Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $G_2$ geometry of $3$-Sasaki structures   We initiate a systematic study of the deformation theory of the second
Einstein metric $g_{1/\sqrt{5}}$ respectively the proper nearly $G_2$ structure
$\varphi_{1/\sqrt{5}}$ of a $3$-Sasaki manifold $(M^7,g)$. We show that
infinitesimal Einstein deformations for $g_{1/\sqrt{5}}$ coincide with
infinitesimal $G_2$ deformations for $\varphi_{1/\sqrt{5}}$. The latter are
showed to be parametrised by eigenfunctions of the basic Laplacian of $g$, with
eigenvalue twice the Einstein constant of the base $4$-dimensional orbifold,
via an explicit differential operator. In terms of this parametrisation we
determine those infinitesimal $G_2$ deformations which are unobstructed to
second order.

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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn