Symmetry inheritance in Riemannian manifolds with physical applications
This paper studies symmetries of geometrical objects \(\Omega\) on a manifold \(M\) (usually the space-time manifold of general relativity) generated by a vector field \(X\) and defined by \({\mathcal L}_ X \Omega= \alpha\Omega\) for some real valued function \(\alpha\) on \(M\). The author, somewhat unusually, refers to the above feature as ``symmetry inheritance''. It is not entirely clear to the reviewer exactly what the author's intentions are in this paper. Many aspects of different types of symmetries are mentioned, but only superficially and mostly in support of the somewhat narrow thesis of this paper. The mathematics is at times confusing and imprecise and it is disconcerting to still see ``infinitesimals'' mentioned at this level. One of the main symmetries studied is ``curvature inheritance symmetry'' described by a vector field \(X\) satisfying \({\mathcal L}_ X R_{bcd}^ a= 2\alpha R_{bcd}^ a\) with respect to the curvature tensor and some function \(\alpha\). In the reviewer's opinion the author does not give any compelling reason to study such symmetries. A similar study is made where the curvature tensor is replaced by the Ricci tensor. The paper also gives the impression of giving some kind of review of symmetries in general relativity but in this respect the bibliography is somewhat then and, at times, strangely slanted and out of date.
scientific article; zbMATH DE number 465046
scientific article