By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if they have the same image in $(M,g)$. 


Manifolds with constant curvature $\leq 0$,
by Cartan's theorem,
cannot have any closed contractible geodesics,
and every riemannian metric on $S^2$ has infinitely many closed geodesics 
(for $n\geq 3$, the analogous theorem for $S^n$ is not known).
Moreover,
if the sequence of Betti numbers of the loops space $\Omega(M)$ is unbounded and $M$ is simply-connected,
then $(M,g)$ contains infinitely many (contractible) closed geodesics.


**Are there any known examples of riemannian manifolds with finitely and positively many closed contractible geodesics (or even just closed geodesics)?**


There is a theorem associated with Gromov asserting that the word problem of $\pi\_1 M$ is solvable if there is a metric $g$ on $M$ with only finitely many contractible closed geodesics. I was wondering if there are any non-trivial examples for this theorem.



A riemannian manifold with finitely many closed contractible geodesics
I think if you take the metric on $\mathbb{R}^2$ obtained by rotating a curve which is $\sqrt{1-x^2}$ for $-1\leq x\leq 0$, and $x^2+1$ for $x\geq 0$ around the $x$-axis, then I think there will be a single closed contractible geodesic obtained by rotating the point $(0,1)$ around the $x$-axis.


