Does there exist an [Einstein manifold](http://en.wikipedia.org/wiki/Einstein_manifold) which is not conformally flat, which is to say one which has non-vanishing [Weyl tensor.](http://en.wikipedia.org/wiki/Weyl_tensor) If so, what is a good example.



Example of Non-Conformally Flat Einstein Manifold?
All conformally flat homogeneous riemannian manifolds are symmetric spaces, by a result of Takagi. All homogeneous riemannian manifolds of dimension $\leq 11$ admit Einstein metrics, by results of Wang and Ziller. Most homogensous riemannian manifolds are not symmetric spaces.


References:
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Takagi, H.: Conformally flat Riemannian manifolds admitting a transitive group of isometries
I, II. Tohoku Math. J. 27, 103–110(I), 445–451(II) (1975)


McKenzie Y. Wang, Wolfgang Ziller: Existence and non-existence of homogeneous Einstein metrics, Inventiones mathematicae, 1986, Volume 84, Issue 1, pp 177-194


