locally conformally flat manifold A Riemannian manifold $(M,g)$ is locally conformally flat if it is locally conformal to $\mathbb{R}^n$ with the flat metric. I learn that Weyl tensor of a locally conformally flat manifold must vanish. I would like to ask: Is there any example of manifold $M$ such that it cannot be equipped with a metric $g$ with $(M,g)$ being locally flat? Is there any topological restriction on locally confomrally flat manifolds? Is there any classification theorem for locally conformally flat manifolds?
 A: The simplest example is $S^n$, it is locally conformally flat with the standard metric,
and is not flat for obvious reasons.
While flat manifolds are precisely quotients of $\mathbb R^n$ by discreet group of isometries, one should not expect to have a classification of conformally flat manifolds in higher dimensions. For example, already in dimension $4$ it was proven by Kapovich in 
M. Kapovich. Conformally ﬂat metrics on 4-manifolds. J. Diﬀerential
Geom. 66 (2004), no. 2, 289–301,
that arbitrary finitely presented group can be a subgroup of a fundamental group of a conformally flat manifold.
The article of Kapovich is and from its introduction you will learn a lot on the question.  $4$-dimensional manifolds with LCF structure have zero signature, in dimension $3$ it is known that some manfiolds don't admit conformally flat structure, first example was constructed in W. Goldman, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for $3$-manifolds, Transactions
of AMS 278 (1983).
One more remark -- all hyperbolic manifolds (of constant negative sectional curvature) are all conformally flat. A connected sum of two conformally flat manifolds is conformally flat and so this already gives you a large collection of examples.
A: A simple obstruction is this:  No compact (without boundary), simply-connected $n$-manifold that is not diffeomorphic to the $n$-sphere carries a locally conformally flat structure.  The reason is that the developing map construction shows that any locally conformally flat metric $g$ on a simply-connected $M^n$ is, up to a conformal factor, the pullback of the standard metric on $S^n$ under some immersion $\phi:M^n\to S^n$.  If $M$ is also compact without boundary, then $\phi$ is a covering map and, hence, a diffeomorphism.
A: Another topological obstruction by Kuiper,1950 is the following: 
Universal cover of a compact, LCF space with an infinite Abelian
fundamental group must be $\mathbb R^n$ or $\mathbb R \times \mathbb S^{n−1}$.
Using this, for example you can show that $\mathbb S^2\times \mathbb T^2$ does not admit any LCF metric. 
See the following for references and introduction to the subject: 
Kalafat, M. - Locally conformally flat and self-dual structures on simple 4-manifolds. Proceedings of the Gökova Geometry-Topology Conference 2012, 111–122, Int. Press, Somerville, MA, 2013.
