In this paper, we consider a Finsler sphere
with dimension
and flag curvature
. The action of the
connected isometry group
on
, together with
the action of
shifting
the parameter
of the
closed curve
, define
an action of
on the
free loop space
of
.
In particular, for each closed geodesic, we have a
-orbit
of closed geodesics. We assume the Finsler sphere
described
above has only finite orbits of prime closed geodesics. Our main theorem claims that, if the
subgroup
of all isometries preserving each close geodesic is of dimension
, then there exists
geometrically distinct orbits
of prime closed geodesics,
such that for each
, the
union
of geodesics in
is a totally geodesic
submanifold in
with
a nontrivial
-action.
This theorem generalizes and slightly refines the one in a previous work, which only
discussed the case of finite prime closed geodesics. At the end, we show that, assuming
certain generic conditions, the Katok metrics, i.e., the Randers metrics on spheres
with
,
provide examples with the sharp estimate for our main theorem.
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