Example of Non-Conformally Flat Einstein Manifold? Does there exist an Einstein manifold which is not conformally flat, which is to say one which has non-vanishing Weyl tensor. If so, what is a good example.
 A: 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:
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
A: K3 surface is Einstein, by Calabi-Yau theorem, but it is not conformally flat; this can be seen e.g. from topology, or from the reference that José Figueroa-O'Farrill has given above.
A: Does your question refer to both Riemannian and semi-Riemannian spaces? If so, then the Schwarzschild metric is an example. It's a vacuum solution to the Einstein field equations (with zero cosmological constant), so it's an Einstein manifold. It has a nonvanishing Weyl tensor.
A: By the standard curvature decomposition, for an Einstein metric, if the Weyl curvature vanishes, then the sectional curvature is constant, so other than space forms, all Einstein manifolds are not conformally flat.
Sorry I didn't see Deane Yang's answer, he already got this answer.
