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556 35 Special vector and tensor fieldsConstant null bivectors A null bivector F ab can be written in the formF ab = p a k b − p b k a , p n p n =1, k a k a =0, p n k n =0. (35.18)If F ab is constant, so is F ab F c b = k a k c , which implies k a;b =0. It can beshown (Ehlers and Kundt 1962) that a V 4 admits a constant null bivectorif and only if it is a pp-wave (§24.5)ds 2 =dx 2 +dy 2 − 2du dv − 2H(x, y, u)du 2 , k a = −u ,a . (35.19)35.2 Complex recurrent, conformally recurrent, recurrent andsymmetric spaces35.2.1 The definitionsA complex recurrent space-time V 4 is a space for which the self-dual Weyltensor (3.53) satisfies the conditionC ∗ abcd;e = C ∗ abcdK e . (35.20)Generally, the recurrence vector K e is complex; if it is real, then the spaceisaconformally recurrent spaceC abcd;e = C abcd K e , (35.21)and if K e is zero, one gets a conformally symmetric space.A recurrent space is a space in which the Riemann tensor satisfiesR abcd;e = R abcd K e . (35.22)For a recurrent space, the identities R abcd;[mn] +R cdmn;[ab] +R mnab;[cd] =0yield K e = K ,e . If instead of (35.22) onlyR ac;e = R ac K e (35.23)holds, the space is Ricci recurrent (for which see Hall 1976b).A recurrent space is said to be symmetric if K e vanishes,R abcd;e =0. (35.24)Obviously, each recurrent (symmetric) space-time is conformally recurrent(conformally symmetric) and hence a complex recurrent space too.Using the canonical forms (§4.2) of the self-dual Weyl tensor C ∗ abcdand evaluating (35.20), it can easily be shown that there are no complexrecurrent space-times of Petrov types I,II and III. Therefore, we haveonly to deal with types D, N and O. We shall list the main results in thefollowing, only indicating the ideas of the proofs which can be found inSciama (1961), Kaigorodov (1971), McLenaghan and Leroy (1972) andthe references given there.