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There is (or was) a notorious open problem (see e.g. this survey by Keith Burns and Vladimir Matveyev from 2013) in differential geometry:

Conjecture. Let $M$ be a closed (compact, with empty boundary) Riemannian manifold of dimension $\ge 2$. Then $M$ contains infinitely many geometrically distinct (nonconstant) closed geodesics.

(Geometrically distinct here means "with distinct images".)

There is an arXiv preprint

"The Existence of Infinitely Many Geometrically Distinct Non-constant Prime Closed Geodesics on Riemannian Manifolds" by Sergio Charles

from August 2018 claiming a proof of this conjecture (see Theorem 6.14 of the preprint). As far as I know, this preprint is still unpublished. The proof is long and technical, a combination of differential geometry, rational homotopy theory and some infinite-dimensional analysis. It is well beyond my comfort zone.

Did anybody succeed going through the entire linked paper? What is the current status of the conjecture? Is it still regarded as an open problem?

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  • $\begingroup$ You should add "differential geometry" tag, it will be more visible I guess... $\endgroup$
    – GSM
    Commented Jan 17, 2022 at 21:21
  • $\begingroup$ @GSM ok................ $\endgroup$
    – Moishe Kohan
    Commented Jan 17, 2022 at 21:53

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