×

A compactness theorem in Riemannian manifolds. (English) Zbl 1394.53044

The author generalizes the classical Myers theorem to the case of a complete and connected Riemannian \(n\)-manifold \(M\) with \(m\)-Bakry-Émery Ricci tensor given by \[ \mathrm{Ric}_{V,m}=\mathrm{Ric} + \frac{1}{2}\mathcal{L}_Vg - \frac{1}{m-n}V^*\otimes V^*, \] where \(\mathcal{L}_V\) and \(V^*\) denote, respectively, the Lie derivative and the metric dual of a smooth vector field \(V\) on \(M\). It is assumed that \(m\geq n\) and that, if \(m=n\), then the vector field \(V\) is zero and the tensor above reduces to the usual Ricci tensor. Assuming certain positive lower bound on \(\mathrm{Ric}_{V,m}\), the author shows that \(M\) must be compact with diameter bounded above by a constant depending on the lower bound. The proof is based on the Riccati comparison theorem.

MSC:

53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
Full Text: DOI

References:

[1] Barros, A., Ribeiro, E.: Integral formulae on quasi-Einstein manifolds and applications. Glasg. Math. J. 54, 213-223 (2012) · Zbl 1235.53047 · doi:10.1017/S0017089511000565
[2] Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, Kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15-53 (1982) · Zbl 0493.53035 · doi:10.4310/jdg/1214436699
[3] Kuwada, K.: A probabilistic approach to the maximal diameter theorem. Math. Nachr. 286, 374-378 (2013) · Zbl 1266.53042 · doi:10.1002/mana.201100330
[4] Limoncu, M.: Modifications of the Ricci tensor and applications. Arch. Math. 95, 191-199 (2010) · doi:10.1007/s00013-010-0150-0
[5] Limoncu, M.: The Bakry-Emery Ricci tensor and its applications to some compactness theorems. Math. Z. 271, 715-722 (2012) · Zbl 1264.53042 · doi:10.1007/s00209-011-0886-7
[6] Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401-404 (1941) · Zbl 0025.22704 · doi:10.1215/S0012-7094-41-00832-3
[7] Qian, Z.: Estimates for weighted volumes and applications. Q. J. Math. Oxf. 48, 235-242 (1997) · Zbl 0902.53032 · doi:10.1093/qmath/48.2.235
[8] Ruan, Q.: Two rigidity theorems on manifolds with Bakry-Emery Ricci curvature. Proc. Jpn. Acad. Ser. A 85, 71-74 (2009) · Zbl 1170.53024 · doi:10.3792/pjaa.85.71
[9] Wang, L.F.: A Myers theorem via m-Bakry-Émery curvature. Kodai Math. J. 37, 187-195 (2014) · Zbl 1314.53072 · doi:10.2996/kmj/1396008254
[10] Zhu, S.: The comparison geometry of Ricci curvature. Comp. Geom MSRI Publ. 30, 221-262 (1997) · Zbl 0896.53036
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.