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44 3 Some topics in Riemannian geometryto point,ĝ ab =e 2U g ab , ĝ ab =e −2U g ab , U = U(x n ). (3.83)The connection coefficients and the covariant derivative of a 1-form σ aretransformed tôΓ c ab =Γ c ab +2δ c (a U ,b) − g abU ,c , U ,c ≡ g cd U ,d ,̂∇ âσ b = ∇ a σ b − U ,b σ a − U ,a σ b + g ab U ,c σ c , ̂σ a = σ a .(3.84)The curvature tensors of the two spaces with metrics ĝ ab and g ab areconnected by the relatione 2U ̂R da bc = R da bc +4Y [a[b δd] c] ,Y a b ≡ U ,a ;b − U ,a U ,b + 1 2 δa bU ,e U ,e ,(3.85)which holds for n-dimensional Riemannian spaces ̂V n and V n (the covariantderivative is taken with respect to g ab ). From (3.85) one obtains theequationĉR ab = R ab +(2− n)Y ab − g ab Y c (3.86)for the Ricci tensors in ̂V n and V n . In three dimensions, this equationtakes the form̂R αβ = R αβ − U ,α;β + U ,α U ,β − g αβ (U ,γ ;γ + U ,γ U ,γ ). (3.87)The application of (3.86) to a (flat) space V 2 yieldsdŝ 2 =e 2U (dx 2 +dy 2 ): ̂R AB = Kĝ AB , K = −e −2U U ,A ,A. (3.88)A space is called conformally flat if it can be related, by a conformaltransformation, to flat space. A space V 2 is always conformally flat. Aspace V 3 is conformally flat if and only if the Cotton tensorC a bc ≡ 2(R a [b − 1 4 Rδa [b) ;c] (3.89)vanishes, see Schouten (1954); York (1971) defined a related conformallyinvariant tensor density. A space V n , n>3, is conformally flat if and onlyif the conformal tensorC abcd ≡ R abcd + R(g ac g bd − g ad g bc )/(n − 1)(n − 2)− (g ac R bd − g bc R ad + g bd R ac − g ad R bc )/(n − 2) (3.90)(for V 4 , see (3.50)) vanishes. The conformal tensor components C a bcd areunchanged by (3.83).