Lengths of geodesics between two points on a Riemannian manifold
Bibliographic record
Abstract
Let x x and y y be two (not necessarily distinct) points on a closed Riemannian manifold M n M^n . According to a well-known theorem by J.-P. Serre, there exist infinitely many geodesics between x x and y y . It is obvious that the length of a shortest of these geodesics cannot exceed the diameter of the manifold. But what can be said about the lengths of the other geodesics? We conjecture that for every k k there are k k distinct geodesics of length ≤<!-- ≤ --> k diam <!-- --> ( M n ) \le k \operatorname {diam}(M^n) . This conjecture is evidently true for round spheres and is not difficult to prove for all closed Riemannian manifolds with non-trivial torsion-free fundamental groups. In this paper we announce two further results in the direction of this conjecture. Our first result is that there always exists a second geodesic between x x and y y of length not exceeding 2 n diam <!-- --> ( M n ) 2n\operatorname {diam}(M^n) . Our second result is that if n = 2 n=2 and M 2 M^2 is diffeomorphic to S 2 S^2 , then for every k k every pair of points of M 2 M^2 can be connected by k k distinct geodesics of length less than or equal to ( 4 k 2 −<!-- − --> 2 k −<!-- − --> 1 ) diam <!-- --> ( M 2 ) (4k^2-2k-1)\operatorname {diam}(M^2) .
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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 | 0.001 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.013 | 0.004 |
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 source (direct Gemma or distilled Codex), 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".