A comment on the integrability conditions of the conformal Killing equation
Let M be a differentiable manifold of dimension \(n\geq 4\) with metric \(g_{ab}\), Weyl tensor \(C^ a_{bcd}\), Ricci tensor \(R_{ab}\), Ricci scalar R and a tensor \(C_{abc}\) defined by \[ C_{abc}=\frac{2}{n- 2}[R_{c[b;a]}-\frac{1}{2(n-1)}R_{;[a}g_{b]c}] \] where square brackets denote skew-symmetrisation and a semicolon denotes a covariant derivative. \textit{K. Yano}, in his book ``The theory of Lie derivatives and its applications'' (1957; Zbl 0077.158) shows that the integrability conditions for the conformal Killing equation \({\mathcal L}_ Xg_{ab}=2\phi g_{ab}\) (where X is a vector field on M, \({\mathcal L}\) denotes a Lie derivative and \(\phi\) is a function on M) can be generated from the two conditions \[ {\mathcal L}_ XC^ a_{bcd}=0\quad and\quad {\mathcal L}_ XC_{abc}=-C^ d_{cab}\phi_{;d}. \] In this paper the author shows how the second of these conditions follows from the first and is, in this sense, redundant.
scientific article; zbMATH DE number 4105768