Some Myers-Type Theorems and Comparison Theorems for Manifolds with Modified Ricci Curvature

Year 2021, Volume: 14 Issue: 1, 174 - 182, 15.04.2021

Issa A. Kaboye Mahamane Mahi Harouna , Mahaman Bazanfaré

https://doi.org/10.36890/iejg.819887

Abstract

 In this paper we establish some new compactness criteria for complete Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below. These results improve or generalize previous ones obtained by H. Tadano [6], J. Wan [7], I.A. Kaboye and M. Bazanfar\'e [3]. We also prove a volume comparison theorem for such manifolds.

Keywords

Bakry-Emery Ricci curvature, Myers-type theorem, Modified mean curvature, volume comparison theorem

References

• [1] Bazanfaré, M.: A volume comparison theorem and number of ends for manifolds with asymptotically nonnegative Ricci curvature. Revista Math. Compl. 13-2, 399–409 (2000).
• [2] Galloway, G. J.: A generalization of Myers’ theorem and an application to relativistic cosmology. J. Differential Geom. 14-1, 105–116, (1979).
• [3] Kaboye, I.A. and Bazanfaré, M.: Manifolds with Bakry-Émery Ricci Curvature Bounded Below. Advances in Pure Mathematics. 6, 754-764 (2016).
• [4] Limoncu, M.: The Bakry-Émery Ricci tensor and its applications to some compactness theorems. Math. Z. 271, 715–722 (2012).
• [5] Soylu, Y.: A Myers-type compactness theorem by the use of Bakry-Émery Ricci tensor. Differ. Geom. Appl. 54, 245–250 (2017).
• [6] Tadano, H.: Some Ambrose and Galloway-type theorems via Bakry-Émery and modfied Ricci curvatures. Pacific J. Math. 294-1, 213-231 (2018).
• [7] Wan, J.: An extension of Bonnet-Myers theorem. Math. Z. 291, 195–197 (2019).
• [8] Wei, G. and Wylie, W.: Comparison geometry for the Bakry-Émery Ricci tensor. J. Differ. Geom. 83, 377–405 (2009).
• [9] Wu, J.Y.: Myers’ type theorem with the Bakry-Émery Ricci tensor. Ann. Global Anal. Geom. 54-4, 541–549 (2018).