Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature. (English) Zbl 1170.53024

Summary: We generalize the Cheng’s maximal diameter theorem and Bishop volume comparison theorem to the manifold with the Bakry-Emery Ricci curvature. As their applications, we obtain some rigidity theorems on the warped product.


53C24 Rigidity results
53C20 Global Riemannian geometry, including pinching
Full Text: DOI


[1] D. Bakry and M. Ledoux, Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator. Duke Math. J. 85 (1996), no. 1, 253-270. · Zbl 0870.60071
[2] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc, 145 (1969) 1-49. · Zbl 0191.52002
[3] D. Bakry and Z. Qian, Volume Comparison Theorems without Jacobi fields, in Current trends in potential theory , pp. 115-122, Theta, Bucharest. · Zbl 1212.58019
[4] S. Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289-297. · Zbl 0329.53035
[5] X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math Pures Appl. (9) 84 (2005), no. 10, 1295-1361. · Zbl 1082.58036
[6] J. Lott, Some geometric properties of the Bakry-Emery-Ricci tensor. Comment. Math. Helv., 78 (2003, no. 4, 865-883. · Zbl 1038.53041
[7] S. B. Myers, Connections between differential geometry and topology. I. Simply connected surfaces, Duke Math. J. 1 (1935), no. 3, 376-391. · Zbl 0012.27502
[8] P. Petersen, Riemannian geometry , Springer, New York, 1998. · Zbl 0914.53001
[9] Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 190, 235-242. · Zbl 0902.53032
[10] G. Wei and W. Wylie, Comparison Geometry for the Bakry-Emery Ricci Tensor,
[11] G. Wei and W. Wylie, Comparison Geometry for the Smooth Metric Measure Spaces, 4th International Congress of Chinese Mathematicians (Hangzhou, China, 2007 , Proceedings of the 4th International Congress of Chinese Mathematicians, Hangzhou, 2007, Vol. II 191-202.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.