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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 itsmulticonformal class $ [\![ g ]\!]$ as the totality $\{f_1^2g_1\oplus\cdots\oplus f_l^2g_l\}$ of all Riemannian metrics obtained from multiplyingthe metric $g_i$ of each factor $M_i$ by a function $f_i^2>0$ on the totalspace $M$. A multiconformal class $ [\![ g ]\!]$ contains not only all warpedproduct type deformations of $g$ but also the whole conformal class$[\tilde{g}]$ of every $\tilde{g}\in [\![ g ]\!]$. In this article, we provethat $ [\![ g ]\!]$ carries a metric of positive scalar curvature if and onlyif the conformal class of some factor $(M_i, g_i)$ does, under the technicalassumption $\dim M_i\ge 2$. We also show that, even in the case where everyfactor $(M_i, g_i)$ has positive scalar curvature, $ [\![ g ]\!]$ carries ametric of scalar curvature constantly equal to $-1$ and with arbitrarily largevolume, provided $l\ge 2$ and $\dim M\ge 3$. In this case, such negative scalarcurvature metrics within $ [\![ g ]\!]$ for $l=2$ cannot be of any warpedproduct type.

Scalar curvature and the multiconformal class of a direct product Riemannian manifold