Necessary and sufficient conditions for (i) Weyl, (ii) Riemann-Cartan connections
For a space with metric \(h_{ab}\) its semimetric connection defined by \(\nabla_ c h_{ab}= h_{ab}\lambda_ c\) necessarily satisfies an algebraic relation of the type \(h_{ai}\breve R^ i_{bcd}+h_{bi}\breve R^ i_{acd}=0\), where \(\breve R^ i_{bcd}\) is related to the curvature tensor \(R^ i_{bcd}\) of the Euclidean connection by \(\breve R^ i_{bcd}=R^ i_{bcd}-{1\over 2}\delta^ a_ bR^ i_{icd}\). For a four-dimensional space time, in particular, it is shown that this relationship also provides a sufficient condition for the local existence of a curvature tensor of a semi-metric connection.
scientific article; zbMATH DE number 569289
scientific article