Geometry (in Chinese),” Publishing Company of Peking University, Beijing, 1993.
[3] M. L. Cai, “Ends of Riemannian Manifolds with NonNegative Ricci Curvature Outside a Compact Set,” Bulletin of the AMS (New Series), Vol. 24, 1991, pp. 371-377. doi:10.1090/S0273-0979-1991-16038-6
[4] H. Karcher, “Riemannian Comparison Constructions,” In: Shiohama (Ed.), Geometry of Manifolds, Academic Press, San Diego, 1989, pp. 171-222.
[5] Z. Shen, “On Complete Manifolds of Nonnegative k-th Ricci Curvature,” Transactions of the AMS—American Mathematical Society, Vol. 338, No. 1, 1993, pp. 289-309. doi:10.2307/2154457

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.