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We present a global conformal invariant on closed six-manifolds whichobstructs the existence of a conformally Einstein metric. We show that thisobstruction is nontrivial and, up to multiplication by a constant, is theunique 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 aconformally Einstein metric on closed six-manifolds with infinite fundamentalgroup.

Global obstructions to conformally Einstein metrics in dimension six