Bibliographic record
Abstract
In this paper we investigate the relationship between the existence of parallel semi-Riemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces in the presence of a specified number of independent parallel semi-Riemannian metrics is completely determined by the the signature of the metrics and the dimension d of the manifold, when d ≠ 4. In particular, the existence of two independent, parallel semi-Riemannian metrics, one of which having signature ( p,q ) with p ≠ q , implies the holonomy group is reducible. The ( p,p ) cases, however, may allow for more than one parallel metric and yet an irreducible holonomy group: for n = 2 m , m ≥ 3, there exist connections on R n with irreducible infinitesimal holonomy and which have two independent, parallel metrics of signature ( m,m ). The case of four-dimensional manifolds, however, depends on the topology of the manifold in question: the presence of three parallel metrics always implies reducibility but reducibility in the case of two metrics of signature (2,2) is guaranteed only for simply connected manifolds. The main theorem in the paper is the construction of a topologically non-trivial four-dimensional manifold with a connection that admits two independent metrics of signature (2,2) and yet has irreducible holonomy. We provide a complete solution to the general problem.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".