Abstract
By using some line integrals in terms of the m-Bakry–Émery and m-modified Bakry–Émery Ricci curvatures, we give various compactness criteria for complete Riemannian manifolds when m is a positive constant, a negative constant, or infinity. Our results generalize previous Myers-type compactness criteria obtained by M. Fernández-López and E. García-Río, M. Limoncu, Z. Qian, G. Wei and W. Wylie, J.-Y. Wu, and W. Wylie, as well as a previous Ambrose-type compactness criterion obtained by K. Kuwae and X.-D. Li. The key ingredients in proving our results are Riccati inequalities obtained from Bochner–Weitzenböck formulas via m-Bakry–Émery and m-modified Bakry–Émery Ricci curvatures.
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