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This note investigates the possibility of converses of the Weyl theorems that  two conformally related metrics on a manifold have the same Weyl conformal  tensor and that two projectively related connections on a manifold have the  same Weyl projective tensor. It shows that, in all relevant cases,  counterexamples to each of Weyl’s theorems exist except for his conformal  theorem in the 4-dimensional, positive definite case, where the converse  actually holds. This (conformal) 4-dimensional problem is then solved  completely for the other possible signatures.

On the converse of Weyl’s conformal and projective theorems