The Manifolds with Ricci Curvature Decay to Zero

Author(s)
Huashui Zhan

ABSTRACT

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.

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.

Cite this paper

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.

H. Zhan, "The Manifolds with Ricci Curvature Decay to Zero,"

References

[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

[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