Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the structure of complete 3-manifolds with nonnegative scalar
  curvature   In this paper we will show the following result: Let $\mathcal{N} $ be a
complete (noncompact) connected orientable Riemannian three-manifold with
nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $ K_{s}
\leq K $. Suposse that $\Sigma \subset \mathcal{N} $ is a complete orientable
connected area-minimizing cylinder so that $\pi_1 (\Sigma) \in \pi_1
(\mathcal{N})$. Then $\mathcal{N}$ is locally isometric either to $\mathbb{S}
^1 \times \mathbb{R} ^2 $ or $\mathbb{S}^1 \times \mathbb{S}^1 \times
\mathbb{R}$ (with the standard product metric).
  As a corollary, we will obtain: Let $\mathcal{N} $ be a complete (noncompact)
connected orientable Riemannian three-manifold with nonnegative scalar
curvature $S \geq 0$ and bounded sectional curvature $ K_{s} \leq K $. Assume
that $\pi_1 (\mathcal{N})$ contains a subgroup which is isomorphic to the
fundamental group of a compact surface of positive genus. Then, $\mathcal{N}$
is locally isometric to $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$
(with the standard product metric).

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The categories of this paper are: stat
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: q-alg