Dimension of the isometry group in three-dimensional Riemannian spaces
aa r X i v : . [ g r- q c ] F e b Dimension of the isometry group inthree-dimensional Riemannian spaces
Joan Josep Ferrando , and Juan Antonio S´aez Departament d’Astronomia i Astrof´ısica, Universitat de Val`encia, E-46100Burjassot, Val`encia, Spain Observatori Astron`omic, Universitat de Val`encia, E-46980 Paterna, Val`encia, Spain Departament de Matem`atiques per a l’Economia i l’Empresa, Universitat deVal`encia, E-46022 Val`encia, SpainE-mail: [email protected]; [email protected]
Abstract.
The necessary and sufficient conditions for a three-dimensionalRiemannian metric to admit a group of isometries of dimension r acting on s-dimensional orbits are obtained. These conditions are Intrinsic, Deductive, Explicitand ALgorithmic and they offer an IDEAL labeling that improves previously knowninvariant studies.PACS numbers: 04.20.-q, 02.20.Sv, 02.40.Ky
1. Introduction
The invariant characterization of the three-dimensional Riemannian metrics admittinga group G r of isometries acting on orbits O s was presented by Bona and Coll yearsago [1, 2]. Their approach is supported by the capital theorems by Eisenhart [3] andKerr [4], and they employ conditions that are expressed in terms of the eigenvalues andeigenvectors of the Ricci tensor. This kind of invariant approach has been revisitedrecently [5] by offering some algorithms for computing the dimension of the isometrygroup. Lately [6] we have presented an IDEAL approach to the transitive case: wehave given the necessary and sufficient (Intrinsic, Deductive, Explicit and ALgorithmic)conditions for a three-dimensional Riemannian metric to admit a transitive group ofisometries, and we have also distinguished the three different groups G , the threedifferent groups G and the ten Bianchi-Behr types G in transitive action. Here weextend our IDEAL labeling to the non-transitive case by distinguishing the action of aG on two-dimensional orbits, the existence of a group G and the action of a group G .We summarize our results in a compact algorithm that uses explicit metric tensorialconcomitants of the Ricci tensor. We also present explicit Ricci concomitants that allowus to distinguish between the three cases of G on O and to discriminate when thegroup G is abelian. Our tensorial approach avoids obtaining the Ricci eigenvectors imension of the isometry group in 3-D Riemannian spaces Z and two of its differential concomitants thatcollect, respectively, the connection coefficient of the frame and their first and seconddirectional derivatives. These tensors allow us to characterize the number of independentfunctions generated by the connection coefficients.The different cases in which a Ricci-frame exists are analyzed in section 3 and,for each of them, we obtain an explicit concomitant T of the Ricci tensor with thisRicci-frame as eigenframe. From a previous result [6], we can determine the connectiontensor Z in terms of this tensor T , and consequently, we obtain Z as a concomitant ofthe Ricci tensor.With the results obtained in sections 2 and 3 we can perform an algorithm,presented in section 4 as a flow diagram, that distinguishes the three-dimensionalRiemannian spaces admitting a group of isometries G r acting on s-dimensional orbits.Finally, section 5 is devoted to expressing in terms of explicit Ricci concomitants theconditions distinguishing when an isometry group G is commutative, and the sign ofthe curvature of the orbits of a G acting on O .
