TY  - JOUR
A1  - Roos, Saskia
A1  - Otoba, Nobuhiko
T1  - Scalar curvature and the multiconformal class of a direct product Riemannian manifold
JF  - Geometriae dedicata
N2  - For a closed, connected direct product Riemannian manifold (M, g) = (M-1, g(1)) x ... x (M-l, g(l)), we define its multiconformal class [[g]] as the totality {integral(2)(1)g(1) circle plus center dot center dot center dot integral(2)(l)g(l)} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a positive function fi 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 [(g) over tilde] of every (g) over tilde is an element of[[ g]]. In this article, we prove that [[g]] contains a metric of positive scalar curvature if and only if the conformal class of some factor (Mi, gi) does, under the technical assumption dim M-i = 2. We also show that, even in the case where every factor (M-i, g(i)) has positive scalar curvature, [[g]] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided l = 2 and dim M = 3.
KW  - Positive scalar curvature
KW  - Constant scalar curvature
KW  - The Yamabe
KW  - problem
KW  - Warped product
KW  - Umbilic product
KW  - Twisted product
Y1  - 2021
U6  - https://doi.org/10.1007/s10711-021-00636-9
SN  - 0046-5755
SN  - 1572-9168
VL  - 214
IS  - 1
SP  - 801
EP  - 829
PB  - Springer
CY  - Dordrecht
ER  -