Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting open negatively curved manifolds up to tangential homotopy
  equivalence   Under mild assumptions on a group G, we prove that the class of complete
Riemannian n-manifolds of uniformly bounded negative sectional curvatures and
with the fundamental groups isomorphic to G breaks into finitely many
tangential homotopy types. It follows that many aspherical manifolds do not
admit complete negatively curved metrics with prescribed curvature bounds.

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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
9
The categories of this paper are: nlin