Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scalar curvature and the multiconformal class of a direct product
  Riemannian manifold   For a closed, connected direct product Riemannian manifold $(M,
g)=(M_1\times\cdots\times M_l, g_1\oplus\cdots\oplus g_l)$, we define its
multiconformal class $ [\![ g ]\!]$ as the totality $\{f_1^2g_1\oplus
\cdots\oplus f_l^2g_l\}$ of all Riemannian metrics obtained from multiplying
the metric $g_i$ of each factor $M_i$ by a function $f_i^2>0$ on the total
space $M$. A multiconformal class $ [\![ g ]\!]$ contains not only all warped
product type deformations of $g$ but also the whole conformal class
$[\tilde{g}]$ of every $\tilde{g}\in [\![ g ]\!]$. In this article, we prove
that $ [\![ g ]\!]$ carries a metric of positive scalar curvature if and only
if the conformal class of some factor $(M_i, g_i)$ does, under the technical
assumption $\dim M_i\ge 2$. We also show that, even in the case where every
factor $(M_i, g_i)$ has positive scalar curvature, $ [\![ g ]\!]$ carries a
metric of scalar curvature constantly equal to $-1$ and with arbitrarily large
volume, provided $l\ge 2$ and $\dim M\ge 3$. In this case, such negative scalar
curvature metrics within $ [\![ g ]\!]$ for $l=2$ cannot be of any warped
product type.

30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat