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Math-2301 · mathematics · Geometry

Closed geodesics in a finite-volume torus ansatz

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Does every complete Riemannian manifold of finite volume have a nonconstant closed geodesic? Must it in fact have infinitely many geometrically distinct closed geodesics? The source additionally asks analogous questions for finite-volume Finsler metrics, distinguishing reversible metrics and the noncompact irreversible case.

Formulation references: [S1], [S2].

Short English statement

Does every complete Riemannian manifold of finite volume have a nonconstant closed geodesic? Must it in fact have infinitely many geometrically distinct closed geodesics? The source additionally asks analogous questions for finite-volume Finsler metrics, distinguishing reversible metrics and the noncompact irreversible case.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2301-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2301

Infinitely many prime closed geodesics: The closed-three-manifold corollary covers the closed dimension-three subcase of finite-volume Riemannian manifolds. It does not include noncompact complete finite-volume manifolds or all dimensions.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Closed geodesics in a finite-volume torus ansatz

OpenAI's claim

Infinitely many prime closed geodesics: Every smooth Riemannian sphere of dimension at least two is claimed to have infinitely many prime closed geodesics with distinct images. The paper also obtains infinitude on every nonempty closed three-manifold.

Mathematical statement of the claimed result

Mathematical claim 1.
Whatever smooth metric is chosen on the standard sphere $S^n$, at each integer dimension $n\ge2$ the manuscript claims infinitely many prime closed geodesics having pairwise distinct images. Prime means not an iterate of a shorter closed geodesic. Its three-manifold corollary asserts the same conclusion for every nonempty closed smooth three-manifold and every smooth metric, allowing nonorientability.

Outline of the argument

Infinitely many prime closed geodesics: Assuming finitely many prime images gives a uniform budget for local loop-space crossings, including degenerate root types. Reflection-equivariant cohomology and half-period translation compare ordinary and root-degree filtrations. Extra classes would force a quadratic discrepancy, but the exact-sequence and filling estimates allow only a linear budget; covering and three-manifold results provide the final transfer.

Formal verification

Infinitely many prime closed geodesics: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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