Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vanishing theorems on complete Riemannian manifold with a parallel
  $1$-form   In this article, we first consider the $L^{2}$ \textit{Morse-Novikov
cohomology} on a complete Riemannian manifold $M$ equipped with a parallel
$1$-form which includes Vaisman manifold. Based on a vanishing theorem of
$L^{2}$ \textit{Morse-Novikov cohomology}, we prove that the $L^{2}$-harmonic
forms on $M$ are identically zero.

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The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph