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126 9 Invariants and the characterization of geometriesof N ), one obtains the basis 1-forms from ω K = I K βdI α , where the componentsI K β form the inverse of I α |L, and the metric is then given byη ij ω i ∧ ω j (Karlhede 1980a). If the matrix (I α |L) is not invertible thereis a continuous symmetry: if the rank of (I α |L) (the number of indexsets in A) isk, one has to give (dim N ) − k 1-forms ω P , introducing additionalcoordinates on N , subject to d 2 ω P = 0 and then one can findω A = I A α(dI α −I α |P ω P ) for 1 ≤ A ≤ k using the inverse of I β A (Bradleyand Karlhede 1990, Bradley and Marklund 1996). When using a reducedframe bundle with an assumed symmetry group, one can give values ina fixed frame together with the generators of the rotations and appropriatelyspecialize the method just outlined. This method has mainly beenapplied to recovering already known solutions (Karlhede and Lindström1983, Bradley and Karlhede 1990, Bradley 1986) but some new solutionshave been found also (see Bradley and Marklund (1996), Marklund (1997),Marklund and Bradley (1999)).A considerable number of papers report applications of the aboveCartan equivalence procedure (see e.g. the references in MacCallum andSkea (1994)), some of them cited in Table 9.1. Special adaptations of themethods can be made for classes of metrics with special properties. Forexample, the spatially homogeneous Bianchi metrics can be dealt within terms of the automorphism group variables described in Chapter 13(Araujo and Skea 1988a) and the subset of metrics with two commutingKilling vectors which are computationally tractable can be widened usingthe factorization method of §20.7 (Seixas 1992a), although such metricscan be made arbitrarily complicated and intractable by repeated applicationof the generating techniques (Chapter 34).These methods can also be applied to classification of the potentialspaces arising in Hamiltonian descriptions, e.g. of those for space-timeswith a G 3 on S 3 (Chapters 13 and 14) or H 3 on T 3 containing a G 2 Ion S 2 (Uggla et al. 1995a). The potential spaces may themselves havesymmetries (see §34.1 and Uggla et al. (1995a)). The general Cartanmethod has been used to classify the null bundle of space-time (Nurowskiet al. 1999), leading to a classification procedure for non-conformally-flatEinstein spaces, and null hypersurfaces (Nurowski and Robinson 2000).9.5 Limits of families of space-timesIt is possible in principle that one could find new solutions of Einstein’sequations as limits of known solutions. In practice, since the limits aregenerally simpler than the space-times in the parent families, this rarelyhappens: the limits are usually already known. The general situation forthe case where the limit is a non-singular (region of) space-time was