Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: General Sobolev Inequality on Riemannian Manifold   Let M be a complete n-dimensional Riemannian manifold, if the sobolev
inqualities hold on M, then the geodesic ball has maximal volume growth; if the
Ricci curvature of M is nonnegative, and one of the general Sobolev
inequalities holds on M, then M is diffeomorphic to $R^{n}$.

29
The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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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: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn