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110 8 Continuous groups of transformationsThe result of Theorem 8.16 is a special case of a phenomenon knownas orthogonal transitivity. This occurs when the orbits of a group of motionsare submanifolds of a V n which have orthogonal surfaces and is ofparticular interest for space-times admitting a group G 2 I (see Chapter 17et seq.). Schmidt proved a number of further theorems on this matter,includingTheorem 8.19 If a group G r of motions of r = 1 2d(d +1), (d >1),parameters has orbits of dimension d the orbits admit orthogonal surfaces(Schmidt 1967).8.7 Homothety groupsThe equation defining a homothetic (or ‘homothetic Killing’) vector,(L ξ g) ab = ξ a;b + ξ b;a =2Φg ab , Φ = const, (8.63)implies related equations for other geometrically defined tensors, e.g.(L ξ R) a bcd =0, L ξ R = −2ΦR, (8.64)and conversely these equations give a sequence of integrability conditionsin the same way as (8.23). Note that from (8.64) Einstein spaces with Λ ̸=0 cannot admit proper homothetic motions. In classifying homotheties,the ‘homothetic bivector’ ξ [a,b] is of importance; for example, if it vanishesand ξ is null, the space-time must be algebraically special (McIntosh andvan Leeuwen 1982).Now consider a homothety group (i.e. a Lie group each of whose elementsis a homothety). A basis of its generators will obey(L ξAg) ab =Φ A g ab , (8.65)where each Φ A is a constant, possibly 0, and in general Φ A ̸= Φ B ifA ̸=B. Since the Lie derivative is linear (over R) in the vector field used,the generators w = C A ξ A satisfy Lwg ab =(σ c w c )g ab for some 1-formσ. Generators satisfying σ c w c = 0 are isometries and so a space-timeadmitting a group H r of homothetic motions necessarily admits a groupG r−1 of motions. From (2.63) the commutator of any two homotheties orisometries must be an isometry, so the G r−1 is an invariant subgroup ofthe H r . The structure constants of the basis (8.65) must satisfyC A BCΦ A =0, (8.66)(Yano 1955) and so dσ = 0. The generators of the H n can thus be chosenso that only one of them is a proper homothety (i.e. a homothetic motion