Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On commuting billiards in higher-dimensional spaces of constant
  curvature   We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with
$C^2$-smooth strictly convex boundaries. We prove that if the corresponding
actions by reflections on the space of oriented lines commute, then the
billiards are confocal ellipsoids. This together with the previous analogous
result of the author in two dimensions solves completely the Commuting Billiard
Conjecture due to Sergei Tabachnikov. The main result is deduced from the
classical theorem due to Marcel Berger saying that in higher dimensions only
quadrics may have caustics. We also prove versions of Berger's theorem and the
main result for billiards in spaces of constant curvature: space forms.

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The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
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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: quant-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
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The categories of this paper are: atom-ph