Lengths of geodesics between two points on a Riemannian manifold
Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y"> <mml:semantics> <mml:mi>y</mml:mi> <mml:annotation encoding="application/x-tex">y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be two (not necessarily distinct) points on a closed Riemannian manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript n"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">M^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. According to a well-known theorem by J.-P. Serre, there exist infinitely many geodesics between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y"> <mml:semantics> <mml:mi>y</mml:mi> <mml:annotation encoding="application/x-tex">y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distinct geodesics of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to k d i a m left-parenthesis upper M Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>k</mml:mi> <mml:mi>diam</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\le k \operatorname {diam}(M^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y"> <mml:semantics> <mml:mi>y</mml:mi> <mml:annotation encoding="application/x-tex">y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of length not exceeding <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n d i a m left-parenthesis upper M Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mi>diam</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2n\operatorname {diam}(M^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our second result is that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 2"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M squared"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">M^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is diffeomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S squared"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> every pair of points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M squared"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">M^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be connected by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distinct geodesics of length less than or equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 4 k squared minus 2 k minus 1 right-parenthesis d i a m left-parenthesis upper M squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4</mml:mn> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mi>diam</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(4k^2-2k-1)\operatorname {diam}(M^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.010 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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".