Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vacuum Einstein field equations in smooth metric measure spaces: the
  isotropic case   On a smooth metric measure spacetime $(M,g,e^{-f} dvol_g)$, we define a
weighted Einstein tensor. It is given in terms of the Bakry-\'Emery Ricci
tensor as a tensor which is symmetric, divergence-free, concomitant of the
metric and the density function. We consider the associated vacuum weighted
Einstein field equations and show that isotropic solutions have nilpotent Ricci
operator. Moreover, the underlying manifold is a Brinkmann wave if it is
$2$-step nilpotent and a Kundt spacetime if it is $3$-step nilpotent. More
specific results are obtained in dimension $3$, where all isotropic solutions
are given in local coordinates as plane waves or Kundt spacetimes.

19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph