We describe conditions under which a spacetime connection and a scaled
Lorentzian metric define natural symplectic and Poisson structures on the
tangent bundle of the Einstein spacetime

Janyška, Josef

English

arXiv

summary:We describe conditions under which a spacetime connection and a scaled Lorentzian metric define natural symplectic and Poisson structures on the tangent bundle of the Einstein spacetime

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumJosef JanyškaGeometric structures on the tangent bundle of the Einstein spacetimeArchivum Mathematicum, Vol. 42 (2006), No. 2, 195--203Persistent URL: http://dml.cz/dmlcz/107996Terms of use:© Masaryk University, 2006Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 42 (2006), 195 – 203GEOMETRIC STRUCTURES ON THE TANGENT BUNDLEOF THE EINSTEIN SPACETIMEJOSEF JANYŠKAAbstract. We describe conditions under which a spacetime connection anda scaled Lorentzian metric define natural symplectic and Poisson structureson the tangent bundle of the Einstein spacetime.IntroductionGeometrical structures induced on the tangent bundle of the Einstein spacetimeplay a fundamental role in the covariant classical and quantum mechanics. Thecovariant classical and quantum mechanics over the Einstein spacetime proposedin [3, 4] is natural in the sense of [7, 8, 10] and independent of the base of scales,so the “spaces of scales” are systematically used. Roughly speaking, a space ofscales has the algebraic structure of IR+ but has no distinguished ‘basis’. The basicobjects of the theory (metric, 2-forms, 2-vectors, etc.) are valued into scaled vectorbundles, that is into vector bundles multiplied tensorially with spaces of scales.In this way, each tensor field carries explicit information on its “scale dimension”.Actually, in this paper, we assume the space of lengths L. Moreover, Lp denotes⊗pL.In [1, 5] the classification of symplectic and Poisson structures on the tangentbundle of a pseudo-Riemannian manifold was given for a non-scaled metric g anda torsion free linear connection K. In this case the metric g and the connectionK admit a family of symplectic 2-forms Υ[g, K] or Poisson 2-vectors Λ[g, K] onthe tangent bundle parametrized by a function µ(g(u, u)) satisfying certain con-ditions. Moreover, g and K are related by the condition that ∇g is a symmetric(0,3)-tensor field. For a scaled metric g and a general spacetime connections theconstructions of the 2-form Υ[g, K] and the 2-vector Λ[g, K] are the same but2000 Mathematics Subject Classification: 53B15, 53B30, 53D05, 53D17, 58A10, 58A20,58A32.Key words and phrases: spacetime, spacetime connection, Schouten bracket, Frölicher–Nijen-huis bracket, symplectic structure, Poisson structure.This research has been supported by the Ministry of Education of the Czech Republic underthe project MSM0021622409 and by the Grant agency of the Czech Republic under the projectGA 201/05/0523.Received August 24, 2005.196 J. JANYŠKAfrom the independence on the base of scales it follows that Υ[g, K] and Λ[g, K]are unique, up to a multiplicative real constant. In this paper we generalize theresults of [1, 5] for a scaled metric and a general spacetime connection. Namely,we shall describe the condition under which the 2-form Υ[g, K] and the 2-vectorΛ[g, K] give symplectic and Poisson structures, respectively.If E is a manifold, then the tangent bundle will be denoted by τ [E] : TE → Eand local coordinates (xλ) on E induce the fibered local coordinates (xλ, ẋλ) onTE. By map(E, E′) we denote the sheaf of smooth maps.1. Geometry of the spacetimeWe recall basic properties of the Einstein spacetime and its tangent bundle.1.1. Spacetime. We assume spacetime to be an oriented and time oriented 4–dimensional manifold E equipped with a scaled Lorentzian metric g : E → L2 ⊗(T ∗E ⊗ T ∗E) with signature (− + ++) . The dual metric will be denoted byḡ : E → L∗2 ⊗ (TE ⊗ TE). Let us note that the dimension is not relevant. Ourresults are valid for any dimension n ≥ 3 and a pseudo-Riemannian metric of thesignature (1, n − 1).A spacetime chart is defined to be an ordered chart (x0, xi) ∈ map(E, IR× IR3)of E , which fits the orientation of spacetime and such that the vector ∂0 is timelikeand time oriented and the vectors ∂1, ∂2, ∂3 are spacelike. In the following weshall always refer to spacetime charts. Latin indices i, j, . . . will span spacelikecoordinates, while Greek indices λ, µ, . . . will span spacetime coordinates.We have the coordinate expressionsg = gλµ dλ ⊗ dµ , with gλµ ∈ map(E, L2 ⊗ IR)ḡ = gλµ ∂λ ⊗ ∂µ , with gλµ ∈ map(E, L∗2 ⊗ IR) .1.2. Spacetime connections. We define a (general) spacetime connection to bea connection K of the bundle τ [E] : TE → E . We recall that a connection Kof the bundle TE → E can be expressed, equivalently, by a tangent valued formK : TE → T ∗E ⊗TTE , which is projectable over 1 : E → T ∗E ⊗TE , or by thevertical valued form ν[K] : TE → T ∗TE ⊗ V TE . Their coordinate expressionsare of the typeK = dλ ⊗ (∂λ + Kλν ∂̇ν) , ν[K] = (ḋν − Kλν dλ) ⊗ ∂̇ν ,(1.1)where Kλν ∈ map(TE, IR) and (∂λ, ∂̇λ) or (dλ, ḋλ) are the induced bases of localsections of TTE → TE or T ∗TE → TE, respectively.The connection K is said to be linear if it is a linear fibred morphism over1 : E → T ∗E ⊗ TE . Moreover, the connection K is linear if and only if itscoordinate expression is of the typeKλν = Kλνµ ẋµ , with Kλνµ ∈ map(E, IR) .The torsion of the connection K is defined to be the vertical valued 2–formτ [K] =: −[ϑ, K] : TE → Λ2T ∗E ⊗ V TE ,GEOMETRIC STRUCTURES 197where [ , ] is the Frölicher-Nijenhuis bracket and ϑ : TE → T ∗E ⊗ V TE is thenatural vertical valued 1–form with the coordinate expression ϑ = dλ ⊗ ∂̇λ . Wehave the coordinate expressionτ [K] = ∂̇µKλν dλ ∧ dµ ⊗ ∂̇ν .(1.2)In the linear case, the torsion can be identified with a section τ [K] : E →Λ2T ∗E ⊗ TE and its coordinate expression turns out to be the usual formulaτ [K] = Kλνµ dλ ∧ dµ ⊗ ∂ν . Thus, the connection K is linear and torsion free ifand only if its coordinate expression is of the typeKλν = Kλνµ ẋµ , with Kλνµ = Kµνλ ∈ map(E, IR) .We shall denote by K[g] the canonical torsion free linear spacetime metricconnection given by ∇g = 0. We haveK[g]µλν = −12gλρ (∂µgρν + ∂νgρµ − ∂ρgµν) .(1.3)The curvature of the connection K is defined to be the vertical valued 2–formR[K] =: −[K, K] : TE → Λ2T ∗E ⊗ V TE ,(1.4)where [ , ] is the Frölicher-Nijenhuis bracket. We have the coordinate expressionR[K] = R[K]λµν dλ ∧ dµ ⊗ ∂̇ν(1.5)= −2 (∂λKµν + Kλρ ∂̇ρKµν) dλ ∧ dµ ⊗ ∂̇ν .In the linear case, the coordinate expression turns out to be the usual formulaR[K] = R[K]λµνσ ẋσdλ ∧ dµ ⊗ ∂̇ν(1.6)= −2 (∂λKµνσ + Kλρσ Kµνρ) ẋσdλ ∧ dµ ⊗ ∂̇ν .Hence, in the linear case, the curvature can be identified with a sectionR[K] : E → Λ2T ∗E ⊗ TE ⊗ T ∗E ,with the usual coordinate expressionR[K] = R[K]λµνσ dλ ∧ dµ ⊗ ∂ν ⊗ dσ(1.7)= −2 (∂λKµνσ + Kλρσ Kµνρ) dλ∧ dµ ⊗ ∂ν ⊗ dσ .1.3. The Lie derivative and the exterior covariant differential with re-spect to a spacetime connection. A (general) spacetime connection K con-sidered as a tangent valued 1-form on TE admits as usual, [8], the Lie derivativeof forms on TE. Namely,L[K] φ =(i(K) d − d i(K))φ : TE → Λr+1T ∗TEfor any r-form φ : TE → ΛrT ∗TE. Similarly we can define the Lie derivativeL[R[K]]φ =(i(R[K]) d + d i(R[K]))φ : TE → Λr+2T ∗TE .On the other hand a linear spacetime connection K admits covariant exteriordifferential, [8], of vector-valued forms on E. We apply this operation on T ∗E-valued forms on E and compare it with the Lie derivative.198 J. JANYŠKALet φ be an T ∗E-valued r-form on E, or equivalently φ : E → ΛrT ∗E⊗E T∗Ebe a section. The covariant exterior differential of φ with respect to K is thendefined to be the T ∗E-valued (r + 1)-form dKφ on E given bydKφ(X1, · · · , Xr+1)(Y ) =r+1∑i=1(−1)i+1∇Xi(φ(X1, · · · , X̂i, · · · , Xr+1))(Y )(1.8)+∑i<j(−1)i+jφ([Xi, Xj], X1, . . . , X̂i, . . . , X̂j, . . . , Xr+1)(Y ),for any vector fields Y, X1, · · · , Xr+1 on E, the vector fields X̂i being omitted.Any T ∗E-valued r-form on E can be considered to be a linear horizontal r-formon TE. Then we haveLemma 1.1. Let φ be a linear horizontal r-form on TE and K be a spacetimeconnection. Then the Lie derivative L[K] φ is a linear horizontal (r + 1)-form onTE if and only if K is linear. Moreover, (r + 1)L[K] φ and dKφ coincides.Proof. Let φ = φρλ1...λr ẋρ dλ1 ∧ . . . ∧ dλr , φρλ1...λr ∈ map(E, IR), be a linearhorizontal r-form on TE. Then we haveL[K] φ = (∂µφρλ1...λr ẋρ + φσλ1...λr Kµσ) dµ ∧ dλ1 ∧ . . . ∧ dλr ,i.e., in the linear spacetime connection case,L[K] φ = (∂µφρλ1...λr + φσλ1...λr Kµσρ) ẋρ dµ ∧ dλ1 ∧ . . . ∧ dλr ,which implies that L[K] φ is a linear horizontal (r + 1)-form.On the other hand φ can be considered to be a T ∗E-valued r-form on E withcoordinate expression φ = φρλ1...λr dρ ⊗ (dλ1 ∧ . . . ∧ dλr ). ThendKφ = (r + 1) (∂µφρλ1...λr + φσλ1...λr Kµσρ) dρ⊗ (dµ ∧ dλ1 ∧ . . . ∧ dλr ) . Remark 1.2. Now we shall apply L[K] and dK on specific situation of the scaledmetric g. The metric g can be considered to be a L2 ⊗ T ∗E-valued 1-form on E.Then the covariant exterior differential dKg is a L2 ⊗ T ∗E-valued 2-form definedfor any vector fields X, Y, Z by(dKg)(X, Y )(Z) =(∇X(Y♭) −∇Y (X♭) − ([X, Y ]♭))(Z) ,where ♭ denotes the musical mapping g♭ : TE → L2⊗T ∗E. We have the coordinateexpressiondKg = 2 (∂λgρµ + gσµ Kλσρ) dρ ⊗ (dλ ∧ dµ) .(1.9)On the other hand the musical mapping g♭ can be considered as a linear hori-zontal 1-form on TE with the coordinate expression g♭ = gλµ ẋλ dµ . Then we havethe coordinate expressionL[K] g♭ = (∂λgρµ ẋρ + gρµ Kλρ) dλ ∧ dµ(1.10)and, if K is linear,L[K] g♭ = (∂λgρµ + gσµ Kλσρ) ẋρ dλ ∧ dµ ,(1.11)GEOMETRIC STRUCTURES 199i.e., in the linear case, L[K] g♭ is a linear horizontal 2-form on TE which can beconsidered to be a L2 ⊗ T ∗E valued 2-form on E which coincides with 12dK g. 1.4. Spacetime 2–forms and 2–vectors. The map Tτ [E] : TTE → TE , canbe regarded as a vector valued 1-form υ : TE → T ∗TE ⊗TETE , with coordinateexpression υ = dλ ⊗ ∂λ .We define the spacetime 2–form of TE associated with g and a spacetime con-nection K to be the scaled 2–formΥ[g, K] =: g y(ν[K] ∧ υ): TE → L2 ⊗ Λ2T ∗TE .We have the coordinate expressionΥ[g, K] = gλµ (ḋλ − Kνλ dν) ∧ dµ(1.12)and, if K is linear,Υ[g, K] = gλµ (ḋλ − Kνλρ ẋρ dν) ∧ dµ(1.13)We define the spacetime 2–vector of TE associated with g and a spacetimeconnection K to be the scaled 2–vectorΛ[g, K] =: ḡ y(K ∧ ϑ): TE → L∗2 ⊗ Λ2TTE .We have the coordinate expressionΛ[g, K] = gλµ (∂λ + Kλν ∂̇ν) ∧ ∂̇µ(1.14)and, if K is linear,Λ[g, K] = gλµ (∂λ + Kλνρ ẋρ ∂̇ν) ∧ ∂̇µ .(1.15)Lemma 1.3. We havei(Λ[g, K])Υ[g, K] = −4 .Proof. We havei(Λ[g, K])Υ[g, K] = −gλµ gλµ = −4 . 2. Induced structures on the tangent bundle of the spacetimeWe study symplectic and Poisson structures induced on the tangent bundle ofthe spacetime by the metric g and a spacetime connection K.2.1. General spacetime connection case. Let us assume a spacetime connec-tion K given by (1.1) , the spacetime 2–form Υ[g, K] given by (1.12) and thespacetime 2–vector Λ[g, K] given by (1.14).Lemma 2.1. Υ[g, K] is closed if and only if the following two conditions aresatisfied∂νgλµ + gρµ ∂̇λKνρ − ∂µgλν − gρν ∂̇λKµρ = 0(2.1)R[K]λµν + R[K]µνλ + R[K]νλµ = 0 ,(2.2)where we have set R[K]λµν = gρν R[K]λµρ.200 J. JANYŠKAProof. It follows immediately from the coordinate expressiondΥ[g, K] = −(∂λgρν Kµρ + gρν ∂λKµρ) dλ ∧ dµ ∧ dν− (∂µgλν + gρν ∂̇λKµρ) ḋλ ∧ dµ ∧ dν=12R[K]λµν dλ ∧ dµ ∧ dν− (∂µgρν + gσν ∂̇ρKµσ) (ḋρ − Kλρ dλ) ∧ dµ ∧ dν . Now we shall describe the geometrical interpretation of the equations (2.1) and(2.2). Let us consider the Liouville vector field I = ẋλ ∂̇λ .Lemma 2.2. The conditions (2.1) or (2.2) are equivalent withL[I] L[K] g♭ = 0(2.3)orL[K] L[K] g♭ = 0 ,(2.4)respectively.Proof. We haveL[I] L[K] g♭ =(i(I) d + d i(I)) ((∂λgρµ ẋρ + gρµ Kλρ) dλ ∧ dµ)= ẋρ (∂λgρµ + gσµ ∂̇ρKλσ) dλ ∧ dµ .It is easy to see that L[I] L[K] g♭ = 0 if and only if the condition (2.1) is satisfied.Further from (1.5) we haveL[R[K]]g♭ = gρν R[K]λµρ dλ ∧ dµ ∧ dνi.e., the condition (2.2) is equivalent with L[R[K]]g♭ = 0. But, from (1.4),L[R[K]]g♭ = −L[[K, K]]g♭ = −2 L[K] L[K] g♭.Hence (2.2) is equivalent with L[K] L[K] g♭ = 0. Lemma 2.3. The Schouten bracket[Λ[g, K], Λ[g, K]]: TE → L∗4 ⊗ Λ3TTEhas the coordinate expression[Λ[g, K], Λ[g, K]]= 2 gρν (∂ρgλµ− gσλ ∂̇σKρµ) (∂λ + Kλκ ∂̇κ) ∧ ∂̇µ ∧ ∂̇ν+ R[K]κµν ∂̇κ ∧ ∂̇µ ∧ ∂̇ν ,where we have set R[K]λµν = gλρ gµσ R[K]ρσν .Proof. We havei([Λ[g, K], Λ[g, K]])β = 2 i(Λ[g, K]) di(Λ[g, K])βfor any closed 3-form β.GEOMETRIC STRUCTURES 201Then[Λ[g, K],Λ[g, K]]= 2 gρν (∂ρgλµ− gσλ ∂̇σKρµ) ∂λ ∧ ∂̇µ ∧ ∂̇ν+(gων gρµ Rρωκ + 2 gσν Kρκ (∂σgρµ − gρω ∂̇ωKσµ))∂̇κ ∧ ∂̇µ ∧ ∂̇ν= 2 gρν (∂ρgλµ − gσλ ∂̇σKρµ) (∂λ + Kλκ ∂̇κ) ∧ ∂̇µ ∧ ∂̇ν+ R[K]κµν ∂̇κ ∧ ∂̇µ ∧ ∂̇ν . Lemma 2.4.[Λ[g, K], Λ[g, K]]= 0 if and only if the conditions (2.1) and (2.2)are satisfied, i.e., if and only if the conditions (2.3) and (2.4) are satisfied.Proof. From Proposition 2.3 it follows that[Λ[g, K], Λ[g, K]]= 0 if and only ifgρν (∂ρgλµ − gσλ ∂̇σKρµ) − gρµ (∂ρgλν + gσλ ∂̇σKρν) = 0(2.5)R[K]κµν + R[K]µνκ + R[K]νκµ = 0 .(2.6)But by lowering the indices in (2.5) we get from ∂ρgλµ = −gλτ gµω ∂ρgτω just (2.1)and by lowering indices in (2.6) we get just (2.2) . Theorem 2.5. The metric g and a general spacetime connection K induce onTE natural symplectic and natural Poisson structures if and only if the conditions(2.3) and (2.4) are satisfied.Proof. The regularity of g implies that Υ[g, K] and Λ[g, K] are non degenerate.Lemmas 2.1, 2.2 and 2.4 then imply that Υ[g, K] and Λ[g, K] define symplecticand Poisson structures, respectively, if and only if (2.3) and (2.4) are satisfied. 2.2. Linear spacetime connection case. We assume a linear spacetime con-nection K.Lemma 2.6. Let K be a linear spacetime connection. Υ[g, K] is closed if andonly if L[K] g♭ = 0.Proof. By Lemmas 2.1 and 2.2 Υ[g, K] is closed if and only if (2.3) and (2.4) aresatisfied. But for a linear spacetime connection K the horizontal 2-form L[K] g♭is linear. Moreover, for any linear horizontal r-form φ, we have L[I]φ = φ, i.e.,L[I] L[K] g♭ = L[K] g♭ .Then the condition (2.3) is equivalent with L[K] g♭ = 0 which implies (2.4). Remark 2.7. In [1] we have proved that the spacetime 2-form Υ[g, K] is closedif and only if dKg = 0. By Remark 1.2 it coincides with Lemma 2.6. Lemma 2.8. Let K be a linear spacetime connection then[Λ[g, K], Λ[g, K]]= 0if and only if L[K] g♭ = 0.Proof. This follows immediately. 202 J. JANYŠKATheorem 2.9. Let K be a linear spacetime connection. Then the following iden-tities are equivalent:(1) L[K] g♭ = 0 .(2) dKg = 0 .(3) dΥ[g, K] = 0 .(4)[Λ[g, K], Λ[g, K]]= 0 .Proof. This follows from Remark 1.2 and Lemmas 2.6 and 2.8. Corollary 2.10. A linear spacetime connection K and the metric g induce onTE natural symplectic and Poisson structures if and only if dKg = 0 = L[K] g♭ .Lemma 2.11. Let K be a linear spacetime connection then the following threeidentities are equivalent:(1) dKg = 0.(2) L[K] g♭ = 0.(3) (∇Xg)(Y, Z) − (∇Y g)(X, Z) = 2 g(τ [K](X, Y ), Z) .Proof. (1) ⇔ (2). This follows immediately from Remark 1.2.(1) ⇔ (3). Let us recall that for a linear connection K we have2 τ [K](X, Y ) = ∇Y X −∇XY + [X, Y ] .(2.7)Then, by Remark 1.2,(dKg)(X, Y )(Z) = (∇Xg)(Y, Z) + g(∇XY, Z) − (∇Y g)(X, Z)− g(∇Y X, Z) − g([X, Y ], Z)= (∇Xg)(Y, Z) − (∇Y g)(X, Z) + g(∇XY −∇Y X − [X, Y ], Z)= (∇Xg)(Y, Z) − (∇Y g)(X, Z) − 2 g(τ [K](X, Y ), Z) . Corollary 2.12. If K is a torsion free connection then dKg = 0 = L[K] g♭ isequivalent to (∇Xg)(Y, Z) = (∇Y g)(X, Z), i.e., for a torsion free linear connec-tion, dKg = 0 = L[K] g♭ is equivalent with the symmetry of the (0,3)-tensor field∇g. Theorem 2.13. Let K be a linear torsion free spacetime connection. Then thefollowing identities are equivalent:(1) ∇g is a symmetric (0,3)-tensor field.