Strongly integrable harmonic SU(3)-structures in cohomogeneity one
Bibliographic record
Abstract
Harmonic geometric structures 𝜉 are the critical points of the energy functional ∫ 𝑀 |∇𝜉 | 2 vol 𝑔 and provide examples of special metrics with torsion, with 𝜉 stabilized under the action of a Lie group 𝐻.When 𝐻 = 𝑆𝑈 (𝑚), the associated structures 𝐽 are almost complex structures which are metric-compatible and come with a complex orientation, and the harmonic condition [∇ * ∇𝐽, 𝐽] = 0 gives us examples of non-Kähler metrics in a given isometry class [𝑔].We impose the harmonic condition on almost complex structures 𝐽, and we consider in particular the almost complex structures arising from integrable 𝑆𝑈 (3)-structures with holomorphic complex volume form on compact manifolds 𝑀 under the cohomogeneity one action of a compact Lie group 𝐺, where 𝑀/𝐺 [-1, 1].We prove a non-existence result when the Lie algebra of the isotropy group and that of the symmetry group are the pair 𝔰𝔲(2) ⊂ 𝔰𝔲(3) = 𝔤.In this setting, we find that there exist local solutions on the union of principal orbits 𝑀 princ , with 𝑀 princdense as a subset of 𝑀, and we describe the differential conditions necessary to obtain local Kähler solutions.Finally, we provide an argument to show that no solutions can extend from the principal orbits to the singular orbits in the two possible cases for the singular orbit types.This proves that no almost complex structures in this setting can satisfy the harmonic condition globally on compact 𝑀.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
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