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Affine conformal vector fields in semi-Riemannian manifolds. (English) Zbl 0734.53023

This paper studies what the author calls Affine conformal vector fields. Such a vector field X on a manifold M satisfies, by definition, ${ℒ}_{x}{g}_{ab}=2\sigma {g}_{ab}+{K}_{ab}$ where g is the metric on M, a semicolon denotes the covariant derivative from g, $\sigma$ is a real valued function and ${K}_{ab;c}=0$. It seems to the reviewer that either K is a (constant) multiple of g, in which case X is conformal or, K is not a multiple of g in which case M is locally metrically decomposable. A study of such vector fields can then proceed in a systematic way (if one should want to study them which, since the set of all such vector fields does not form a Lie algebra, is not clear).

The approach taken by the author is rather unsystematic. The techniques employed and the bibliography suggest that the author is unaware of several (not necessarily recent) works which would greatly simplify his task. Also the remarks about reducibility at the bottom of p. 278 (which are false) and similar remarks on p. 279 (before theorem 3) suggest a certain amount of confusion. In theorem 5, the term semi-Euclidean is used but not defined. This could mean flat but of indefinite signature, or semi-Riemannian. If either is the case the theorem (which claims that a global affine conformal vector field on a semi-Euclidean space is necessarily affine) is false. The author does not stress the restrictive nature of the condition ${K}_{ab;c}=0$ (K$\not\approx g\right)$ and the resulting confusion is demonstrated by the “models” described on pages 280-282. Such restrictions have been discussed e.g. in the reviewer and J. da Costa [J. Math. Phys. 29, No.11, 2465-2472 (1988; Zbl 0661.53017), and the reviewer, ibid. 32, No.1, 181-187 (1991; Zbl 0731.53024)]. Another puzzling feature is the collection of results on pages 284-286 which deal with general n-dimensional manifolds but quote results from 4-dimensional space-times!! Finally the definition of a Killing vector field, equation (7.1) has been misprinted - round brackets around the two indices are omitted.

##### MSC:
 53B99 Local differential geometry
##### Keywords:
affine collineation; symmetry; conformal vector fields