DEFORMATIONS BETWEEN BIANCHI GEOMETRIES IN CLASSICAL AND QUANTUM COSMOLOGY M. MOHAZZAB yz, M. RAINER xy, and H.-J. SCHMIDT x x Projektgruppe Kosmologie Institut fur Mathematik, Universitat Potsdam P.O.Box 601553, D-14415 Potsdam, Germany y Institute for Studies in Theoretical Physics and Mathematics P.O.Box 5746, Tehran 19395, Iran z Physics Dept. Alzahra University Tehran 19834, Iran 1
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Abstract
We systematically investigate the possible transitions between classical homogeneous Riemannian 3-hypersurfaces of cosmological models on one hand, and between homogeneous Lorentzian minisuperspace 3-geometries on the other hand. For the Riemannian case, in contrast to our earlier approaches where we only evaluated the three scalar invariants from the Ricci tensor, here we additionally evaluate a scalar invariant constructed from the rst covariant derivative of the Ricci tensor. Coincidence of these four invariants implies local isometry in the set of homogeneous Riemannian 3-manifolds, whereas the three eigenvalues of the Ricci tensor do not suce. Possible deformations between homogeneous 3-dimensional factor spaces can be classi ed this way. The analogous classi cation in the Lorentzian case gives a di erent result for each signature of the metric. This can be applied to investigate homogeneous deformations of conformal classes of 3-dimensional minisuperspaces. The latter are a fundamental ingredient for the canonical quantization of a factor space cosmology. 1 2 3
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This work was nancially supported by the DAAD and DFG grant Schm 911/6-2
1
1 Introduction
Multidimensional space-times M = IRM Mn, composed from a nite number n of homogeneous Riemannian factor spaces Mi, i = 1; : : : ; n have received some interest recently [?]-[?], both as classical cosmological models including additional internal spaces (generalizing the original Kaluza-Klein idea), and also regarding the canonical quantization on the n-dimensional minisuperspaces M associated to them. At any time t 2 IR, the factor M (t) of M should be a homogeneous Riemannian 3-space corresponding to a spacelike hypersurface of IR M , although (since we do not impose any constraint on the second fundamental form of the hypersurfaces M in IR M ) for t 6= t the factor M (t ) might be di erent from M (t ). The latter di erence might then either be due to di erent anisotropies (i.e. di erent scales in di erent directions) or even due to a change of their transitive isometry subgroup. The latter is known either to be simply transitive like for the Bianchi types or to be the 4-dimensional one of Kantowski-Sachs type. The canonical quantization of the dilatonic degrees of freedom of a cosmological model M with several factor spaces can be performed conformally equivariant on the corresponding minisuperspace M (see e.g. [?]). For a cosmological model with 2 internal factor spaces, M and M , M is a homogeneous Lorentzian 3-space. For a direct product M M M , M is the at 3-dimensional Minkowski space. If M M M is a more general warped product, then the associated scale factors ai = e i remain independent variables, but the operators @ @ i no longer commute with each other and M will be curved. So, a speci cally interesting class of geometries on M are the homogeneous Lorentzian ones. Like the corresponding Riemannian geometries, the latter admit either a simply transitive isometry subgroup of some Bianchi type or to the 4-dimensional Kantowski-Sachs isometry. Furthermore they also allow di erent anisotropies. However, unlike than in the Riemannian case, here the 3 directions of the triads carry di erent signs. Hence, besides a global sign changing the determinant of the metric, we have to consider 3 di erent possibilities of signature that can be assigned to the directions of the Lie algebra generators. So, homogeneous 3-spaces appear in both, classical and quantum cosmological models. For both kind of signatures, a homogeneous 3-space is either of KantowskiSachs type or it has a Bianchi geometry, admitting a transitive isometry Lie group of Bianchi type I-IX. The limit transitions between associated real 3-dimensional Lie algebras are classi ed systematically by a natural topology of the space K of all such Lie algebras. For a xed choice of the signature, the homogeneous 3-spaces corresponding to the Bianchi types can be classi ed by their characteristic scalar invariants which, besides the usual invariants R; N; S from the Ricci tensor, essentially include 1
1
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3
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a fourth invariant Y related to the York tensor. The invariants R; N; S; Y are calculated with an orthogonal frame triad and the structure constants of the corresponding Lie algebra of the Bianchi type. In the space of scalar invariants, the di erent Bianchi types occupy characteristic regions, which are related to each other in agreement with K . In Sec. 2 we describe this procedure in general, independently of the signature of the metric. In Sec. 3 we present the resulting scalar invariants for all non- at homogeneous Bianchi geometries. In 3.1 the Riemannian case is analysed in detail. For Lorentzian signatures, in 3.2 partial results are given and compared to the Riemannian case. Sec. 4 brie y shows the possible application of these results to a generalized factor space cosmology and its minisuperspace quantization. Sec. 5 resumes the results. 3
2 Bianchi geometries When we consider homogeneous Riemannian 3-manifolds, the Kantowski-Sachs spaces are the only ones, which do not admit a 3-dimensional transitive subgroup of their isometry group. The latter then is 4-dimensional, with corresponding Lie algebra IX IR. For all other Riemannian homogeneous 3-spaces there exists a 3-dimensional subgroup of the isometry group. This 3-dimensional Lie group G is given by one of the Bianchi types I-IX. Actually the Kantowski-Sachs spaces appear as limit cases of certain Bianchi IX spaces. Hence, in order to study homogeneous 3-spaces, we can restrict to the Bianchi geometries. The following considerations refer to both, Riemannian and pseudo-Riemannian homogeneous 3-spaces. The Lie algebra A of a Bianchi group G is determined by the commutators [ea; eb] = Cabc ec (2.1) of its generators ea. These may be represented w.r.t. coordinates fx g as leftinvariant vector elds, ea = e a @x@ : (2.2) Then, their duals are the triad 1-forms 3
3
ea = ea dx ; 3
3
(2.3)
and the coordinate components of the rst fundamental form ds = dx dx may be expressed by the anholonomic components w.r.t. the triad, = gabea eb : (2.4) While depend in general on x , we can choose the anholonomic representation without loss of generality such that gab is constant and diagonal, i.e. with s; t; w 2 IR and i 2 f1; ?1g 3 2 s e 0 0 7 6 (2.5) (gab ) = 664 0 es w?t 0 775 : 0 0 es?t In the case of Riemannian 3-spaces gab 1is de nite. This corresponds to an orthog1 s w ?t 1 s?t s 2 2 2 ;e de ne the linear scales onal triad frame of reference, where e ; e of measurement in the 3 orthogonal directions. In the Lorentzian case the corresponding frame is pseudo-orthogonal. The real parameters s; t; w are adapted to simplify the following calculations in a speci c choice of the triad basis which is consistent with [?] (see Sec. 8.6, p. 106, cf. also [?]). The Lie algebras of Bianchi type are represented by matrices (ea ) with anholonomic coordinate rows of generator index a = 1; 2; 3 and columns = 1; 2; 3 w.r.t. holonomic coordinates x , denoted also as x =: x; x =: y; x =: z. Bianchi I: 3 2 1 7 6 (2.6) (ea ) = 664 1 775 ; 1 Bianchi II: 3 2 1 ?z 7 6 7 7; (2.7) (ea ) = 664 0 1 5 1 Bianchi IV: 3 2 1 7 6 (2.8) (ea ) = 664 ex 0 775 ; 2
1
2
+
3
( +
)
1
Bianchi V:
(
2
)
3
xex ex
2 6 (ea ) = 664
1
4
ex
ex
3 7 7 7; 5
(2.9)
Bianchi VIh , h = A : 2
2 6 (ea ) = 664
Bianchi VIIh, h = A :
1
eAx cosh x ?eAx sinh x ?eAx sinh x eAx cosh x
3 7 7 7; 5
(2.10)
2
Bianchi VIII:
Bianchi IX:
3 7 7 7; 5
2 6 (ea ) = 664
1
2 6 (ea ) = 664
cosh y cos z ? sin z 0 7 cosh y sin z cos z 0 775 ; sinh y 0 1
eAx cos x ?eAx sin x eAx sin x eAx cos x
(2.11)
3
(2.12)
3
2 6 (ea ) = 664
cos y cos z ? sin z 0 7 cos y sin z cos z 0 775 : (2.13) ? sin y 0 1 p For each of the lines fVIhg and fVIIhg the parameter range is given by h = A 2 [0; 1[. Let us also remind that, III := VII . The structure constants and metrical connection coecients could be calculated from ds and the triad. 1
2
Cijk = ds ([ei; ej ]; ek ); 2
Cijk = Cijr grk ;
(2.14)
?kij = 21 gkr (Cijr + Cjri + Cirj ): W.r.t. the triad basis, the curvature operator is de ned as
0 for any gab, Riemannian or Lorentzian. Then for the latter case there remain 3 subcases of signature := ( ; ; ) to be studied: = (+ ? ?); (? + ?); (? ? +), cf. Eq. (??). For the gures in the following psections, p ^ we also consider the^ rescaled cube ^ spanned by the scalar invariants R= 3; 6S 2 [?1; 1] and 2 tanh Y 2 [0; 2]. 1
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3.1 Riemannian geometries
First we list the scalar invariants for all non- at Riemannian Bianchi geometries. p R^ II = ? 33 p 16 S^II = 81 3 p ^YII = 8 3 (3.1) 9 7
12 ew + 1 48 e w + 16 ew + 3 16 + 72 ew S^IV = 9 (48 e w + 16 ew + 3) = w ew Y^IV = 8 +w8 e + 32 (48 e + 16 ew + 3) =
R^IV = ? p
2
3 2
2
2
(3.2)
3 2
2
p
R^ V = ? 3 S^V = 0 Y^V = 0
(3.3)
In the next formulas, D represents an expression to simplify the following equations.
DVIh := 3 + 4 (4 h + 1) ew + 2 1 + 16 h + 24 h e +4 (4 h + 1)e w + 3 e w R^VIh = ?(DVIh )? 21 1 + 2 (1 + 6 h) ew + e w 3
2
2
w
4
2
S^VIh = 98 (DVIh )? 23 (ew + 1) 2 + (9 h ? 5) ew + 2 e Y^VIh = 8 (DVIh )? 23 (ew + 1) e w + (h ? 1) e w +2 2 h ? 5 h + 2 e w + (h ? 1) ew + 1 4
2
2
4
2
w
3
(3.4)
2
DVIIh := 3 + 4 (4 h ? 1) ew + 2 1 ? 16 h + 24 h e w +4 (4 h ? 1) e w + 3 e w R^VIIh = ?(DVIIh )? 21 1 + 2 (6 h ? 1) ew + e w S^VIIh = 89 (DVIIh )? 32 (ew ? 1) 2 + (9 h + 5) ew + 2 e Y^VIIh = 8 (DVIIh )? 23 (ew ? 1) e w + (h + 1) e w +2 2 h + 5 h + 2 e w + (h + 1) ew + 1 3
2
2
4
2
4
2
2
4
2
w
3
(3.5)
2
DVIII := 2 e? w t + 3 e? w t + 4 e? w t ? 4 et + 4 ew + 3 e w +4 e?w + 3 e? w ? 4 e?w t + 4 e? w t + 2 + 4 e?w t ?4 ew t ? 4 e?w t + 2 e t 2
+2
2
+4
2
2
+
+2
+3
2
8
+
2
2
+3
+
R^VIII = ?(DVIII)? 12 (2 e?w t + e?w + e?w t + ew + 2 ? 2 et) S^VIII = ? 98 (DVIII)? 23 6 ew t + 3 et w + 3 e t? w + 14 e? w t +6 e? w t + 6 e? w t ? 3 e? w t + 6 e?w t ? 14 e t ?3 e? w t ? 2 e w ? 2 e? w ? 15 e?w t ? 3 e? w +15 e? w t + 18 ew t ? 15 e?w t ? 15 e t ? 3 e w +15 et + 18 e? w t ? 18 e? w t + 6 e?w ? 42 e?w t +6 ew ? 15 e? w t ? 2 e? w t + 14 Y^VIII = 8 (DVIII)? 32 3 ew t ? et w ? e t? w + 6 e? w t +3 e? w t + 3 e? w t + e? w t + 3 e?w t ? 6 e t +e? w t + e w + e? w ? 6 e?w t + e? w + 6 e? w t +5 ew t ? 6 e?w t ? 6 e t + e w + 6 et + 5 e? w t ?5 e? w t +3 e?w ? 18 e?w t + 3 ew ? 6 e? w t + e? w t + 6 (3.6) +
3
+2
3
+5
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+2
+2
+4
3
3
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+2
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+2
+5
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+6
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+
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+4
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+
+6
DIX := 2 + 3 e w + 3 e? w ? 4 e?w ? 4 e? w t + 3 e? w t + 2 e t +2 e? w t + 4 e?w t ? 4 e? w t ? 4 ew + 4 et +4 e?w t ? 4 e?w t ? 4 ew t R^IX = (DIX)? 21 ?ew ? e?w t ? e?w + 2 e?w t + 2 + 2 et S^IX = ? 98 (DIX)? 23 6 ew t + 3 et w + 3 e t? w ? 14 e? w t +6 e? w t + 6 e? w t + 3 e? w t + 6 e?w t ? 14 e t +3 e? w t ? 2 e w ? 2 e? w + 15 e?w t + 3 e? w +15 e? w t ? 18 ew t + 15 e?w t + 15 e t + 3 e w +15 et ? 18 e? w t ? 18 e? w t + 6 e?w ? 42 e?w t +6 ew + 15 e? w t ? 2 e? w t ? 14 Y^IX = ?8 (DIX)? 23 ?3 ew t + et w + e t? w + 6 e? w t ?3 e? w t ? 3 e? w t + e? w t ? 3 e?w t + 6 e t +e? w t ? e w ? e? w ? 6 e?w t + e? w ? 6 e? w t +5 ew t ? 6 e?w t ? 6 e t + e w ? 6 et + 5 e? w t + 5 e? w t ?3 e?w + 18 e?w t ? 3 ew ? 6 e? w t ? e? w t + 6 (3.7) In order to verify that, for given (R; N; S; Y ) there exists only one Bianchi geometry, we x N andpexamine of any Bianchi geometry in the cube p ^ the position ^ ^ scalar invariants (R= 3; 6S; 2 tanh Y ). Figs. 1 and 2 below show the points corresponding to Bianchi types VIh=VIIh and VIII/IX respectively. 2
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At the branch points of the curve L the Ricci tensor has a triple eigenvalue, which is negative for geometries of Bianchi type V, and positive for type IX geometries with parameters (t; w) = (0; 0). These constant curvature geometries are all conformally at with Y^ = 0. Besides the at Bianchi I geometry, the remaining p pcon^ ^ ^ ^ formally at spaces with Y are the Kantowski-Sachs space (R; S; Y ) = ( 2; ? ; 0) and, point re ected, the Bianchi type IIIc , corresponding to the end point of the line of Bianchi III geometries in Fig. 2. In Fig. 1 the point (?1; 0; 0) admits both types, Bianchi V and VIIh with h > 0. Nevertheless, this point corresponds only to one homogeneous space, namely the space of constant negative curvature. This is possible, because this space has a 6-dimensional Lie group, which contains the Bianchi V and VIIh subgroups. Note that in the at limit V ! I, the additional Bianchi groups VIIh change with h ! 0. Similarly in Fig. 2, the points which admit Bianchi III lie on the curve L of the R^ ? S^ diagram. However, these points are also of Bianchi type VIII. In fact, each of them correspond to one homogeneous geometry only. However, the latter admits a 4-dimensional isometry group, which has two 3-dimensional subgroups, namely Bianchi III and VIII, both containing the same 2-dimensional non-Abelian subgroup. Altogether, the location of Riemannian Bianchi (and KS) spaces is consistent with the topology of the space of 3-dimensional real Lie algebras (see [?]-[?]). 2
2 18
2
3.2 Lorentzian geometries
For the pseudo-Riemannian homogeneous spaces, the analysis can be done quite analogously, but up to now, only partial results are known (see e.g. [?]). It is not yet clear in general, under which condition the 3 subcases mentioned at the beginning of Sec. 3 yield isometric spaces. A further problem is that here some of the invariants (??) might become singular for some non- at spaces with N = 0. Note also that the Lorentzian at space accommodates not only the transitive Abelian Bianchi type I and the isometry group VII of its space-like hyperplanes, but also further left-invariant Bianchi groups, which do not appear in the at Riemannian case (cf. also [?]): Naturally, one of them is the isometry group VI of its Minkowskian hyperplanes. Further admissible Bianchi groups on the Minkowski space are the types II and V. Actually Lorentzian Bianchi type II geometries are completely known: There exists a at left-invariant Lorentz metric, and all non- at ones are homothetically equivalent to each other. Non- at Lorentzian Bianchi II geometries are in all 3 signature cases the same as the Riemannian ones, Eq. (??). Bianchi geometries with signature = (+ ? ?) of type IV, V, VIh and VIIh are the same as the Riemannian ones, Eqs. (??), (??), (??) and (??), respectively. 0
0
10
Similarly, the analogous geometries of signature = (? + ?) and = (?? +) have ^ S; ^ Y^ , although here these are di erent from Eqs. both exactly the same values of R; (??), (??), (??) and (??), respectively. For type V this p from constant negative p di erence is just given by a re ection ^ ^ curvature R = ? 3 to constant positive curvature R = 3. Although for Bianchi type V the value of R^ changes with the signature , its property Y^ = 0 is preserved for all 3 Lorentzian signatures as in the Riemannian case. Actually Lorentzian Bianchi V geometries are known to exist for any constant curvature, including the
at case [?], which however is excluded for our invariants (??). According to the natural topology in the space K of real Lie algebras, type IV interpolates between II and V, and both, VIh and VIIh, converge to IV for h ! 1. So IV, VIh and VIIh have to behave under a change of signature consistently with the xed point II and the re ection of V. Fig. 3 shows the location of Bianchi types II, IV, V, VIh, VIIh for di erent signatures. Finally Bianchi VIII and IX geometries are di erent for all signatures. Fig. 4 shows the location of Bianchi types II, V, VIa, VI , VII , VIII, IX for di erent signatures. 3
1
0
4 Application to cosmological models In this section we show up the possible application of our results to both, classical and quantum cosmological models. In the former case, the traditionally considered 1 + 3-dimensional inhomogeneous cosmological models with homogenous Riemannian 3-hypersurfaces (see [?], [?], [?]), and the more general class of multidimensional geometries, admit deformations between Riemannian Bianchi geometries. This can especially be applied to investigate the change of the spatial anisotropy of the universe, essentially a ecting physical quantities like its tunnelling rate [?]. In the case of quantum cosmological models, minisuperspaces of generalized multidimensional geometries with 2 internal factor spaces admit deformations of their Lorentzian Bianchi geometries.
4.1 Classical factor spaces
Multidimensional geometries are usually given on a multifactor manifold
M = IR M : : : Mn ; D := dim M = 1 + d + : : : + dn ; n X g ds = ?e dt dt + ai dsi : 1
1
2
2
2
i=1
11
2
(4.1)
Here ai = e i is the scale factor of the factor space Mi of dimension di 2 IN, and dsi = gkli dxki dxl i : where M is the usual external space, usually with d = 3, and M ; : : :; Mn are additional internal factorspaces. For dk = 3 any non- atphomogeneous geometry g k on Mk can change continp ^ 2 tanh Y^ ) moves continuously within the cube ^ 3; 6S; uously in time only if (R= [?1; 1] [0; 2]. A path in the latter determines a 1-parameter family of homogeneous Riemannian 3-spaces, where at certain (time-)parameter values the Bianchi isometry subgroup may change. For dim Mk > 3 we know at least that continuous deformations of 3-dimensional Bianchi subgeometries of g k induce a continuous deformation of g k itself. However there might be some 3-dimensional Bianchi subgeometries of g k which cannot be deformed into each other without passing a third Bianchi type, di erent from both of them. Nevertheless, their respective embeddings into a d -dimensional homogeneous geometry with 3 < d dk might admit a direct 1-parameter family containing no third isometry type between them. Actually, for dim Mk > 3, the complete information on all possible deformations of the geometry gk would require both, a control over the space of all dk -dimensional Lie algebras, and additional scalar invariants which involve the Weyl tensor. So, for the general multidimensional geometry, our results provide a complete control over the possible continuous deformations of the homogeneous external space M with d = 3. Furthermore we obtain a partial control over the possible continuous homogeneous deformations on the internal spaces M ; : : :; Mn . ( )
2
( )
1
( )
1
2
( )
2
( )
( )
( )
0
0
1
1
2
4.2 Quantum minisuperspaces
In order to apply our results also to quantum cosmological models, let us now consider minisuperspaces of geometries. Any multidimensional geometry M with n factor spaces has a natural n-dimensional minisuperspace M, given by the scale exponents i = ln ai, i = 1; : : : ; n. The minisuperspace coordinates ; : : :; n are subject to the principle of general covariance w.r.t. minisuperspace coordinate transformations. The minisuperspace geometry is then given by the metric G = Gij d i d j : (4.2) When the factor spaces Mi are independent of each other, i.e. at any time t the space-like hypersurface of M is given as their direct product, then f @ @ i g is a commuting set. Hence, for n > 1 the metric G on M is the at Minkowskian one, which is trivially homogeneous. (Usually, in coordinates i, i = 1; : : :; n, then Gkl = dk kl ? dk dl, with k; l = 1; : : : ; n; cf. also [?].) 1
12
Let M now be still a homogeneous Lorentzian n-space, but non- at. This means that f @ @ i g are no longer commuting, and the space-like hypersurfaces of the underlying geometry on M is now a warped product of the geometries on M ; : : :; Mn . The factor spaces Mi are no longer independent. In this case we call corresponding geometry on M a generalized multidimensional geometry. For n = 3 we have some control over the space of non- at homogeneous Lorentzian geometries G. Each of those which have a 3-dimensional transitive subgroup of the isometry group corresponds to a non- at Lorentzian Bianchi geometry. The canonical quantization scheme has been considered e.g. in [?]. In [?] it has been applied to minisuperspaces of arbitrary dimension. An essential feature there is the conformal equivariance, which essentially means that, under conformal rescaling of the minisuperspace metric G !f G = e f G the Wheeler-deWitt equation H^ = 0 transforms in a well-de ned way according to certain conformal representation given by the function f . More precisely, with conformal weight b = ? n? , any wavefunction of the functional space is transformed into f = ebf . With potential V, Laplace-Beltrami operator and the conformal coupling c = nn?? , the Wheeler-deWitt operator is given as H^ = ? [ ? c R] + V and transforms conformally to H^ f = e? f ebf H^ e?bf + V f : (4.3) Altogether, the conformally equivalent Wheeler-deWitt equation is H^ f f = 0: (4.4) 1
2
2
4(
2
2 1)
1 2
2
So it is clear that for the quantized theory only conformal equivalence classes [G] rather than minisuperspace geometries G themselves can have an invariant meaning. This implies that non- at minisuperspaces, which are conformally at, yield a quantum cosmological model which is equivalent to one on a usual Minkowskian minisuperspace. Especially in the case n = 3 there exist minisuperspaces of any Bianchi geometry. If Y^ = 0, the geometry is conformally at. This happens independently of the signature for Bianchi type V. Furthermore a Lorentzian Bianchi V minisuperspace can even be at, if also the possibility of null eigendirections of the triads is taken into account. The only invariant property of Lorentzian Bianchi V spaces is to be of constant curvature. Hence a Bianchi V minisuperspace naturally yields a quantum cosmological model, which is conformally equivalent to at quantum cosmological model. However in general neither the property Y^ = 0 nor conformal atness can be expected to be invariant under transformation between the 3 Lorentzian signatures 13
= (+ ? ?); (? + ?); (? ? +). This means that in the underlying general-
ized multidimensional geometry it might be relevant which of factorspaces carries the distinguished sign (possibly selecting the external space). More generally, the eigendirections of the triads must essentially been taken into account.
5 Conclusion We systematically investigated the possible transitions between classical homogeneous Riemannian 3-hypersurfaces of cosmological models on the one hand, and between homogeneous Lorentzian minisuperspace 3-geometries on the other hand. For the Riemannian case, in contrast to our earlier approaches where we only evaluated the three scalar invariants from the Ricci tensor, here we additionally evaluate a scalar invariant constructed from the rst covariant derivative of the Ricci tensor. Coincidence of these four invariants implies local isometry in the set of homogeneous Riemannian 3-manifolds, whereas the three eigenvalues of the Ricci tensor do not suce. This classi cation can be used to study homogeneous deformations of 3-dimensional factor spaces of classical cosmological models. An analogous classi cation in the Lorentzian case can be applied to investigate homogeneous deformations of conformal classes of 3-dimensional minisuperspaces. The latter are a fundamental ingredient in the canonical quantization of a generalized factor space cosmology. In the case of Bianchi V (minisuper-)spaces, their Lorentzian signature allows them to have a constant curvature of any value. So, this example gives support to a conformally equivariant quantization, where only conformal equivalence classes of geometries might maintain a physical meaning. Although we obtained some results on the conformal atness of Lorentzian geometries, especially of Bianchi type V, further investigations are needed. Especially changes between di erent Lorentzian signatures, i.e. the relationship between the triad of the tangent frame and the signature, have to be examined in more detail, and also more generally than here, such that e.g. also null eigendirections of the triad can be taken into account systematically. Recent results of [?] are a warning that, unlike the Riemannian case, a Lorentzian signature might result unexpected e ects if some of the invariants become null structures. This may happen if an eigendirection is lightlike. Finally, attempts to relate local and global topology of homogeneous spaces should be driven further on. In the Riemannian case, [?] have already indicated some relations between Bianchi and Thurston geometries.
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References
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Figure Captions Fig. II, IV, V, VIh(w = 0), p p 1: Riemannian Bianchi geometries VIh ( h = 0; ; ; ; ; 1; 2), VIIh ( h = 0; ; ; ; ; ; 1); w.r.t. p the common origin, p ^ the axes of the^ 3 planar diagrams, are: ^ R= 3 to the right, 6S up, and 2 tanh Y both, left and down. 1 5
1 4
1 3
1 7
5 8
1 5
1 4
1 3
1 2
Fig. 2: Riemannian Bianchi geometries II, V, VI , VI , VII , VIII(t; w) (t = ?5; ?1; 0; 1; 5), IX(t; w) (t = 0; ; 1; 2; 5); w.r.t. p the common origin, p ^ the axes of the^ 3 planar diagrams, are: ^ R= 3 to the right, 6S up, and 2 tanh Y both, left and down. 1 2
0
1
0
Fig. 3: Bianchi types II, IV, V, VIh, VIIh (h and w as in Fig. 1) for: (+ + +) lower left, (+ ? ?) top left, (? + ?) top right, (? ? +) lower right; ^ p3 to the right, p6S^ to the left, and 2 tanh Y^ up. the axes are: R= Fig. 4: Bianchi types II, V, VI , VI , VII , VIII, IX (t; w as in Fig. 2) for: (+ + +) lower left, (+ ? ?) top left, (? + ?) top right, (? ? +) lower right; ^ p3 to the right, p6S^ to the left, and 2 tanh Y^ up. the axes are: R= 0
1
17
0
