Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature

Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature

Qi-hua Ruan

Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 85, Number 6 (2009), 71-74.

Abstract

In this paper, 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.

Primary Subjects: 53C21

Secondary Subjects: 53C20

Keywords: Maximal diameter; Bakry-Emery Ricci curvature; volume comparison; warped product

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.pja/1244037800
Digital Object Identifier: doi:10.3792/pjaa.85.71
Zentralblatt MATH identifier: 05598750

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References

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.

Mathematical Reviews (MathSciNet): MR1412446

Digital Object Identifier: doi:10.1215/S0012-7094-96-08511-7

Project Euclid: euclid.dmj/1077243045

Zentralblatt MATH: 0870.60071

R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc, 145 (1969) 1-49.

Mathematical Reviews (MathSciNet): MR251664

Digital Object Identifier: doi:10.2307/1995057

Zentralblatt MATH: 0191.52002

D. Bakry and Z. Qian, Volume Comparison Theorems without Jacobi fields, in Current trends in potential theory, pp. 115--122, Theta, Bucharest.

Mathematical Reviews (MathSciNet): MR2243959

S. Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289--297.

Mathematical Reviews (MathSciNet): MR378001

Digital Object Identifier: doi:10.1007/BF01214381

Zentralblatt MATH: 0329.53035

X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math Pures Appl. (9) 84(2005), no. 10, 1295--1361.

Mathematical Reviews (MathSciNet): MR2170766

Digital Object Identifier: doi:10.1016/j.matpur.2005.04.002

Zentralblatt MATH: 1082.58036

J. Lott, Some geometric properties of the Bakry-Emery-Ricci tensor. Comment. Math. Helv., 78(2003, no. 4, 865--883.

Mathematical Reviews (MathSciNet): MR2016700

Digital Object Identifier: doi:10.1007/s00014-003-0775-8

S. B. Myers, Connections between differential geometry and topology. I. Simply connected surfaces, Duke Math. J. 1 (1935), no. 3, 376--391.

Mathematical Reviews (MathSciNet): MR1545884

Digital Object Identifier: doi:10.1215/S0012-7094-35-00126-0

Project Euclid: euclid.dmj/1077489099

Zentralblatt MATH: 0012.27502

P. Petersen, Riemannian geometry, Springer, New York, 1998.

Mathematical Reviews (MathSciNet): MR1480173

Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 190, 235--242.

Mathematical Reviews (MathSciNet): MR1458581

Digital Object Identifier: doi:10.1093/qmath/48.2.235

Zentralblatt MATH: 0902.53032

G. Wei and W. Wylie, Comparison Geometry for the Bakry-Emery Ricci Tensor, arxiv:math.DG/0706.1120.

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.

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