There is some well known obstruction for a manifold to admit Einstein metric. Moreover it is clear that any metric conformal to an einstein metric is Bach flat, but all the Bach flat metrics are not conformal to an Einstein one. So my question is, is there some obstruction for a manifold to get a Bach flat metric? In particular, I am interested in this question for 4-manifold.
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2$\begingroup$ A minor correction: the only obstruction known for an Einstein metric is in dimensions 0,1,2,3,4. $\endgroup$– Ben McKayCommented Nov 3, 2022 at 10:10
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$\begingroup$ I realize that I even don't know any example of Bach flat metric which is not conformal to an Einstein one? I guess it is hard to have a concrete example, but does it proved it exists? $\endgroup$– PaulCommented Nov 6, 2022 at 16:09
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3$\begingroup$ Self-dual and anti-self dual metrics (and hence locally conformally flat) are also Bach flat. So, for example, the product metric on $S^1 \times S^3$ is Bach flat, even though there is no Einstein metric on $S^1 \times S^3$. $\endgroup$– Jeffrey CaseCommented Nov 8, 2022 at 14:57
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