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35.2 Recurrent and symmetric spaces 557The canonical form (cp. Table 4.2)35.2.2 Space-times of Petrov type DC ∗ abcd =2Ψ 2 (V ab U cd + U ab V cd + W ab W cd ) (35.25)of a type D Weyl tensor is compatible with (35.20) only ifΨ 2,e =Ψ 2 K e , U ab;e W ab = V ab;e W ab = 0 (35.26)holds. Equation (35.26) implies W ab = const, which means that a type Dcomplex recurrent space-time is necessarily decomposable and a product(35.11) of two two-dimensional spaces.Because of (35.20) and (35.26), C ∗ abcd;[ef]= 0 holds. Inserting (35.25)and using the decomposition (3.45) of the curvature tensor one obtainsΨ 2 = −R/12, 4E abcd = EW ab W cd ,4R ab = −(R + E)(l a k b + k a l b )+(R − E)(m a m b + m a m b ).(35.27)One sees that the recurrence vector K e =Ψ 2,e /Ψ 2 is real; all complexrecurrent space-times of type D are conformally recurrent. Furthermore,(35.27) shows that the scalar curvatures of the (m, m)- and the (k, l)-spaces are given by(1)R(x 1 ,x 2 )=(R − E)/4, (2)R(x 3 ,x 4 )=(R + E)/4. (35.28)If the space is recurrent, with K e non-zero, then R 2 = E 2 holds,i.e. the space-time is a product of a flat and a curved two-dimensionalspace.For a conformally symmetric space-time, R is constant, and the full setof Bianchi identities yields E = const: the space is symmetric and has theline element (12.8),ds 2 =2dζdζ2du dv[] 2−1+(R − E)ζζ/8 [1 − (R + E)uv/8] 2 . (35.29)35.2.3 Space-times of type NInserting the canonical form Cabcd ∗ = −4V abV cd of a Weyl tensor of typeN into (35.20), one obtains k a;b = k a p b , the (eigen-) null vector k isrecurrent. Conversely, these two relations yield (35.20): a space-time oftype N is complex recurrent exactly if it contains a recurrent null vector.