Local isometric immersions of pseudo-spherical surfaces and k-th order\n evolution equations
Bibliographic record
Abstract
We consider the class of evolution equations that describe pseudo-spherical\nsurfaces of the form u\\_t = F (u, $\\partial$u/$\\partial$x, ..., $\\partial$^k\nu/$\\partial$x^k), k $\\ge$ 2 classified by Chern-Tenenblat. This class of\nequations is characterized by the property that to each solution of a\ndifferential equation within this class, there corresponds a 2-dimensional\nRiemannian metric of curvature-1. We investigate the following problem: given\nsuch a metric, is there a local isometric immersion in R 3 such that the\ncoefficients of the second fundamental form of the surface depend on a jet of\nfinite order of u? By extending our previous result for second order evolution\nequation to k-th order equations, we prove that there is only one type of\nequations that admit such an isometric immersion. We prove that the\ncoefficients of the second fundamental forms of the local isometric immersion\ndetermined by the solutions u are universal, i.e., they are independent of u.\nMoreover, we show that there exists a foliation of the domain of the parameters\nof the surface by straight lines with the property that the mean curvature of\nthe surface is constant along the images of these straight lines under the\nisometric immersion.\n
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.006 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| 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".