In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison theorem for manifolds with nonnegative Bakry-émery Ricci curvature which allows us to prove a topolological rigidity theorem for such manifolds.
References
[1]
Wei, G. and Wylie, W. (2009) Comparison Geometry for the Bakry-Emery Ricci Tensor. Journal of Differential Geometry, 83, 377-405
[2]
Limoncu, M. (2012) The Bakry-émery Ricci Tensor and Its Applications to Some Compactness Theorems. Mathematische Zeitschrift, 271, 715-722. http://dx.doi.org/10.1007/s00209-011-0886-7
[3]
Tadano, H. (2016) Remark on a Diameter Bound for Complete Riemannian Manifolds with Positive Bakry-émery Ricci Curvature. Differential Geometry and Its Applications, 44, 136-143. http://dx.doi.org/10.1016/j.difgeo.2015.11.001
[4]
Mahaman, B. (2001) Un théorème de Myers pour les variétés à courbure de Ricci minorée par une constante negative. Afrika Matematika, 12, 39-51.
[5]
Itokawa, Y. (1990) Distance Sphere and Myers-Type Theorems for Manifolds with Lower Bounds o, the Ricci Curvature. Illinois Journal of Matematics, 34, 693-705.
[6]
Abresch, U. and Gromoll, D. (1990) On Complete Manifolds with Nonnegative Ricci Curvature. Journal of the American Mathematical Society, 3, 355-374. http://dx.doi.org/10.1090/S0894-0347-1990-1030656-6
[7]
Mahaman, B. (2000) A Volume Comparison Theorem and Number of Ends for Manifolds with Asymptotically Nonnegative Ricci Curvature. Revista Matemática Complutense, 13, 399-409.
[8]
Mahaman, B. (2005) Open Manifolds with Asymptotically Nonnegative Curvature. Illinois Journal of Mathematics, 49, 705-717.
[9]
Xia, C. (1999) Open Manifolds with Nonnegative Ricci Curvature and Large Volume Growth. Commentarii Mathematici Helvetici, 74, 456-466. http://dx.doi.org/10.1007/s000140050099
[10]
Shen, Z. (1996) Complete Mnifolds with Nonnegative Ricci Curvature and Large Volume Growth. Inventiones Mathematicae, 125, 393-404. http://dx.doi.org/10.1007/s002220050080
