We present a global conformal invariant on closed six-manifolds which obstructs the existence of a conformally Einstein metric. We show that this obstruction is nontrivial and, up to multiplication by a constant, is the unique such invariant. This also gives rise to a (possibly trivial) diffeomorphism invariant which obstructs the existence of an Einstein metric. We also discuss global conformal invariants which obstruct the existence of a conformally Einstein metric on closed six-manifolds with infinite fundamental group.

I thank Misha Gromov and Claude LeBrun for helpful comments and encouragement. I also thank the anonymous referee for many helpful comments. This work was partially supported by the Simons Foundation (Grant #524601).