On the converse of Weyl’s conformal and projective theorems | Zendy

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On the converse of Weyl’s conformal and projective theorems

Author(s) -

Graham Hall

Publication year - 2013

Publication title -

publications de l institut mathematique

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.246

H-Index - 17

eISSN - 1820-7405

pISSN - 0350-1302

DOI - 10.2298/pim1308055h

Subject(s) - weyl transformation , counterexample , conformal map , converse , weyl tensor , mathematics , pure mathematics , manifold (fluid mechanics) , conformal gravity , conformal geometry , tensor (intrinsic definition) , conformal field theory , mathematical analysis , discrete mathematics , riemann curvature tensor , geometry , mechanical engineering , curvature , engineering

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.

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