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# Infinitely many closed geodesic images on every Riemannian sphere



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Abstract: 
We solve the sphere case of the closed-geodesic infinitude problem:
every smooth Riemannian metric on the standard sphere S^n, nge 2,
has infinitely many prime closed geodesics with pairwise distinct
images. The two-sphere case is classical; the new content is every
dimension nge 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.


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# Introduction
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A closed geodesic on a Riemannian manifold (M,g) is a
nonconstant map gamma:R/bbR/Z M that satisfies the
geodesic equation and is periodic. It is prime if it travels
around its trajectory exactly once. Two prime geodesics are the same
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.


Let nge 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.


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


Let N be a closed manifold that admits a finite smooth cover
diffeomorphic to the standard sphere S^n with nge 2. Every smooth
Riemannian metric on N has infinitely many geometrically distinct
prime closed geodesics.


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}.

[Closed 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.


Proof. 
Work on one connected component of M, so assume M is connected.

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.

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^3to S^3/Gamma is
the quotient, then phi composed with a diffeomorphism S^3/Gammato M
is a finite smooth cover S^3to M. Cref{cor:finite-cover} applies, and
only the smooth covering is used, so the given metric remains arbitrary.


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