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Myers' theorem for complete Riemannian hypersurfaces

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In this paper we study the Myers' theorem for orientable Riemannian hypersurfaces under some restrictions on the mean curvature, the second-order mean curvature, or the divergence of the shape operator and give some estimates for the diameter of such hypersurfaces.

Abstract

In this paper we study the Myers' theorem for orientable Riemannian hypersurfaces under some restrictions on the mean curvature, the second-order mean curvature, or the divergence of the shape operator and give some estimates for the diameter of such hypersurfaces.

참고문헌 (13)

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  3. F. Fang, "Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature", Ann. Inst. Fourier (Grenoble), 59(2), pp.563~573 (2009)
  4. Akito Futaki, "On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons", Annals of Global Analysis and Geometry, 44(2), pp.105~114 (2013), Springer Science and Business Media LLC[DOI]
  5. Xiang-Dong Li, "Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds", Journal de Mathématiques Pures et Appliquées, 84(10), pp.1295~1361 (2005), Elsevier BV[DOI]
  6. Murat Limoncu, "The Bakry–Emery Ricci tensor and its applications to some compactness theorems", Mathematische Zeitschrift, 271(3-4), pp.715~722 (2012), Springer Science and Business Media LLC[DOI]
  7. Ovidiu Munteanu, "Smooth metric measure spaces with non-negative curvature", Communications in Analysis and Geometry, 19(3), pp.451~486 (2011), International Press of Boston[DOI]
  8. S. B. Myers, "Riemannian manifolds with positive mean curvature", Duke Math. J, 8, pp.401~404 (1941)[DOI]
  9. Homare Tadano, "Remark on a diameter bound for complete Riemannian manifolds with positive Bakry–Émery Ricci curvature", Differential Geometry and its Applications, 44, pp.136~143 (2016), Elsevier BV[DOI]
  10. G. Wei, "Comparison geometry for the Bakry-Emery Ricci tensor", J. Dif-ferential Geom, 83(2), pp.377~405 (2009)[DOI]
  11. Jia-Yong Wu, "A note on the splitting theorem for the weighted measure", Annals of Global Analysis and Geometry, 43(3), pp.287~298 (2013), Springer Science and Business Media LLC[DOI]
  12. Jia-Yong Wu, "Myers’ type theorem with the Bakry–Émery Ricci tensor", Annals of Global Analysis and Geometry, 54(4), pp.541~549 (2018), Springer Science and Business Media LLC[DOI]
  13. Shijin Zhang, "A theorem of Ambrose for Bakry–Emery Ricci tensor", Annals of Global Analysis and Geometry, 45(3), pp.233~238 (2014), Springer Science and Business Media LLC[DOI]

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