Projective Transformations of Pseudo-Riemannian Manifolds

Abstract

A projective transformation of a pseudo-Riemannian manifoldM is an automorphism of the induced Riemannian connection of the projective structure that takes geodesics in M to geodesics again. For the first time, the problem of determining Riemannian spaces admitting continuous transformation groups preserving geodesics was considered by Sophus Lie for the case of two-dimensional surfaces (see [306]). However, as Fubini wrote in the preface to [262], “the famous mathematician did not succeed in solving this problem” (which Fubini called the “Lie problem”). Having criticized Lie’s method, which, to Fubini’s opinion, could not be used for the general solution of the problem, Fubini developed his own approach based on the infinitesimal calculus, later named the “Lie differentiation.” The subsequent development of the theory of projective transformations on linear connection spaces is connected with the names of E. Cartan, Eisenhart, Thomas, Knebelman, Schouten, Yano, Egorov, Vranceanu, Kobayashi, and others. It is known that in the spaces of constant curvature S, when considered in the small, the complete projective group coincides with the projective group of pseudo-Euclidean space, i.e., with the group of bilinear substitutions, and depends on n(n+ 2) parameters. In the spaces V n of nonconstant curvature, the order of the complete projective group does not exceed n(n−2)−5 (Egorov [91]), and, moreover, in the majority of cases, this group consists of similarity transformations (homotheties) or isometries. In 1903, in “Turin Academy Notes,” Fubini’s paper “Groups of geodesic transformations” was published [262], which as mentioned above, laid the foundations for a systematic definition and study of Riemannian spaces admitting infinitesimal projective transformations. Later on, Solodovnikov [156–158] continued Fubini’s research and completely solved the problem posed; the works of Fubini and Solodovnikov contain a classification of the Riemannian spaces V , n ≥ 3, in terms of (local) groups of projective transformations which are larger than homothety groups. The conclusions of Fubini and Solodovnikov are based on the assumption that the metrics considered are positive definite. Taking a given signature as a condition considerably complicates the problem and requires a basically new approach for its solution. It was proposed in the papers of the author [5– 11], where the problem of defining all pseudo-Riemannian manifolds with Lorentz signature (+ − . . .−) (Lorentz manifolds) of dimension n ≥ 3 admitting nonhomothetic infinitesimal projective and affine transformations was solved, and for each of them, the maximal projective and affine Lie algebras, together with the homothetic and isometric subalgebras, were defined. This paper includes a survey of the results in the theory of projective transformations of pseudoRiemannian manifolds, in particular, the solution of the classical geometrical problem of determining the pseudo-Riemannian metrics with corresponding geodesics (Sec. 5) and the Lie problem (Sec. 6). By using the technique of skew-normal frames developed by the author in [31] (see Sec. 1), all two-dimensional