Space of conformal classes Let $M^n$ be a smooth, closed, simply connected manifold. 
Question: "How many" distinct conformal classes of metrics does $M$ admit?
For example, when $n=2$, then $M^2=S^2$. The uniformization theorem tells us that any metric on $M^2$ is conformally diffeomorphic to the constant curvature metric on $M^2$. Therefore, in this case, $M^2$ has a unique conformal class up to diffeomorphism.
What is known in higher dimensions, and for specific cases like $M^n=S^n$? 
I would appreciate references. Thanks.
 A: The space of conformal classes (up to conformal isomorphism) is infinite-dimensional starting with dimension 3. I will prove this only in 3-dimensional case. I will consider a family of Riemannian metrics on $R^3$ of the form
$$
dx^2 + (1+f) dy^2 + (1+h)dz^2,
$$
where $f=f(z) > 0$ and $h=h(y) > 0$. 
Remark. This choice of metrics is not random. It is proven in 
D. DeTurck, D. Yang, Existence of elastic deformations with prescribed principal strains and triply orthogonal systems. Duke Math. J. 51 (1984), no. 2, 243–260. 
that every Riemannian metric in dimension 3 can be locally conformally transformed to one of the above form with $f=f(x,y,z), h=h(x,y,z)$. I made further assumptions $f=f(z), h=h(y)$ in order to have cleaner computations. 
In order to distinguish conformal classes of these metrics I will be using the Cotton tensor (due to its conformal invariance). 
A direct computation of the Cotton tensor of metrics from this family (cf. Appendix C here for a general computation of the Cotton tensor; such explicit computations are also done in Eisenhart's "Riemannian Geometry" which is a bit dated and, thus, hard to read) yields:
$$
C= \frac{1}{4} \left[ \begin{array}{ccc}
0&f'''& - h'''\\
f''' & 0 & 0\\
-h''' & 0 & 0
\end{array}\right]
$$
Assuming that $f''', h'''$ are nowhere vanishing, every conformal isomorphism of two metrics from this family has the form 
$$
\Phi: (x,y,z)\mapsto (\phi(x), \psi(y), \eta(z)),
$$
which implies linearity of $\Phi$ (since I am assuming that $f$ is independent of $x, y$ and $h$ is independent of $x, z$). From this, we see that the space of conformal isomorphism classes of metrics in this family is infinite-dimensional. The construction is purely local, hence, can be done on any 3-dimensional manifold.  
