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Weakly Einstein and weakly η-Einstein structures on almost contact metric three-manifolds

  • Sun Hyang Chun
  • Yunhee Euh
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Abstract

We study weakly Einstein and weakly η-Einstein structures on almost contact metric three-manifolds. We first obtain a pointwise structural description of the Ricci operator Q on a weakly η-Einstein almost contact metric three-manifold. More precisely, at each point, either Q has the η-Einstein form, Qξ=0, or the metric is weakly Einstein. We then prove that if a contact metric three-manifold satisfies Qξ=0, then either Q=0 or rankQ=1. Hence, such a metric is either flat or weakly Einstein but not Einstein. We also construct examples showing that the contact metric assumption is essential: there exist almost contact metric three-manifolds satisfying Qξ=0 whose Ricci operators have rank two. Finally, we classify simply connected homogeneous contact metric three-manifolds that are weakly Einstein.

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Cite

BibTeXRIS
BibTeX
@misc{chun2026weakly,
  title = {{Weakly Einstein and weakly $η$-Einstein structures on almost contact metric three-manifolds}},
  author = {Sun Hyang Chun and Yunhee Euh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26024},
  eprint = {2609.26024},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26024},
}
RIS
TY  - UNPB
TI  - Weakly Einstein and weakly η-Einstein structures on almost contact metric three-manifolds
AU  - Sun Hyang Chun
AU  - Yunhee Euh
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26024
UR  - https://arxiv.org/abs/2609.26024
AB  - We study weakly Einstein and weakly η-Einstein structures on almost contact metric three-manifolds. We first obtain a pointwise structural description of the Ricci operator Q on a weakly η-Einstein almost contact metric three-manifold. More precisely, at each point, either Q has the η-Einstein form, Qξ=0, or the metric is weakly Einstein. We then prove that if a contact metric three-manifold satisfies Qξ=0, then either Q=0 or rankQ=1. Hence, such a metric is either flat or weakly Einstein but not Einstein. We also construct examples showing that the contact metric assumption is essential: there exist almost contact metric three-manifolds satisfying Qξ=0 whose Ricci operators have rank two. Finally, we classify simply connected homogeneous contact metric three-manifolds that are weakly Einstein.
ER  -