Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Variations of the Godbillon--Vey invariant of transversely
  parallelizable foliations   We consider a $(2q+1)$-dimensional smooth manifold $M$ equipped with a
$(q+1)$-dimensional, a priori non-integrable, distribution ${\cal D}$ and a
$q$-vector field ${\bf T}=T_1\wedge\ldots\wedge T_q$, where $\{T_i\}$ are
linearly independent vector fields transverse to~${\cal D}$. Using a $q$-form
$\omega$ such that ${\cal D} = \ker\,\omega$ and $\omega({\bf T})=1$, we
construct a $(2q+1)$-form analogous to that defining the Godbillon--Vey class
of a $(q+1)$-dimensional foliation, and show how does this form depend on
$\omega$ and ${\bf T}$. For a compatible Riemannian metric $g$ on $M$, we
express this $(2q+1)$-form in terms of ${\bf T}$ and extrinsic geometry
of~${\cal D}$ and normal distribution ${\cal D}^\bot$. We find Euler-Lagrange
equations of associated functionals: for variable $(\omega,{\bf T})$ on
$(M,g)$, and for variable metric on $(M,{\cal D})$, when
distributions/foliations and forms are defined outside a "singularity set"
under additional assumption of convergence of certain integrals. We show that
for a harmonic distribution ${\cal D}^\bot$ such $(\omega,{\bf T})$ is
critical, characterize critical pairs when ${\cal D}$ is integrable and find
sufficient conditions for critical pairs when variations are among foliations,
calculate the index form and consider examples of critical foliations among
twisted products, Reeb foliations and transversely holomorphic flows.

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The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph