Roos, Saskia and Otoba, Nobuhiko (2021) Scalar curvature and the multiconformal class of a direct product Riemannian manifold. GEOMETRIAE DEDICATA, 214 (1). pp. 801-829. ISSN 0046-5755, 1572-9168
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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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | CONFORMAL DEFORMATION; WARPED PRODUCTS; DIFFERENTIAL-EQUATIONS; HARMONIC MORPHISMS; TWISTED PRODUCT; METRICS; EXISTENCE; Positive scalar curvature; Constant scalar curvature; The Yamabe problem; Warped product; Umbilic product; Twisted product |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 18 Aug 2022 10:57 |
| Last Modified: | 18 Aug 2022 10:57 |
| URI: | https://pred.uni-regensburg.de/id/eprint/46780 |
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