Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive intermediate Ricci curvature on products of homogeneous spaces   We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci
curvature, i.e. $\mathrm{Ric}_2>0$, on products of positively curved
homogeneous spaces. Using these examples, we demonstrate that the Hopf
conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and
Hsiang-Kleiner theorem for positively curved manifolds do not hold in the
$\mathrm{Ric}_2>0$ setting. These observations indicate that the class of
manifolds with $\mathrm{Ric}_2>0$ is vastly different from the class of
positively curved manifolds.

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The categories of this paper are: mtrl-th
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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: alg-geom
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: quant-ph