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PreprintCC BY 4.0arXiv

A complete classification of left-invariant Einstein metrics on S3× S3

  • Sixuan Gu
  • Wei Qi
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Abstract

We complete the classification of compact simply connected homogeneous Einstein manifolds in dimension six by resolving the remaining cases for left-invariant Einstein metrics on G=SU(2)×SU(2)≅ S3× S3. Previous work leaves two cases for the isotropy group K, namely K={e} and K≅ℤ2. We first rule out K={e}. Then we show that any ℤ2 symmetry generated by an inner involution σ with trσ=−2 necessarily extends to a ℤ2×ℤ2 symmetry. Together with the previously known classification results, these theorems imply that every left-invariant Einstein metric on G is, up to homothety and isometry, either the standard product metric gcan or the Jensen nearly Kähler metric gNK.

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Cite

BibTeXRIS
BibTeX
@misc{gu2026complete,
  title = {{A complete classification of left-invariant Einstein metrics on $S^3\times S^3$}},
  author = {Sixuan Gu and Wei Qi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06425},
  eprint = {2609.06425},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06425},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A complete classification of left-invariant Einstein metrics on S^3× S^3
AU  - Sixuan Gu
AU  - Wei Qi
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06425
UR  - https://arxiv.org/abs/2609.06425
AB  - We complete the classification of compact simply connected homogeneous Einstein manifolds in dimension six by resolving the remaining cases for left-invariant Einstein metrics on G=SU(2)×SU(2)≅ S^3× S^3. Previous work leaves two cases for the isotropy group K, namely K={e} and K≅ℤ_2. We first rule out K={e}. Then we show that any ℤ_2 symmetry generated by an inner involution σ with trσ=−2 necessarily extends to a ℤ_2×ℤ_2 symmetry. Together with the previously known classification results, these theorems imply that every left-invariant Einstein metric on G is, up to homothety and isometry, either the standard product metric g_can or the Jensen nearly Kähler metric g_NK.
ER  -