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?



locally conformally flat manifold
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](https://projecteuclid.org/euclid.jdg/1102538612)*,


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)](http://www.ams.org/journals/tran/1983-278-02/S0002-9947-1983-0701512-8/).


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.


