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

Abstract For a closed, connected direct product Riemannian manifold $$(M, g)=(M_1, g_1) \times \cdots (M_l, g_l)$$ ( M , g ) = 1 × ⋯ l , we define its multiconformal class $$ [\![ g ]\!]$$ [ ] as the totality $$\{f_1^2g_1\oplus \oplus f_l^2g_l\}$$ { f 2 ⊕ } of all metrics obtained from multiplying metric $$g_i$$ i each factor $$M_i$$ by positive function $$f_i$$ on total space M . A contains not only warped type deformations but also whole conformal $$[\tilde{g}]$$ ~ every $$\tilde{g}\in ∈ In this article, prove that scalar curvature if and some $$(M_i, g_i)$$ does, under technical assumption $$\dim M_i\ge 2$$ dim ≥ We show that, even in case where has curvature, constantly equal to $$-1$$ - with arbitrarily large volume, provided $$l\ge M\ge 3$$ 3