Closed geodesics

Definitions and mathematical statement

In the usual closed-geodesic setting, a metric is bumpy if every closed geodesic, including its iterates, is nondegenerate modulo the unavoidable reparametrization direction. Equivalently, in the standard Riemannian formulation there is no nontrivial periodic Jacobi field beyond the tangent-field degeneracy. The source does not specify the metric category, so this definition is given only as the usual meaning of the named assumption.

The supplied question asks: is the preceding result true without the bumpy assumption? That preceding result is not included in the record. Hence its manifold, curvature, reversibility, geodesic-count, length, index, and conclusion data cannot be recovered. No exact removal-of-hypothesis statement can be formulated until that antecedent is identified.

The research summary speculates about Javaloyes--Ziller-style closed-geodesic results, reversibility, and index parity, while expressly saying the precise claim could not be verified. Those possibilities are not selected as a replacement theorem. The target retained here is necessity of bumpiness for the source's omitted closed-geodesic assertion.

Short English statement

Can the bumpy-metric hypothesis be removed from the closed-geodesic result referenced in the original source? The antecedent theorem is missing, so its intended scope and conclusion remain unknown. The supplied summary explicitly calls the statement incomplete and unverified. General closed-geodesic existence results mentioned there do not identify the omitted assertion or settle this question.

Sources

• Local UnsolvedMath registry, permanent ID 6800028, source code catalogue entry, "Closed geodesics," proposed by Wolfgang Ziller, University of Pennsylvania, and Miguel Angel Javaloyes Victoria, Murcia, 2018. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Morgan and Pansu, A List of Open Problems in Differential Geometry (2018), Question environment 28, https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source-TeX extraction, accessed 2026-07-29.
• The antecedent theorem and its bibliographic reference are absent. The supplied summary mentions Javaloyes--Ziller and Lyusternik--Schnirelmann context without exact citations.
• Source-specific provenance: the item is labelled a question, current status NEEDS_REVIEW; the public source TeX is on a coauthor's publication page, with redistribution terms NEEDS_REVIEW, separately from the dataset licence.