Weakly Einstein conformal products
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912749941948416 |
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| author | Derdzinski, Andrzej Park, JeongHyeong Shin, Wooseok |
| author_facet | Derdzinski, Andrzej Park, JeongHyeong Shin, Wooseok |
| contents | One says that a Riemannian four-manifold is \emph{weakly Einstein} if the three-index contraction of its curvature tensor against itself equals a function times the metric. Since this includes all four-manifolds that are Einstein, or conformally flat and scalar-flat, the term \emph{proper} may be used for weakly Einstein manifolds (or metrics) not belonging to the latter two classes. We establish two classification-type results about proper weakly Einstein metrics conformal to Riemannian products. This includes constructions of new examples, among them -- some of (local) cohomogeneity two, in contrast with the two previously known narrow classes of examples, having cohomogeneity zero and one. We also exhibit a simple coordinate description of one of the known examples, the EPS space, which shows that it is a conformal product and constitutes a single local-homothety type. Finally, we prove that there exist no proper weakly Einstein manifolds with harmonic curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05173 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weakly Einstein conformal products Derdzinski, Andrzej Park, JeongHyeong Shin, Wooseok Differential Geometry 53C25, 53C18, 53B20 One says that a Riemannian four-manifold is \emph{weakly Einstein} if the three-index contraction of its curvature tensor against itself equals a function times the metric. Since this includes all four-manifolds that are Einstein, or conformally flat and scalar-flat, the term \emph{proper} may be used for weakly Einstein manifolds (or metrics) not belonging to the latter two classes. We establish two classification-type results about proper weakly Einstein metrics conformal to Riemannian products. This includes constructions of new examples, among them -- some of (local) cohomogeneity two, in contrast with the two previously known narrow classes of examples, having cohomogeneity zero and one. We also exhibit a simple coordinate description of one of the known examples, the EPS space, which shows that it is a conformal product and constitutes a single local-homothety type. Finally, we prove that there exist no proper weakly Einstein manifolds with harmonic curvature. |
| title | Weakly Einstein conformal products |
| topic | Differential Geometry 53C25, 53C18, 53B20 |
| url | https://arxiv.org/abs/2512.05173 |