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102 8 Continuous groups of transformationson b and d, yield (n − 2)(K ,a g ce − K ,c g ae ) = 0, and contracting again ona and e gives (n − 2)(n − 1)K ,c = 0. Thus, if n ≥ 3, K is constant andthe space admits a G r (r = 1 2n(n + 1)) of motions. Conversely if a spaceV n admits a G r (r = 1 2 n(n + 1)) of motions then by (8.20) it admits an I s(s = 1 2n(n − 1)) of isotropies and is thus of constant curvature if n ≥ 3.If n =2,aG 2 (or G 3 ) of motions must be transitive and then L ξ R =0for the Riemann tensor leads to K = constant.Collecting together these arguments we find we have provedTheorem 8.13 A Riemannian space is of constant curvature if and onlyif it (locally) admits a group G r of motions with r = 1 2n(n +1).Theorem 8.14 A Riemannian space V n (n ≥ 3) is of constant curvatureif and only if it (locally) admits an isotropy group I s of s = 1 2n(n − 1)parameters at each point.Theorem 8.15 A two-dimensional Riemannian space admittinga G 2 ofmotions admits a G 3 of motions.Substituting (8.24) into the definition (3.50) of the Weyl tensor we findC a bcd = 0, and so (3.85) can be solved to find the factor e 2U in (3.83)relating the metric to that of a flat space of the same dimension andsignature, ˚g ab = diag(ε 1 ,...,ε n ), where ε 1 ,...,ε n = ±1 as appropriate.Equations (3.85) are satisfied if2(e −U ) ,ab = K˚g ab , (e −U ) ,a (e −U ) ,a = K(e −U − 1), (8.27)the solution of which can be transformed toe −U =1+ 1 4 K˚g abx a x b . (8.28)Hence the metric of a space V n of constant curvature can always be writtenasds 2 dx a dx a=) 2(8.29)(1+ 1 4 Kx bx b(indices raised and lowered with ˚g ab ), for any value of K or signature ofV n , and any two metrics of the same constant curvature and signaturemust be locally equivalent.A space V n of non-zero constant curvature, K ̸= 0, can be consideredas a hypersurfaceZ a Z a + k(Z n+1 ) 2 = kY 2 , K = kY −2 , k = ±1, (8.30)inan(n + 1)-dimensional pseudo-Euclidean space with metricds 2 =dZ a dZ a + k(dZ n+1 ) 2 . (8.31)