I know from two sources
that it is (or at least was) unknown whether there are infinitely
many geometrically distinct closed geodesics
for every Riemannian metric on $S^3$, the 3-sphere
(Weinberger, *Computers, Rigidity, and Moduli*, 1995, p.101;
Berger, *A Panoramic View of Riemannian Geometry*, 2003, p.461).
And from the same two sources, that it is known that
there is at least one closed geodesic on any compact
Riemannian manifold (Lusternick and Fet).
My question is whether or not it is known that
there are at least *two* distinct closed geodesics on $S^3$?

On $S^2$ it is known there are at least three simple closed geodesics (Lusternick and Schnirelmann), and infinitely many periodic geodesics (Bangert, Franks, Hingston). It might help an idea I'm considering if it were known there is more than one closed geodesic on $S^3$. Thanks for pointers!