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\title{Infinitely many closed geodesic images\\ on every Riemannian sphere}
\author{OpenAI}
\date{2026}

\begin{document}
\maketitle

\begin{abstract}
We solve the sphere case of the closed-geodesic infinitude problem:
every smooth Riemannian metric on the standard sphere $S^n$, $n\ge 2$,
has infinitely many prime closed geodesics with pairwise distinct
images. The two-sphere case is classical; the new content is every
dimension $n\ge 3$, including degenerate metrics. Using finite covers,
Perelman's geometrization theorem, and a theorem of
Rademacher--Taimanov, we get the same conclusion for every smooth
Riemannian metric on a nonempty closed smooth three-manifold.
\end{abstract}

\tableofcontents

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\section{Introduction}\label{sec:introduction}
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A \emph{closed geodesic} on a Riemannian manifold $(M,g)$ is a
nonconstant map $\gamma:\mathbb R/\mathbb Z\to M$ that satisfies the
geodesic equation and is periodic. It is \emph{prime} if it travels
around its trajectory exactly once. Two prime geodesics are the same
\emph{geometric object} if their images coincide; changing the starting
point, reversing the direction, or iterating an existing geodesic does
not create a new image. The closed-geodesic infinitude problem asks
whether every closed Riemannian manifold of dimension at least two has
infinitely many geometrically distinct prime closed geodesics. We settle
the sphere case.

\begin{theorem}\label{thm:spheres}
Let $n\ge 2$ and let $g$ be any smooth Riemannian metric on the standard
smooth sphere $S^n$. Then $(S^n,g)$ has infinitely many prime closed
geodesics with pairwise different images.
\end{theorem}

Two immediate corollaries come from finite covers and from
geometrization in dimension three.

\begin{corollary}\label{cor:finite-cover}
Let $N$ be a closed manifold that admits a finite smooth cover
diffeomorphic to the standard sphere $S^n$ with $n\ge 2$. Every smooth
Riemannian metric on $N$ has infinitely many geometrically distinct
prime closed geodesics.
\end{corollary}

This includes all real projective spaces and all smooth spherical space
forms. The passage between a closed manifold and a finite cover is
classical (see \cite[p.~633--634]{ViguePoirrierSullivan1976}); we spell
it out in \cref{sec:covers}.

\begin{corollary}[Closed three-manifolds]\label{cor:three-manifolds}
Let $M$ be a nonempty closed smooth three-manifold, orientable or not.
Every smooth Riemannian metric on $M$ has infinitely many prime closed
geodesics with pairwise distinct images.
\end{corollary}

\begin{proof}
Work on one connected component of $M$, so assume $M$ is connected.

\emph{Case 1: infinite fundamental group.}
Rademacher and Taimanov \cite[Theorem~2]{RademacherTaimanov2018} produce
infinitely many geometrically distinct closed geodesics for the given
metric. Their notion of geometric distinctness is exactly "the images
coincide" \cite[Section~3]{RademacherTaimanov2018}. Each closed geodesic
has a least positive period, and replacing it by its prime root does not
change its image. So the case is settled.

\emph{Case 2: finite fundamental group.}
The elliptization part of Perelman's geometrization theorem says that
$M$ is diffeomorphic to a spherical space form $S^3/\Gamma$, where
$\Gamma$ is a finite group acting freely and smoothly on $S^3$
\cite[Corollary~0.2(b)]{MorganTian2007}. If $\phi:S^3\to S^3/\Gamma$ is
the quotient, then $\phi$ composed with a diffeomorphism $S^3/\Gamma\to M$
is a finite smooth cover $S^3\to M$. \Cref{cor:finite-cover} applies, and
only the smooth covering is used, so the given metric remains arbitrary.
\end{proof}

No quantitative length-counting is asserted in the finite-fundamental-group
case; only the infinitude.

\subsection*{What this section does, and the idea of the proof}

The argument has three big pieces, and each piece is useful on its own.

\paragraph{Local control.}
Assume the metric has only finitely many prime geodesic images, and try to
derive a contradiction. To do that we must understand the geometry of the
loop space near the iterates of those finitely many trajectories. The
first ingredient (\cref{sec:local}) shows that every iterate contributes
a cohomology group whose total rank is bounded by a single constant, with
support concentrated near a line in the $({\rm index},{\rm length})$
plane, and that a specific cup product vanishes at every reflection
crossing. This holds no matter how degenerate the metric is; in
particular, primes with zero mean index (which can occur at arbitrarily
large lengths) cause no trouble.

\paragraph{Uniform filling.}
The second ingredient (\cref{sec:shortening,sec:cutjoin,sec:ordinary})
controls the classes that \emph{appear temporarily} at a sublevel and
then disappear at a slightly higher level. Square-root energy
$F(\gamma)=\sqrt{E(\gamma)}$ is the natural filtration variable. There
is a single constant $D$ such that, in every degree, every cycle below
$F<a$ that bounds in the full loop space already bounds below $F<a+D$.
The classical shortcut gives a constant depending on the degree; here a
cut operation lowers the degree into a fixed finite range while removing
a bounded-length slice of loop, then a loop product restores the degree
without adding length. The relevant quantitative facts about the sphere
come from the resonance theorem of Hingston and Rademacher
\cite{HingstonRademacher2013}.

\paragraph{Reflection and doubling.}
The third ingredient (\cref{sec:equivariant,sec:budget,sec:transients,sec:activation})
uses the $O(2)$ symmetry of the loop space, restricted to the reflection
$r:\gamma(t)\mapsto\gamma(-t)$. Let $B_a$ be reflection-equivariant
cohomology below $F<a$ relative to constant loops, and let $B_\infty$ be
the global group. The quantity
\[
  T(a) = \dim\coker\bigl(B_\infty\longrightarrow B_a\bigr)
\]
(summed over degrees) counts extra reflection classes at level $a$.
Under the finite-prime assumption (\cref{thm:bounded-transients}) this
quantity is uniformly bounded in $a$.

On the other hand, doubling the parameter of a loop
$\delta:\gamma(t)\mapsto\gamma(2t)$ identifies the $\tfrac12$-translation
fixed locus with a second copy of the loop space and produces
$T(2a)\ge T(a)$ (\cref{thm:doubling}). A bounded nonnegative integer nice
function of this type attains its maximum on an open interval, and
doubling spreads that maximum to all scales. So every regular point in
every expanded interval attains the maximum, which forces an equality
condition at every doubling. The equality condition demands that every
nonzero bottom restriction from an \emph{odd} round stage be partnered by
an independent top restriction in the same degree. But evaluation cup
vanishes at an isolated reflection crossing of odd stage
(\cref{thm:local-control}(iv)), so the top is still zero at the first
moment the bottom appears. Contradiction. Hence finitely many prime
images are impossible, and \cref{thm:spheres} follows.

The bookkeeping in the middle -- \cref{sec:budget,sec:transients} -- is an
exact-sequence calculation that says: a long string of extra reflection
classes would force the "extra budget" $T(a)$ to grow like $a^2$, while
the local defect comparison makes it grow at most like $a$. The uniform
filling estimate rules out the remaining escape route.

\subsection*{Organization}
\Cref{sec:setup} fixes conventions. \Cref{sec:local} gives the uniform
local theorem (including degenerate metrics and vanishing of evaluation
cup). \Cref{sec:shortening,sec:cutjoin,sec:ordinary} prove the uniform
filling statement in every degree. \Cref{sec:equivariant} computes the
global reflection cohomology of the sphere and constructs the classes
used in the doubling argument. \Cref{sec:budget,sec:transients,sec:activation}
give the defect estimate, the bounded-transient theorem, and the final
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