Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Einstein extensions of Riemannian manifolds   Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric
endomorphisms on $N$, we define the extension $M$ of $N$ by $D$ to be the
Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction
similar to extending a Lie group by a derivation of its Lie algebra. We find
the conditions on $N$ and $D$ which imply that the extension $M$ is Einstein.
In particular, we show that in this case, $D$ has constant eigenvalues;
moreover, they are all integer (up to scaling) if $\det D \ne 0$. They must
satisfy certain arithmetic relations which imply that there are only finitely
many eigenvalue types of $D$ in every dimension (a similar result is known for
Einstein solvmanifolds). We give the characterisation of Einstein extensions
for particular eigenvalue types of $D$, including the complete classification
for the case when $D$ has two eigenvalues, one of which is multiplicity free.
In the most interesting case, the extension is obtained, by an explicit
procedure, from an almost K\"{a}hler Ricci flat manifold (in particular, from a
Calabi-Yau manifold). We also show that all Einstein extensions of dimension
four are Einstein solvmanifolds. A similar result holds valid in the case when
$N$ is a Lie group with a left-invariant metric, under some additional
assumptions.

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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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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: cond-mat