Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Remark on a diameter bound for complete manifolds with positive
  Bakry-\'{E}mery Ricci curvature   In this paper, we shall give a new upper diameter estimate for complete
Riemannian manifolds in the case that the Bakry-\'Emery Ricci curvature has a
positive lower bound and the norm of the potential function has an upper bound.
Our diameter estimate improves previous ones obtained by Wei and Wylie (J.
Differential Geom. 83, 377--405, 2009) and Limoncu (Math. Z. 271, 715--722,
2012). As an application, we shall give an upper diameter bound for compact
Ricci solitons in terms of the maximum value of the scalar curvature. By using
such a diameter bound, we shall provide some new sufficient conditions for
four-dimensional compact Ricci solitons to satisfy the Hitchin-Thorpe
inequality.

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The categories of this paper are: stat
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio