Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., p.466 of Berger's A Panoramic view of Riemannian Geometry, or
this Encyc.Math. article.
(Apparently details in the L.-S. 1929 proof were not resolved until 1978 1993.)
Q. Is there an extension to any Riemannian metric for $\mathbb{S}^3$?
E.g., there exist at least $k > 1$ simple, closed geodesics in $\mathbb{S}^3$
under every Riemannian metric?
(Added.) As Igor Rivin points out, $k=1$ is known for any
smooth metric on $\mathbb{S}^n$ via a 1927 proof of Birkoff.
Added a bounty.
Much remains unclear to me, despite Igor's (extremely) useful citations.
The bounty is offered for clarifying the situation (comments cited in several instances):
(1) Klingenberg proves (Mathematische Zeitschrift}) in 1981 there are 4 closed geodesics on $\mathbb{S}^3$.
(2) Long&Duan prove (Advances in Mathematics) in 2009 that there at least 2 closed geodesics on $\mathbb{S}^3$—What happened in
the 28 yrs between? Was Klingenberg's proof not accepted, or did he prove something
nuancedly different?
(3) And it seems that none of these authors are addressing
simplicity, as @alvarezpaiva noted.
The L.&S. theorem explicitly proves non-self-intersection on $\mathbb{S}^2$.
Best Answer
Yes, there is at least one simple closed geodesic for every smooth metric on $\mathbb{S}^n.$ This is a result of G. D. Birckhoff (1927) - ("Dynamical Systems, AMS Coll. Pub. vol 9).
It was shown by Lyusternik that there were at least $n$ closed geodesics on $\mathbb{S}^n,$ and the sharp result (Alber-Klingenberg) is that there are $2n-s - 1,$ where $s = n - 2^{\lfloor \log_2 n\rfloor}.$
Klingenberg, Wilhelm, On the existence of closed geodesics on spherical manifolds, Math. Z. 176, 319-325 (1981). ZBL0469.53042, MR610213.
@article {MR610213,
AUTHOR = {Klingenberg, Wilhelm},
TITLE = {On the existence of closed geodesics on spherical manifolds},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {176},
YEAR = {1981},
NUMBER = {3},
PAGES = {319--325},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {58E10 (49F99)},
MRNUMBER = {610213 (82d:58024)},
MRREVIEWER = {Y. Mut{^o}},
DOI = {10.1007/BF01214609},
URL = {https://doi.org/10.1007/BF01214609}, }