Geodesic
On the Converse of Weyl's Conformal and Projective Theorems
Publications de l'Institut Mathématique, _N_S_94 (2013) no. 108, p. 55 .

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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.
Classification : 53A20 53A30 53C50 
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     author = {Graham Hall},
     title = {On the {Converse} of {Weyl's} {Conformal} and {Projective} {Theorems}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {55 },
     publisher = {mathdoc},
     volume = {_N_S_94},
     number = {108},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_2013_N_S_94_108_a5/}
}
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Graham Hall. On the Converse of Weyl's Conformal and Projective Theorems. Publications de l'Institut Mathématique, _N_S_94 (2013) no. 108, p. 55 . http://geodesic.mathdoc.fr/item/PIM_2013_N_S_94_108_a5/