Weakly Einstein curvature tensors
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917233751490560 |
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| author | Derdzinski, Andrzej Park, JeongHyeong Shin, Wooseok |
| author_facet | Derdzinski, Andrzej Park, JeongHyeong Shin, Wooseok |
| contents | We classify weakly Einstein algebraic curvature tensors in an oriented Euclidean 4-space, defined by requiring that the three-index contraction of the curvature tensor against itself be a multiple of the inner product. This algebraic formulation parallels the geometric notion of weakly Einstein Riemannian four-manifolds, which include conformally flat scalar-flat, and Einstein manifolds. Our main result provides a complete classification of non-Einstein weakly Einstein curvature tensors in dimension four, naturally dividing them into three disjoint five-dimensional families of algebraic types. These types are explicitly constructed using bases that simultaneously diagonalize both the Einstein tensor and the (anti)self-dual Weyl tensors, which consequently proves that such simultaneous diagonalizability follows from the weakly Einstein property. We also point out that our classification has immediate applications, and describe how some known geometric examples that are neither Einstein, nor conformally flat scalar-flat (namely, the EPS space and certain Kähler surfaces) fit within our classification framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weakly Einstein curvature tensors Derdzinski, Andrzej Park, JeongHyeong Shin, Wooseok Differential Geometry 53B20, 15A69 We classify weakly Einstein algebraic curvature tensors in an oriented Euclidean 4-space, defined by requiring that the three-index contraction of the curvature tensor against itself be a multiple of the inner product. This algebraic formulation parallels the geometric notion of weakly Einstein Riemannian four-manifolds, which include conformally flat scalar-flat, and Einstein manifolds. Our main result provides a complete classification of non-Einstein weakly Einstein curvature tensors in dimension four, naturally dividing them into three disjoint five-dimensional families of algebraic types. These types are explicitly constructed using bases that simultaneously diagonalize both the Einstein tensor and the (anti)self-dual Weyl tensors, which consequently proves that such simultaneous diagonalizability follows from the weakly Einstein property. We also point out that our classification has immediate applications, and describe how some known geometric examples that are neither Einstein, nor conformally flat scalar-flat (namely, the EPS space and certain Kähler surfaces) fit within our classification framework. |
| title | Weakly Einstein curvature tensors |
| topic | Differential Geometry 53B20, 15A69 |
| url | https://arxiv.org/abs/2504.18752 |