Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some inequalities on Finsler manifolds with weighted Ricci curvature
  bounded below   We establish some important inequalities under a lower weighted Ricci
curvature bound on Finsler manifolds. Firstly, we establish a relative volume
comparison of Bishop-Gromov type. As one of the applications, we obtain an
upper bound for volumes of the Finsler manifolds. Further, when the S-curvature
is bounded on the whole manifold, we obtain a theorem of Bonnet-Myers type on
Finsler manifolds. Finally, we obtain a sharp Poincar\'{e}-Lichnerowicz
inequality by using integrated Bochner inequality, from which we obtain a sharp
lower bound for the first eigenvalue on the Finsler manifolds.

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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th