Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A General Inequality for Warped Product $CR$-Submanifolds of K\"ahler
  Manifolds   In this paper, warped product contact $CR$-submanifolds in Sasakian, Kenmotsu
and cosymplectic manifolds are shown to possess a geometric property; namely
$\mathcal{D}_T$-minimal. Taking benefit from this property, an optimal general
inequality for warped product contact $CR$-submanifolds is established in both
Sasakian and Kenmotsu manifolds by means of the Gauss equation, we leave
cosyplectic because it is an easy structure. Moreover, a rich geometry appears
when the necessity and sufficiency are proved and discussed in the equality
case. Applying this general inequality, the inequalities obtained by Munteanu
are derived as particular cases, whereas the inequality obtained in [1] is
corrected. Up to now, the method used by Chen and Munteanu can not extended for
general ambient manifolds, this is because many limitations in using Codazzi
equation. Hence, Our method depends on the Gauss equation. The inequality is
constructed to involve an intrinsic invariant (scalar curvature) controlled by
an extrinsic one (the second fundamental form), which provides an answer for
Problem [1]. As further research directions, we have addressed a couple of open
problems arose naturally during this work and depending on its results.

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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin