Isometry groups of three-dimensional Riemannian metrics

Isometry groups of three-dimensional Riemannian metrics Carles Bona, and Bartolomé Coll Citation: Journal of Mathematical Physics 33, 267 (1992); View online: https://doi.org/10.1063/1.529960 View Table of Contents: http://aip.scitation.org/toc/jmp/33/1 Published by the American Institute of Physics Isometry groups of three-dimensional Riemannian metrics Caries Bona Departament de Fisica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain Bathroom Coll Laboratoirede Physique Thborique, Gravitation et Cosmologie Relativistes, Institut Henri Poincark, 11, Rue Pierre et Marie Curie, 75231 Paris Cedex 05, France (Received 5 November 1990; accepted for publication 15 August 1991) The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group Gr of isometries acting on s-dimensional orbits are given. This provides the list of (abstract) groups that can act isometrically and maximally on such metrics. The conditions are expressed in terms of the eigenvalues and eigenvectors of the Ricci tensor. In any case, the order of differentiability of these data necessary to determine the isometry group is less than 4. 1.INTRODUCTION In general relativity there are many interesting problems involving an equation of the form 2(v) g = Q, where g is a three-dimensional Riemannian metric, 2(v) indicates the Lie derivative with respect to the vector field v, and Q is a certain (second-order symmetric) tensor related to the problem in question. This is the case, for example, in the standard evolution formalism' where one is interested in the isometries admitted by the gravitational field at every instant:2 there, g is the induced metric on the instant (spacelike hypersurface), Q vanishes, and the isometries are characterized by the vector fields v verifying 2(v) g = 0 (Ref. 3); slightly more generally, the vector fields characterizing the conformal symmetries at every instant verify 2(v) g = g. In the same evolution formalism, one can ask for stronger symmetry conditions, namely, that the isometric or conformal invariant properties of the gravitational field be space-time symmetries. Then, as was shown in Ref. 4, the spacelike projections v of the vector fields characterizing these space-time symmetries have to verify 2 (v) g = 2ar K + 0 g, where a- is a suitable scalar function, K is the extrinsic curvature of the instant, and 4 vanishes in the isometry case. It is well known that Born's rigidity notion in relativity is excessively severe (Herglotz-Noether theorem, infinitesimal perturbations at infinite velocity). A smoother alternative approach, adapted to the synchronization defined by the instants of the evolution formalism, is the relative rigidity notion introduced in Ref. 5 and considered in Ref. 6 under a slighter restricted form. One can show that a rigid congruence with respect to a given synchronization is characterized by a velocity field whose projection v on every instant acts like a space-time symmetry on that instant: 25(v) g = 2a K. Clearly, a relative conformal rigidity notion alternative to the conformal J. Math. Phys. 33 (1), January 1992 Born's one7 will add a term of the form 0 g to the second member of the above equation. Another physical formalism usual in relativity is that of the quotient metric q, obtained from Cattaneo's technique of projections associated to a system of observers (timelike congruence) 8 Although the quotient metric is not a metric (unless the observers are Born's rigid ones), it is possible to associate with it Ricci-like tensors9 and, for some purposes, to handle the quotient metric as a true three-dimensional Riemannian metric. It is then natural to ask for its isometrics or conformities, i.e., for those vector fields v that are orthogonal to the observers and such that 2(v) q equals 0 or q q, respectively. Finally, there is an interesting question that remains open in the matching problem in general relativity: that of the conditions governing the absorption or creation of isometrics by the surface of a medium. The only known general result follows directly from the Darmoisl° matching conditions and the isometry initial conditions: 4 It states that an isometry group can pass continuously through the matching surface only if this surface is an orbit of the group. If that is not the case, a set of necessary conditions for the group to pass (discontinuously) is that, in the matching surface, an equation of the form 23(v) g = 2oa K admits nontrivial solutions in v. All of these examples show the interest of the study of three-dimensional equations of the form 25(v) g = Q. Their space of solutions depends drastically on the isometry group admitted by the metric g. Of course, the corresponding integrability conditions have been known for a long time'1 but they have never been completely and covariantly solved. The purpose of this paper is to do that. We give, in terms of the eigenvalues and eigenvectors of the Ricci tensor, the necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group of isometries.12"13 Our main results (Theorems 1 to 5) are well adapted to theoretical considerations and are easy to apply to concrete examples: 0022-2488/92/01 0267-06$03.00 M 1991 American Institute of Physics 267 268 C. Bona and S. Coll: Three-dimensional Riemannian metrics They should certainly simplify the present numerical programs for obtaining the Killing vector fields and the problem of equivalence of metrics. In a study of certain space-times, the simple case of two-dimensional metrics has been considered recently by one of us.14 As a by-product of our analysis, the abstract groups that are allowed to act maximally as isometry groups of three-dimensional Riemannian metrics have been obtained. With intransitive actions. They are: GI, the two G2 , S0(3), SO(2,1) and E(2); with transitive actions they are; Bianchi's groups (except for BI, BII, BIII, and BV), SO(3)xU(l), SO(2,l)XU(l), IIIq=o (Ref. 15), S0(4), SO(3,1), and E(3). The results corresponding to the dimension of the group appeared in Ref. 16; for this reason, those proofs are avoided here. II. GENERALITIES In what follows, tensors and cotensors associated by the metric will be noted with the same letter, and tensorial expressions will be given in its covariant form. Let g be a three-dimensional Riemannian metric, R being its Ricci tensor, Gr being the maximal isometry group admitted by g, with dimension r, and 0, denoting the orbits, with dimension s, of G, The necessary and sufficient conditions for a metric to admit a G6 are known from long time ago: g admits a G6 iff R admits a triple eigenvalue, R =a g, and G 6= SO(4), G6 E(3), or G6 SO(3, 1) according to the relationsa> 0, a = 0, or a <0, respectively. Our goal here is to obtain results analogous to the above for all the other (maximal) isometry groups G. that a three-dimensional Riemannian metric g can admit. An old theorem by Bianchi17 forbids the (maximal) isometric action of a 05 so that one has to consider the five following cases: G4, G3 over 03, G3 over 02, G2, and GI. As in the case of the G6'S, the starting point for our analysis is the Eisenhartt 8 theorem about the integrability of the Killing vector fields: if the system {5(v)R = 0,..., L(v)JDPR = 0,...}, DP denoting the set of covariant derivatives of order p, considered as an algebraic system in v and Dv, is constituted by e algebraically independent equations, then the dimension r of the (maximal) isometry group Gr admitted by g is r = 6 - e, and the structure constants of (the Lie algebra of) Gr are given by the commutators of any basis of the vector space of solutions to the above system. In every case, from a theorem by Kerr, 19 the dimension of the orbits °s is given by s = 3 - i, where i is the number of functionally independent scalar invariants that may be obtained as differential concomitants of the metric g. Let u be a unit vector field and D be the covariant derivative operator; in three dimensions, we have the fol- lowing covariant decomposition of Du: Du=uXa(u) + fl(u)*u + a(u) + e(u)/2(g- uXu), (1) where we have noted by an asterisk (*) the Hodge dual operator, fQ (u), E (u) are the rotation and expansion scalars of u, respectively, a ( u) is the acceleration vector of u, and a(u) is the shear two-tensor of u. All these quantities can be defined covariantly in terms of Du by using the interior product (contraction) with u, i(u), the orthogonal projection i(u), and the trace operator tr on the two tensors in the usual way. All the isometry groups that will appear in what follows are well known and we will use the standard notation for them. The only exception is the G4 whose Lie algebra can be generated by the set of vectors (u,vA) (A = 1,2,3) with the nonvanishing structure constants given by I[v1 V2 1 =U, [V2,V3 =1~V, [V3,VI I=V 2. (2) We will follow in this case the notation of Petrov in denoting by IIIq=O this G4. III. G4 Theorem 1: The necessary and sufficient condition for a three-dimensional metric g to admit a G4 as maximal isometry group is that the Ricci tensor be of the form R= (a-f3)uXu+13 g, (3) with constant eigenvalues a, 13 (a=Af3) and the unit eigenvector u be shear-free. The group G4 is locally S0(3) XU(l) whena+13>0,SO(2,1)XU(l) whena+,f<0 and III=0 when a + 13 = 0. Proof: The proof of the statement about the dimension of the isometry group has been given in Ref. 16, and we will not repeat it here. Let us remark that the Bianchi identities of (3) imply that u must be also geodesic and expansion-free so that u is actually a Killing vector of g and can then be chosen as one of the G4 generators. Also, u must be invariant by the isometry group (it is intrinsically given by the Ricci tensor) and, therefore, if we denote by vA the remaining three generators, we have [u,vAI=O (A=1,2,3). (4) Note that, in the generic case, u is not vorticity-free: Du=fi1(u)*u, (5) and we get from the integrability conditions of (5) that 2a = Q(u)2. J. Math. Phys., Vol. 33, No. 1, January 1992 269 C. Bona and B. Coll: Three-dimensional Riemannian metrics Let us consider now the quotient metric q = g - u X u. It follows from (4) that q is invariant by G4; in particular, we have (6) L(vA )q=0 and, allowing for (4), (7) L('vA)q=0, where we have noted for short I VAm(u) (VA). This means that the two-dimensional quotient manifold admits a three-dimensional isometry group and then must be of constant curvature. Its Gauss curvature can be computed in a straightforward way by using Cattaneo's projection formalism relative to u (Ref. 8): (8) +13, 4tr(Ric(q))=a and we have proven the following Lemma. Lemma 1: The two-dimensional quotient manifold associated to the geodesic invariant Killing vector u is of constant curvature given by (8). Its isometry group is a G3 generated by iL(U)(vA), where G 3 SO(3) when a + ,j >0, G 3 E(2) when a +3= 0, and G 3 zSO(2,1) when a +,6<0. Allowing for the invariance of u, Eq. (4), the commutators of the VA generators are related with the G3 structure constants KCAB in the following way: (9) [VA,VB] =KC.4BVC+ EABU, with the extra constant terms EAB arising from the parallel component of the vA relative to u. The K terms can always be put into a canonical form by a suitable linear combination of the VA; if k = + 1, this means If k = 0, we have [vl,v2 ] =E [V3 ,VI 1 =V 12 U, 2 [V 2,V31 =VI + E 2 3 U, + E 31 U, where both E23 and E31 can be eliminated defining vl, v2 according to (11). The remaining extra constant E12, however, must be different from zero because E 12 = 0 would imply that the three-dimensional manifold admits three commuting Killing vectors (u, vl, and V2) and it would be therefore flat so that the G4 would not be maximal. This allows us to divide both vl and v2 by JE12 11/2 and redefine u by multiplying it by the sign of E 12 to get (dropping the primes) IV1 ,V2 1 =U, [V2 ,V31 =VI, [v 3 ,v 1 ] =V 2 , [V 2 ,V3] =VI + E2 3 u, (10) [V3,VI] V2 + E 3 1u, so that we can define a new set of Killing vectors vl=V + E 23 u, V2=V 2 + E 31 u, V3=V 3 VU + E 12 u, (11) which generate an SO(3) subalgebra of the G4. This amounts to taking (dropping the primes) EAB = 0 in (10) and then G4=SO( 3 ) XU(1). The same argument can be repeated ifk= -1 to show that G4 SO(2,1 ) XU( 1 ) in that case. (13) which, together with (4) gives the full set of structure constants of the G4. This isometry group G4 is precisely IIIq=o and admits a three-dimensional subgroup of Bianchi type II (Refs. 20 and 21) generated by (u,vi,v2 ). This completes the proof of Theorem 1. Note that the covariant derivatives of the Ricci tensor in the conditions of Theorem 1 are at most of involved R first order (because of the computation of the shear of u). IV.G3 ON 02 Theorem 2: The necessary and sufficient conditions for a three-dimensional Riemannian metric g to admit a G3 acting on two-dimensional orbits 02 as the maximal isometry group is that the Ricci tensor be of the form R=(a-f3)uXu+13 g, (14a) with the unit eigenvector u being shear-free and vorticityfree and verifying da A u=dP3A u=0, [VI,V 21 =V3 + E 12u, (12) (14b) where da and d13 do not vanish simultaneously. The group G3 is (locally) S0(3) when k= + 1, SO(2,1) when k = - 1 and E(2) when k = 0, whith k defined as k=sign(O (U)2/A +81- a/2). (14c) Proof: The proof concerning the first part of the Theorem is given in Ref. 16. Let us note that the Bianchi identities of (14a) imply that u must be geodesic so that Du=O(u)/2 (g-uXu), (15) where the expansion factor 0(u) is given by22 2 (u) (a/2 -13) = (13 - a) 0(u).- J. Math. Phys., Vol. 33, No. 1, January 1992 (16) 270 C. Bona and B. Cool: Three-dimensional Riemannian metrics As it is well known, the orbits 02 will be constant curvature surfaces and the structure of the isometry groups G3 will depend only on the sign k of the Gauss curvature of these surfaces. In our case, the orbits are orthogonal to the invariant vector u at every point so that it is enough to compute the curvature of the quotient metric q = g - uX u, and this can be done in a straightforward way by using the projection formalism 8 to get the result (14c). Note that the covariant derivatives of R involved in the statements (14) of Theorem 2 are at most of first order as in Theorem 1. V. G3 ON 03 All the statements in this section and in the remaining ones, unless otherwise stated, will refer to a principal triad (UA, A = 1,2,3), that is to say, an orthonormal set of eigenvectors of the Ricci tensor R. The principal triad is uniquely defined if R is nondegenerate, but we will need some additional prescription in the degenerate case, where only the unit eigenvector u corresponding to the simple eigenvalue is intrinsically defined. If u is not shearfree, a natural choice is to complete the triad with the eigenvectors of the shear of u. If u is shear-free but condition (14b) does not hold, there exists a function ti of the Ricci eigenvalues such that d/l A u=/0, and we can use d/i to complete the triad by the orthonormalization procedure. We shall call a simple triad the principal triad which is uniquely defined in this way. Note that if a simple triad does not exist for a given metric g, Theorems I and 2 will determine the dimension and structure of the isometry group of g; conversely, if g admits a simple triad, then Theorems I and 2 do not apply to g. Theorem 3a: The necessary and sufficient conditions for a three-dimensional Riemannian metric g to admit a G3 acting transitively as the maximal isometry group is that g admit a simple triad and that all the Ricci eigenvalues aA and all the spin coefficients rCAB of the triad be constant. The structure constants CCAB of the G3 can be expressed as follows: CCAB=2rTlABI (17) where the brackets indicate antisymmetrization. Proof: The proof of the statement concerning the dimension of the isometry group is given in Ref. 16. On the other hand, the vectors uA of the simple triad are intrinsically defined from the Ricci tensor: They form a complete set of invariant vectors under the isometry group G3. Moreover, as G3 acts transitively, the vector fields UA generate the reciprocal group (which is locally isomorphic to G3 ) and their commutators can be related to the G3 structure constants as given in (17). An invariant classification of the G3 structures can be made by the method developed by Bianchi,2 0 which is based in the algebraic structure of the matrix A AE= 11,A B0 _BCE (18) ) where eABC is the Levi-Civita symbol, or by an analogous method which starts using the matrix NAE=~1eACrBCE (19) which can be easily related with A (Ref. 21). It is well known that, for every Bianchi class, one can construct a G3 belonging to this class which acts transitively on 03. If R is not degenerate, Theorem I does not apply and consequently the G3 is therefore maximal. If R admits a double eigenvalue, R= (a-13)uXu±13 g, (20) the Bianchi identities imply Du=f1(u)*u + a(vXv - v'X V%) (21) where a is different from zero (otherwise, G3 would not be maximal) and the simple triad is formed by u, v, and V. The integrability conditions of (21) imply that both v and v' are expansion-free and their respective accelerations a(v), a(v') are directed along u, namely, a(v) = -a (v') =(a/2 _fl(U) 2 )/c. U, (22) so that, allowing for the fact that u, v, and v' are orthonormal, one gets the following relation: a/2-=O(u)2 _ ,2 (23) and the spin coefficients are completely determined by Dv=fl(u)*v - ovXu, Dv'= il (u) *v' ± ov'X u. (24a) (24b) The integrability conditions of (24) amount to say that fl(u)=0, f3=0, (25) where we have used that o#0. The computation of the Bianchi type is now a simple algebraic exercise: G3 belongs to the Bianchi VIo class. We have therefore proven the following theorem. Theorem 3b: If an isometry group G3 acts maximally on 03 and the Ricci tensor R of 03 is degenerate, then R is of rank one and the G3 belongs to Bianchi class V10. Corollary: The isometry groups G3 belonging to Bianchi classes BI, 1RI, BIII, or BV cannot act maximally on a Riemannian three-dimensional manifold. 2 0 Proof: In both the BI and BV cases, it is well known that the orbits 03 are constant curvature manifolds and therefore the maximal isometry group is a G6 [locally J. Math. Phys., Vol. 33, No. 1, January 1992 C. Bona and B. Coll: Three-dimensional Riemannian metrics isomorphic to E(3), SO(3,1), respectively]. In tihe BII R) .... d and BIII cases, one can compute [by using (18D ) andlU (17)] the spin coefficients and, then, the Ricci ternsor of the orbits. The Ricci tensor results to be degenerate !rate in both cases so that, by Theorem 3b BII and BIII G33cannot be maximal, although they can act on 03 ais subgroups of G4 . A different proof of the same resu lt was given by Bianchi.2 0 Note that, as a further consequence of Theore te obr only derivatives of the Ricci tensor up to the first orde) are needed to decide whether or not g admits a G. (,id 3) as the maximal isometry group, even in the dege.nerate case. VI. G2 AND G1 Theorem 4: The necessary and sufficient consditions for a Riemannian metric g to admit a G2 as the m; aximal isometry group is that a simple triad (UA) exist, tthat all the Ricci eigenvalues and the spin coefficients of th e triad depend effectively on a single function [i and that dA,_i(UA) (dll)=f,(Yll) (26) where f, (A = 1,2,3) are arbitrary functions of, ju. The choosing structure of the group G 2 can be determined by chloosing a new triad (vA) (no longer a principal one) such tihat the unit vector vl is parallel to dlt; then G2 is (resp. iis not) Abelian if and only if the remaining vectors V2 ;and V3 conmute (resp. do not commute). Theorem 5: The necessary and sufficient coneEditions for a Riemannian metric g to admit a GI as the m;maximal isometry group is that a simple triad (Us) exist to)gether with two functions tt and - such that all the Ricci i eigenvalues and the spin coefficients of the triad depen(Id only on por and that dAU~i(uA) (dWY) =fAQI,(r), dAr =_i(UA) (dr) = hA (,sr), (27a) (27b) where fA, hA (A = 1,2,3) are arbitrary functions o)f their arguments. Proof: The proofs of Theorem 5 and the first part of Theorem 4 are given in Ref. 16. Concerning the second s part of Theorem 4, note that the vectors v2 and V3 of the new triad are constructed to be tangent to the orbbits 02 (the surfaces ft = const) and, allowing for (28), thhey iey are are also invariant under G2. Moreover, as G2 acts transsitively on 02, the vector fields V2 , V3 generate the reciiprocal group (which is locally isomorphic to G2 ). As tihere is only one possible non-Abelian structure for G2 (Uilp to a local isomorphism), it is enough to check whether or not v2 and V3 conmute, as stated in Theorem 4. 271 VII. A FINAL REMARK The successive checking of Theorems 1-5 provides an algorithmic procedure to obtain the dimension and structure of the maximal isometry group of a Riemannian three-dimensional manifold. The set of invariant objects (scalars and vectors) that are needed to verify the theorems is obtained algebraically in terms of the Ricci tensor R of g and its covariant derivatives up to a given order p. The value of p is limited by the fact that, as it follows easily from Theorems 1-5, the procedure concludes if the covariant derivative of order p gives no new information (no supplementary invariant objects) ;23 in the worst case, when every successive derivative adds only one new object, one could expect that p<4. Note, however, that, as a consequence of Theorems 3a, 3b, either the simple triad and one of the scalars appear before taking the second derivative of R or the algorithm concludes: This lowers the bound to p<3. We have then proven the following Theorem. Theorem 6: The full set of independent scalars and vectors that are invariant under the isometry group G of a Riemannian three-dimensional metric g can be algebraically obtained from the Ricci tensor of g and its successive covariant derivatives up to the third order. ACKNOWLEDGMENT This work is partially supported by the DGICYT of Spain under project No. PB87-0583-CO2-01. .In the evolution formalism of general relativity, the space-time is described by a system of observers (congruence of timelike lines) endowed with a synchronization (foliation by spacelike hypersurfaces called instants), which is characterized by a time-dependent threedimensional geometry. The standardevolution formalism is the one in which every instant of the synchronization is characterized by its first and second fundamental forms. 2 These symmetries are called intrinsic symmetries, and were first considered by C. B. Collins, Gen. Rel. Grav. 10, 925 (1979). 3These vector fields, which generate the algebra of the intrinsic symmetry group, were introduced by C. Bona and B. Coll, J. Math. Phys. 26, 1583 (1985). 4 The necessary and sufficient conditions for a set of Cauchy data to generate a space-time admitting a group G, of isometries are given, for vacuum and Einstein-Maxwell energy tensors in B. Coll, J. Math. Phys. 18, 1918 (1977), for dust matter in B. Coll, C. R. Acad. Sci. Paris Ser. A 286, 1163 (1978), and for perfect fluids in B. Coll, C. R. 5Acad. Sci. Paris Ser. I 292, 461 (1981). B. Coll, C. R. Acad. Sci. Paris Ser. II 295, 103 (1982). Bona, Phys. Rev. D 27, 1243 (1983). 7Conformal Born's rigidity was considered, at least, by L. Bel and J. C. Escard, Rend. Sc. fis. mat. nat. Lincei 16, 476 (1966). 8See A. Lichnerowicz, Theories Relativistes de la Gravitation et de P~lectromagetisme (Masson, Paris, 1955); C. Cattaneo, Ann. Mat. Pura Appl. 48, 361 (1959); L. Bel and J. C. Escard, Ref. 7. 9 See L. Bel in Recent Developments in Gravitation, edited by J. Cbspedes, J. Garriga, E. Verdaguer (World Scientific, Singapore, 1991) and references therein. 10G. Darmois, Les equations de la gravitation einsteinienne, Mem. des Sc. Math. (Gauthier-Villars, Paris, 1927). 'See L. P. Eisenhart, Riemannian Geometry (Princeton U.P., Prince6C J. Math. Phys., Vol. 33, No. 1, January 1992 272 C. Bona and B. Coll: Three-dimensional Riemannian metrics ton, NJ, 1926); K. Yano, The Theory of Lie Derivatives and Its Applications (North-Holland, Amsterdam, 1955). 12AII our considerations are of local character, and the groups considered are the Lie groups determined by the Lie algebra of Killing vector fields admitted by the metric. 3 1 The case of a Lorentzian metric will be considered elsewhere. 14 C. Bona, J. Math. Phys. 29, 2462 (1988). 5G4 III,=0 is Petrov's notation [A. Z. Petrov, Einstein spaces (Pergamon, New York, 1969)]; see below in the text for its definition. 16 C. Bona, B. Coll, C. R. Acad. Sci. Paris Ser. 1 310, 791 (1990). 7 1 L. Bianchi, Mem. Soc. Ital. Sci. (dei XL) 11, 267 (1897). This result is generally known as a particular case of the n-dimensional generalization theorem by G. Fubini, Ann. Mat. Ser. 3 8, 39 (1903). 8 1 See references in Ref. 11. 19 R. P. Kerr, Tensor 12, 74 (1962). 20 See first reference in Ref. 17. 21 F. B. Estabrook, H. D. Wahlquist, and C. G. Behr, J. Math. Phys. 9, 497 (1968). 22 The corresponding expression given in Ref. 16 was not correct: We give the right one here. 23 One actually needs to take the p + I derivative in order to check this by computing the Wronskian of the appropriate functions. J. Math. Phys., Vol. 33, No. 1, January 1992