2. The connection tensor and its differential concomitants
In an oriented three-dimensional Riemannian manifold with metric g and volumeelement η let us consider { e a } , an oriented ( η = e ∧ e ∧ e ) orthonormal frame ofvector fields, and { θ a } , to be its dual basis. The connection coefficients γ cab are definedas usual by ∇ e a = γ cab θ b ⊗ e c . (1)Associated with the frame { e a } we can define its connection tensor Z as Z ≡ ǫ abc h ( ∇ e a ) · e b i ⊗ e c , Z = Z ab θ a ⊗ e b , Z ab = 12 ǫ cdb δ de γ eca , (2)where ǫ abc is the Levi-Civita symbol. In what follows, a · denotes the contraction of theadjacent indexes in the tensorial product, and A = A · A . Tensor Z is invariant whenwe change the orthonormal frame with a constant rotation. Moreover, it holds that ∇ i ( e a ) j = ( e a ) k Z il η lkj . (3)Note that indexes a, b, . . . are used to count the vectors of the frame, and indexes i, j, . . . will indicate components in a coordinate frame. We shall denote with the same symbola tensor and its associated tensors by raising and lowering indexes with the metric g . imension of the isometry group in 3-D Riemannian spaces Z has been considered previously in [6] and, if a transitive group G leavingthe frame invariant exists, it coincides with the structure tensor of G .The last expression in (2) implies that each of the 3 components Z ab of theconnection tensor Z in the frame { e a } corresponds with each of the nine connectioncoefficients γ cab of this frame. Now we define two tensorial differential concomitants of Z , C ( Z ) and D ( Z ), which collect, respectively, the 3 first derivatives, e a ( Z bc ), and the3 second derivatives, e d e a ( Z bc ), of the connection coefficients. More precisely, from thedefinition (2) and expression (3) we obtain: Proposition 1
Let Z be the connection tensor of the frame { e a } and let us define itsdifferential concomitants C = C ( Z ) , C kij ≡ ∇ k Z ij + Z km (cid:16) η nmi Z nj + η nmj Z in (cid:17) , (4) D = D ( Z ) , D ijkl ≡ ∇ i C jkl + Z in (cid:16) η nj m C mkl + η nkm C jml + η nlm C jkm (cid:17) . (5) Then we have C = e a ( Z bc ) θ a ⊗ θ b ⊗ e c , D = e a e b ( Z eg ) θ a ⊗ θ b ⊗ θ e ⊗ e g . (6)The isotropy group is trivial if, and only if, a Ricci-frame exists. In this case, thedimension of the isometry group coincides with that of the orbits and it depends onthe number of independent functions generated by the connection coefficients of theRicci-frame and its first and second derivatives [1, 2]. Now we introduce some algebraicconcomitants of tensors C ( Z ) and D ( Z ) that allow us to characterize this number.Indeed, the number of independent functions appearing in the connection coefficientsand their derivatives is the number of directions generated by the first tensorial index of C and D . This way, we can make them depend on one, two or three functions by simplyimposing the nullity or non-nullity of the tensorial concomitant built as the contractionof the first index of each factor, of a linear, quadratic or cubic expression, with thevolume element η . More precisely, we have: Proposition 2
Let Z be the connection tensor of the frame { e a } and C = C ( Z ) , D = D ( Z ) the tensors given in (4) and (5) . Let us define the concomitants I ( Z ) ijklr = C pij C qkl η pqr , J ( Z ) ijklmr = C pij D qklm η pqr , (7) L ( Z ) = I ( Z ) · C , M ( Z ) = I ( Z ) · D , N ( Z ) = J ( Z ) · D . (8)
Then, it holds(i) All the connection coefficients are constant if, and only if, C ( Z ) = 0 .(ii) All the connection coefficients depend on a single function x , γ abc = γ abc ( x ) , and d x = X a ( x ) θ a if, and only if, C ( Z ) = 0 , I ( Z ) = 0 , J ( Z ) = 0 . (9) (iii) All the connection coefficients depend on two functions x, y , γ abc = γ abc ( x, y ) with d x ∧ d y = 0 , and d x = X a ( x, y ) θ a , d y = Y a ( x, y ) θ a if, and only if, one of the twofollowing conditions hold: { I ( Z ) = 0 , L ( Z ) = 0 , M ( Z ) = 0 } or { I ( Z ) = 0 , J ( Z ) = 0 , N ( Z ) = 0 } . (10) imension of the isometry group in 3-D Riemannian spaces (iv) Otherwise, the connection coefficients define three independent functions,
3. Obtaining the connection tensor Z In order to perform our IDEAL approach when the isotropy group is trivial, we mustobtain the connection tensor Z associated with a Ricci-frame as an explicit concomitantof the Ricci tensor. A cornerstone to achieve this is the following result proved in [6]: Lemma 1 If T is an algebraic general traceless tensor that diagonalizes in theorthonormal frame { e a } , the connection tensor Z corresponding to this frame can beobtained as Z = Z ( T ) ≡ e K · h b T + 6 c T + 12 b g i . (11) where K ji ≡ ( ∇ T · T ) ikl η lkj , b ≡ tr T , c ≡ tr T , e ≡ b − c . (12)When tensor T of the lemma above is a Ricci concomitant, then the orthonormalframe defined by its eigenvectors is a Ricci-frame. Now we analyze the different casesin which this tensor T exists and how it can be obtained from the Ricci tensor.Let R be the Ricci tensor and let us consider the following Ricci concomitants: r ≡ tr R , S ≡ R − r g , s ≡ tr S , t = tr S . (13)If the Ricci tensor is algebraically general, 6 t = s , then its traceless part S defines aRicci-frame, and we can take T = S to compute Z , as considered in [6].In the algebraic special case, 6 t = s , we know [1, 2, 6] that a group G exists ifthe Ricci tensor is proportional to g ( s = 0). And, when the Ricci tensor admits twodifferent eigenvalues ( s = 0) and the simple eigenvector u is not shear-free, a Ricci-framecan be found. Then, tensor T in lemma 1 can be taken as T = Σ, where [6]:Σ ≡ D −
12 (tr D ) h , D ij ≡ ( ∇ R · R ) klm η ml ( i h j ) k , h ≡ t (cid:16) S − s S (cid:17) . (14)The last expression above gives the projector on the Ricci eigenplane, h = g − u ⊗ u , interms of Ricci concomitants.If the Ricci tensor is algebraically special and the simple eigenvector u is shear-free,then a group G acting simply transitively on the whole space does not exist [1, 2, 6].Nevertheless, groups G and G in action simply transitive on the orbits can exist inthis case. Then, the corresponding Ricci-frames can be obtained from the gradient ofthe invariant scalars as we will see next.Note that the Bianchi identities, 2 ∇ · R = d r , imply that the simple eigenvector u is geodesic, ˙ u = 0, if, and only if, d α ∧ u = 0, where α is the simple Ricci eigenvalue.This invariant depends on the scalars r , s and t as: α = 13 r + 2 ts . (15) imension of the isometry group in 3-D Riemannian spaces u isshear-free and non-geodesic ( ˙ u = 0), a Ricci-frame exists. It is the one determined by u and h (d α ). In this case, tensor T in lemma 1 can be taken as T = A , where: A = 2 h (d α ) ⊗ h (d α ) − h (d α, d α ) h , d α = 13 (cid:16) d r + 2 s d t (cid:17) . (16)The expression for d α above can be obtained from (15) by using that 6 t = s .Finally, let us suppose that ˙ u = 0. Then, we have that ( s d r + 2 d t ) ∧ u = 0, and thealgebraically special condition, 6 t = s , implies that all the algebraic scalars associatedwith the Ricci tensor have a gradient in the direction of u if, and only if, d r ∧ u = 0. Ifthis is not the case, namely if d r ∧ u = 0, a Ricci-frame exists. It is the one determinedby u and h (d r ). In this case, tensor T in lemma 1 can be taken as T = Y , where: Y = 2 h (d r ) ⊗ h (d r ) − h (d r, d r ) h . (17)If none of the above considered cases happens, that is, if the Ricci tensor isalgebraically special, if the simple eigenvector u is geodesic and shear-free and ifd r ∧ u = 0, then no Ricci-frame can be built [1, 2]. Then, two possibilities may occur.If d r = d s = 0, we have that a G exists [1, 2, 6]. And, if (d r ) + (d s ) = 0, we havethat a G (2)3 (three-dimensional group acting on two dimensional orbits) exists [1, 2].It is worth remarking that the above analysis on the existence of Ricci-frames wasindicated by Bona and Coll [1, 2]. Here, we have delved into how they can be obtainedand how to determine tensor T , which provides its associated connection tensor Z (seelemma 1). Once this tensor has been obtained, the results in proposition 2 provide anIDEAL labeling that we present in algorithmic form below.
4. Algorithm to determine the dimension of the isometry group
The IDEAL characterization of the Riemannian spaces with a non-trivial isotropy groupis given in the section above: a G when 3 R = rg , and a G (respectively, a G (2)3 ) if3 R = rg and if there is no Ricci-frame and d r = d s = 0 (respectively, (d r ) + (d s ) = 0).We have also considered the different cases in which a Ricci-frame exists. For eachcase, we have given the explicit expression (in terms of Ricci concomitants) of a trace-less algebraically general tensor T that provides the connection tensor Z associated withthe Ricci-frame (lemma 1). Thus, in order to obtain the IDEAL characterization of theRiemannian spaces with a trivial isotropy group, and as consequence of the results byBona and Coll [1, 2], we must impose on Z the constraints C ( Z ) = 0, (9) or (10), whichdenote that the isometry group has dimension three, two or one, respectively. Notethat, if the first condition does not hold ( C ( Z ) = 0), then (9) states:H2 I ( Z ) = 0 , J ( Z ) = 0 . (18)And if the above condition H2 does not hold, then (10) states:H1 { I ( Z ) = 0 , L ( Z ) = 0 , M ( Z ) = 0 } or { J ( Z ) = 0 , N ( Z ) = 0 } . (19) imension of the isometry group in 3-D Riemannian spaces g , the Ricci tensor R , thealgebraic Ricci concomitants S , r , s and t defined in (13), and the first-order Ricciconcomitants Σ, defined in (14), and d α , defined in (16). In subsequent steps we needthe connection tensor Z ( T ) given in (11), where T is a tensor that depends on thedifferent cases: Z ( S ) is of first order in the Ricci tensor, and Z (Σ), Z ( A ) and Z ( Y )are of second order. Concomitants A and Y are given in (16) and (17), respectively.From Z ( T ) we can define the tensors C ( Z ) given in (4) (first-order in Z ), and D ( Z )given in (5) (second-order in Z ). Conditions H1 and H2 are specified in (18) and (19),respectively. They use the algebraic concomitants I, J, L, M, N of C ( Z ) and D ( Z ) givenin (7) and (8). The end horizontal arrows lead to the different G r . G (3)3 and G (2)3 denote ✘✘✘✘❳❳❳❳ g, R, S, r, s, t Σ , d α ❄✟✟✟✟ ❍❍❍❍✟✟✟✟❍❍❍❍ R = rg ✟✟✟✟ ❍❍❍❍✟✟✟✟❍❍❍❍ t = s ❄❄✟✟✟✟ ❍❍❍❍✟✟✟✟❍❍❍❍ Σ = 0 ❄ ∄ ❄✟✟✟✟ ❍❍❍❍✟✟✟✟❍❍❍❍ C ( Z ) = 0 ✟✟✟ ❍❍❍✟✟✟❍❍❍ H2 ✟✟✟ ❍❍❍✟✟✟❍❍❍ H1 ✲✲✲ ❅❅❅(cid:0)(cid:0)(cid:0) ✲ ✲ ✲✲✲ G G (3)3 G G ✟✟✟✟ ❍❍❍❍✟✟✟✟❍❍❍❍ d α ∧ u = 0 ✟✟✟✟ ❍❍❍❍✟✟✟✟❍❍❍❍ d r ∧ u = 0 ✲✲✲ ❅❅❅(cid:0)(cid:0)(cid:0) ✟✟✟ ❍❍❍✟✟✟❍❍❍ H2 ✟✟✟ ❍❍❍✟✟✟❍❍❍ H1 G G ✲✲❄❄ ∄ ❄❄✟✟✟✟ ❍❍❍❍✟✟✟✟❍❍❍❍ d r = 0 = d s ❄ G (2)3 ✲✲ G ❅❅(cid:0)(cid:0) Z ≡ Z (Σ) ❅❅(cid:0)(cid:0) Z ≡ Z ( S ) ❅❅(cid:0)(cid:0) Z ≡ Z ( A ) ❅❅(cid:0)(cid:0) Z ≡ Z ( Y )yesno nonononononoyesyesyesyesyes yes no yesnono yesnono yesyes Figure 1.
This flow diagram distinguishes the dimension of the groups of isometriesof a three-dimensional Riemannian metric. imension of the isometry group in 3-D Riemannian spaces
5. Labeling the several G (2)3 groups and the commutative G group In [6] we have analyzed the groups of isometries in transitive action on the whole space,and we have also distinguished the three different groups G , the three different groupsG and the ten Bianchi-Behr types G . Now we characterize the three different G (2)3 ,and when a group G is commutative.When a three-dimensional group of isometries acts on two-dimensional orbits, theseorbits are of constant curvature k . And the group is SO(3) when k = +1, SO(2,1)when k = −
1, and E(2) when k = 0. Bona and Coll [2] gave the following invariantexpression for this curvature, k = sign (cid:16) θ + β − α (cid:17) , where θ is the expansion of thesimple eigenvector of the Ricci tensor and α and β are, respectively, the simple andthe double Ricci eigenvalues. One can easily obtain these invariants in terms of explicitRicci concomitants already used in this paper, and we can state: Proposition 3
The three-dimensional Riemannian spaces that admit a three-dimensional group of isometries acting on two-dimensional orbits are characterized bythe algorithm in figure . The group is SO(3) when k = +1 , SO(2,1) when k = − , and E(2) when k = 0 , with k defined as: k = sign (cid:26)
14 ( ∇ · h ) + 16 r − ts (cid:27) , (20) where s and t are given in (13) , and h is given in (14) . When the Riemannian space admits a group G of isometries, the isotropy groupis trivial and a Ricci-frame exists. If Z is its associated connection tensor, then thecovariant derivative of any Killing vector ξ can be obtained as [6]: ∇ ξ = ∗ ( ξ · Z ).Consequently, if ξ , ξ are two independent Killing vectors, we obtain:[ ξ , ξ ] = ξ · ∗ ( ξ · Z ) − ξ · ∗ ( ξ · Z ) = ∗ (( ξ ∧ ξ ) · Z ) . (21)Thus, the group G is commutative if, and only if, ∗ (( ξ ∧ ξ ) · Z ) = 0. On the otherhand, we know that the first index of the tensor C ( Z ) defines a sole direction when amaximal group G of isometries exists, and this direction is orthogonal to the orbits,that is, it is given by ∗ ( ξ ∧ ξ ). Therefore, the commutative condition can be writtenas η ijk C kmn Z j p η pil = 0. Then, a straightforward calculations leads to: Proposition 4
The three-dimensional Riemannian spaces that admit a two-dimensional group of isometries are characterized by the algorithm in figure . Thegroup is commutative if, and only if, Z · C − (tr Z ) C = 0 , (22) where Z = Z ( T ) is given in (11) , with T = S, Σ , A, Y , depending on the different cases,and C = C ( Z ) is given in (4) .imension of the isometry group in 3-D Riemannian spaces Acknowledgments
This work has been supported by the Spanish Ministerio de Ciencia, Innovaci´on yUniversidades and the Fondo Europeo de Desarrollo Regional, Projects PID2019-109753GB-C21 and PID2019-109753GB-C22, the Generalitat Valenciana ProjectAICO/2020/125 and the University of Valencia Special Action Project UV-INVAE19-1197312.
References [1] Bona C and Coll B 1990
C. R. Acad. Sci. Paris
J. Math. Phys. Continuous groups of transformations (Princeton University Press, Princeton)[4] Kerr R P 1962
Tensor Class. Quantum Grav. Class. Quantum Grav.37