(2) dΥ[g, K] = 0 .(3)[Λ[g, K], Λ[g, K]]= 0 .Proof. By Corollary 2.12 for a linear torsion free connection the identity dKg =0 = L[K] g♭ is equivalent with ∇g to be fully symmetric. Corollary 2.14. A linear torsion free spacetime connection K and the metric ginduce on TE natural symplectic and Poisson structures if and only if the covariantdifferential ∇g is a symmetric (0,3)-tensor field. GEOMETRIC STRUCTURES 203Remark 2.15. Let us assume the torsion free spacetime metric connection K[g]given by the Christtoffel symbols (1.3). Then we have ∇g = 0, i.e., ∇g is symmetricin the canonical way, and we have the canonical natural symplectic and Poissonstructures on TE given by Υ[g] = Υ[g, K[g]]and Λ[g] = Λ[g, K[g]]. Moreover, inthe metric case, Υ[g] = dg♭. References[1] Janyška, J., Remarks on symplectic and contact 2–forms in relativistic theories, Boll. Un.Mat. Ital. B (7) 9 (1995), 587–616.[2] Janyška, J., Natural symplectic structures on the tangent bundle of a space–time, TheProceedings of the Winter School Geometry and Topology (Srńı, 1995), Rend. Circ. Mat.Palermo (2) Suppl. 43 (1996), 153–162.[3] Janyška, J., Modugno, M., Classical particle phase space in general relativity, DifferentialGeometry and Applications, Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic,Masaryk University, Brno 1996, 573–602.[4] Janyška, J., Modugno, M., On quantum vector fields in general relativistic quantum me-chanics, in: Proc. 3rd Internat. Workshop Differential Geom. Appl., Sibiu (Romania) 1997,General Mathematics 5 (1997), 199–217.[5] Janyška, J., Natural Poisson and Jacobi structures on the tangent bundle of a pseudo-Riemannian manifold, Contemporary Mathematics 288 (2001), Global Differential Geom.:The Math. Legacy of Alfred Gray, eds. M. Fernándes and J. A. Wolf, 343–347.[6] Janyška, J., Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold, Arch. Math. (Brno) 37 (2001), 143–160.[7] Krupka, D., Janyška, J., Lectures on Differential Invariants, Folia Fac. Sci. Natur. Univ.Masaryk. Brun. Math., 1990.[8] Kolář, I., Michor, P. W., Slovák, J., Natural Operations in Differential Geometry, Springer–Verlag 1993.[9] Libermann, P., Marle, Ch. M., Symplectic Geometry and Analytical Mechanics, ReidelPubl., Dordrecht 1987.[10] Nijenhuis, A., Natural bundles and their general properties, Differential Geom., in honourof K. Yano, Kinokuniya, Tokyo 1972, 317–334.[11] Vaisman, I., Lectures on the Geometry of Poisson Manifolds, Birkhäuser Verlag 1994.Department of Mathematics, Masaryk UniversityJanáčkovo nám 2a, 602 00 Brno, Czech RepublicE-mail: janyska@math.muni.cz

Geometric structures on the tangent bundle of the Einstein spacetime

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Geometric structures on the tangent bundle of the Einstein spacetime

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