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564 35 Special vector and tensor fields35.4 Collineations and conformal motionsBesides Killing vectors (motions)35.4.1 The basic definitionsand homothetic vectors (homothetic motions)L ξ g nm = ξ n;m + ξ m;n = 0 (35.57)L ξ g nm =2ag nm , a = const, (35.58)which originate in the symmetries of Einstein’s equations in the processof a similarity reduction (cp. §10.2.3), there are several other vector fieldswhich originate in the symmetries of e.g. the equation of geodesic motion(so that their existence puts severe restrictions upon space-time) andwhich can also be used to characterize space-times. These areconformal motions: L ξ g nm = 2φ(x)g nm , (35.59)projective collineations: L ξ Γ i jk = δ i jϕ ,k + δ i kϕ ,j , (35.60)affine collineations: L ξ Γ i jk = 0, (35.61)Ricci collineations: L ξ R nm = 0, (35.62)curvature collineations: L ξ R a bnm = 0. (35.63)Conformal motions (Petrov 1966) preserve angles between two directionsat a point and map null geodesics into null geodesics. Projectivecollineations map geodesics into geodesics; affine collineations preserve,in addition, the affine parameters on geodesics (Katzin and Levine 1972).Obviously, motions, affine collineations and homothetic motions are automaticallycurvature collineations. The connection between those vectorfields and geodesic first integrals is discussed e.g. in Katzin andLevine (1981). A collection of formulae on the incorporation of all thesesymmetries into the field equations can be found in Zafiris (1997).For perfect fluids, the notion of a homothetic vector has been generalizedto that of a similarity vector (kinematic self-similarity) satisfyingL ξ (g mn + u n u m )=2a(g mn + u n u m ), L ξ u n = αu n , a, α = const,(35.64)which for α = a is equivalent to (35.58). For applications see Carter andHenriksen (1989), Coley (1997a), Benoit and Coley (1998), and Sintes(1998), and for a further generalization Collins and Lang (1987).We will discuss in some detail only the proper curvature collineationsand the conformal motions. Definition (35.61) implies that proper affine