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the space-time metric15.1 Metric, Killingvectors, and Ricci tensor 227ds 2 = Y 2 (u, v)[(dx 1 ) 2 +Σ 2 (x 1 ,k)(dx 2 ) 2 ] − 2G(u, v)du dv (15.4)(cp. (15.18) and (15.24) below).In the S 2 case, we have to distinguish between Y ,m Y ,m > 0(R-region),Y ,m Y ,m < 0(T -region) (see McVittie and Wiltshire (1975) and referencescited therein) and Y ,m Y ,m = 0. The case Y ,m Y ,m < 0 cannot occur fortimelike orbits T 2 , because of the Lorentzian signature of the metric.Coordinate transformations which preserve the form (15.3) of the metriccan be used to reduce the number of functions in the line element(15.3). For instance, for Y ,a Y ,a ̸= 0 we can always setY = x 3 e λ (isotropiccoordinates), and for Y ,m Y ,m > 0 we can choose Y = x 3 (canonicalcoordinates).In the coordinate system (15.3), the Killing vectors ξ A are given byξ 1 = cos x 2 ∂ 1 − sin x 2 Σ ,1 Σ −1 ∂ 2 , ξ 2 = ∂ 2 ,ξ 3 =sinx 2 ∂ 1 + cos x 2 Σ ,1 Σ −1 ∂ 2(15.5a)for spacelike orbits andξ 1 = cosh x 2 ∂ 1 − sinh x 2 Σ ,1 Σ −1 ∂ 2 , ξ 2 = ∂ 2 ,ξ 3 = − sinh x 2 ∂ 1 + cosh x 2 Σ ,1 Σ −1 ∂ 2(15.5b)for timelike orbits. The group types are: for S 2 , IX (k = 1), VII 0 (k = 0),VIII (k = −1), and for T 2 , VIII (k = 1), VI 0 (k = 0), and VIII (k =−1). These Bianchi types belong to class G 3 A ( §8.2). The spacelike andtimelike metrics on V 2 , and their corresponding groups, are related bycomplex transformations.The existence of a higher-dimensional group of motions, G r , r > 3,imposes further restrictions on the functions ν, λ and Y in the metric(15.3). The de Sitter universe (8.34), the hypersurface-homogeneousspace-times (13.1)–(13.3) with a spatial rotation isotropy, the Friedmannmodels (§14.2), the Kantowski–Sachs solutions (14.15), and the staticspherically-symmetric perfect fluid solutions (§16.1) are solutions admittinga G r , r>3, with a subgroup G 3 on V 2 . For certain Ricci tensortypes, a group G 3 on V 2 implies a G 4 on V 3 (§15.4).For the metric (15.3), the non-zero components G b a of the Einstein tensorare (prime and dot denote differentiation with respect to the coordinatesx 3 and x 4 respectively)G 4 4 = − kY 2 + 2 (Y e−2λ Y ′′ − Y ′ λ ′ + Y ′22Y)∓ 2 Y e−2ν (Ẏ ˙λ + Ẏ 22Y), (15.6a)