Projective transformations of pseudo-Riemannian manifolds
type: Item
instance of: scholarly article
title: Projective transformations of pseudo-Riemannian manifolds
zbMATH Open document ID: 1043.53054
DOI: 10.1023/A:1021041802041
author: Yanyan Li
published in: Journal of Mathematical Sciences (New York)
publication date: 2003-01-01T00:00:00Z
review text: This paper is a survey of the results on the theory of projective transformations of pseudo-Riemannian manifolds. A projective transformation of a pseudo-Riemannian manifold \(M^ n\) is an automorphism of the induced Riemannian connection of the projective structure that takes geodesics of \(M^ n\) into geodesics. The theory of projective transformations on spaces endowed with a linear connection was developed by É.~Cartan, I.~P.~Egorov, L.~P.~Eisenhart, G.~Fubini, M.~S.~Knebelman, S.~Kobayashi, S.~Lie, I.~ Mikesh, I.~A.~Schouten, P.~A.~Shirokov, N.~S.~Sinyukov, A.~S.~Solodovnikov, T.~Y.~ Thomas, G.~G.~Vranceanu, and others. In particular, A.~S.~Solodovnikov, following G.~Fubini's papers, found a classification of the Riemannian spaces \(M^ n, n\geq 3\), in terms of local groups of projective transformations. The present author [Tr. Geom. Semin. 6, 295-316 (1974; Zbl 0309.53046)] proposed a new approach for solution of this problem when the metric is not positive definite. She determined all pseudo-Riemannian manifolds with Lorentz signature \((+,-,\dots,-)\) (Lorentz manifolds) of dimension \(n \geq 3\) admitting nonhomothetic infinitesimal projective and affine transformations, and for each of them the maximal projective and affine Lie algebras along with the homothetic and isometric subalgebras were found. The author finds the solution of the classical geometrical problem of determining the pseudo-Riemannian metrics with corresponding geodesics and the Lie problem. By using the technique of skew-normal frames [Sov. Math. 26, No. 6, 76--81 (1982; Zbl 0509.53019)] she determines all 2-dimensional pseudo-Riemannian manifolds that admit Lie algebras of infinitesimal projective transformations. Special attention is given to concircular geometry, which is closely related to projective transformations, as well to applications in physics, mechanics, and the theory of differential equations. Finally, the author studies the projective geometry of differential systems. She …
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 53B10
zbMATH DE Number: 1897076
zbMATH Keywords: projective transformation
zbMATH Keywords: infinitesimal projective transformation
zbMATH Keywords: concircular geometry
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1897076