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- 1. Hidden symmetries of the ﬁve-dimensional Sasaki-Einstein metrics Mihai Visinescu Department of Theoretical Physics National Institute for Physics and Nuclear Engineering ”Horia Hulubei” Bucharest, Romania BALKAN WORKSHOP 2013 – Beyond the Standard Models – Vrnjaˇcka Banja, Serbia, 25 – 29 April, 2013
- 2. Outline 1. Symmetries and conserved quantities 2. Killing forms 3. K¨ahler, Sasaki manifolds 4. Killing forms on Kerr-NUT-(A)dS spaces 5. Y(p, q) spaces 6. Killing forms on mixed 3-Sasakian manifolds 7. Outlook
- 3. Symmetries and conserved quantities (1) Let (M, g) be a n-dimensional manifold equipped with a (pseudo-)Riemmanian metric g and denote by H = 1 2 gij pipj , the Hamilton function describing the motion in a curved space. In terms of the phase-space variables(xi, pi) the Poisson bracket of two observables P, Q is {P, Q} = ∂P ∂xi ∂Q ∂pi − ∂P ∂pi ∂Q ∂xi .
- 4. Symmetries and conserved quantities (2) A conserved quantity of motions expanded as a power series in momenta: K = K0 + p k=1 1 k! Ki1···ik (x)pi1 · · · pik . Vanishing Poisson bracket with the Hamiltonian, {K, H} = 0, implies K(i1···ik ;i) = 0 , Such symmetric tensor Ki1···ik k is called a St¨ackel-Killing (SK) tensor of rank k
- 5. Killing forms (1) A vector ﬁeld X on a (pseudo-)Riemannian manifold (M, g) is said to be a Killing vector ﬁeld if the Levi-Civita connection of g satisﬁes g( Y X, Z) + g(Y, Z X) = 0, for all vector ﬁelds Y, Z on M.
- 6. Killing forms (2) A natural generalization of Killing vector ﬁelds is given by the conformal Killing vector ﬁelds , i.e. vector ﬁelds with a ﬂow preserving a given conformal class of metrics. More general, a conformal Killing-Yano tensor ( also called conformal Yano tensor or conformal Killing form or twistor form ) of rank p on a (pseudo-) Riemannian manifold (M, g) is a p -form ω which satisﬁes: X ω = 1 p + 1 X−| dω − 1 n − p + 1 X∗ ∧ d∗ ω, for any vector ﬁeld X on M, where n is the dimension of M , X∗ is the 1-form dual to the vector ﬁeld X with respect to the metric g , −| is the operator dual to the wedge product and d∗ is the adjoint of the exterior derivative d.
- 7. Killing forms (3) If ω is co-closed then we obtain the deﬁnition of a Killing-Yano tensor, also called Yano tensor or Killing form : ωi1...ik−1(ik ;j) = 0 . Moreover, a Killing form ω is said to be a special Killing form if it satisﬁes for some constant c the additional equation X (dω) = cX∗ ∧ ω , for any vector ﬁeld X on M . These two generalizations of the Killing vectors could be related. Given two Killing-Yano tensors ωi1,...,ir and σi1,...,ir it is possible to associate with them a St¨ackel-Killing tensor of rank 2 : K (ω,σ) ij = ωii2...ir σ i2...ir j + σii2...ir ω i2...ir j .
- 8. K¨ahler, Sasaki manifolds (1) Even dimensions (1) An almost Hermitian structure on a smooth manifold M is a pair (g, J), where g is a Riemannian metric on M and J is an almost complex structure on M, which is compatible with g, i.e. g(JX, JY) = g(X, Y), for all vector ﬁelds X, Y on M. In this case, the triple (M, J, g) is called an almost Hermitian manifold. If J is parallel with respect to the Levi-Civita connection of g, then (M, J, g) is said to be a K¨ahler manifold. On a K¨ahler manifold, the associated K¨ahler form, i.e the alternating 2-form Ω deﬁned by Ω(X, Y) = g(JX, Y) is closed.
- 9. K¨ahler, Sasaki manifolds (2) Even dimensions (2) In local holomorphic coordinates (z1, ..., zm), the associated K¨ahler form Ω can be written as Ω = igj¯k dzj ∧ d¯zk = X∗ j ∧ Y∗ j = i 2 Z∗ j ∧ ¯Z∗ j , where (X1, Y1, ..., Xm, Ym) is an adapted local orthonormal ﬁeld (i.e. such that Yj = JXj ), and (Zj, ¯Zj) is the associated complex frame given by Zj = 1 2 (Xj − iYj), ¯Zj = 1 2 (Xj + iYj) .
- 10. K¨ahler, Sasaki manifolds (3) Even dimensions (3) Volume form is dV = 1 m! Ωm , where dV denotes the volume form of M, Ωm is the wedge product of Ω with itself m times, m being the complex dimension of M . Hence the volume form is a real (m, m)-form on M. If the volume of a K¨ahler manifold is written as dV = dV ∧ d ¯V then dV is the complex volume holomorphic (m, 0) form of M. The holomorphic form of a K¨ahler manifold can be written in a simple way with respect to any (pseudo-)orthonormal basis, using complex vierbeins ei + Jei. Up to a power factor of the imaginary unit i, the complex volume is the exterior product of these complex vierbeins.
- 11. K¨ahler, Sasaki manifolds (4) Odd dimensions (1) Let M be a smooth manifold equipped with a triple (ϕ, ξ, η), where ϕ is a ﬁeld of endomorphisms of the tangent spaces, ξ is a vector ﬁeld and η is a 1-form on M. If we have: ϕ2 = −I + η ⊗ ξ , η(ξ) = 1 then we say that (ϕ, ξ, η) is an almost contact structure on M.
- 12. K¨ahler, Sasaki manifolds (5) Odd dimensions (2) A Riemannian metric g on M is said to be compatible with the almost contact structure (ϕ, ξ, η) iff the relation g(ϕX, ϕY) = g(X, Y) − η(X)η(Y) holds for all pair of vector ﬁelds X, Y on M. An almost contact metric structure (ϕ, ξ, η, g) is a Sasakian structure iff the Levi-Civita connection of the metric g satisﬁes ( X ϕ)Y = g(X, Y)ξ − η(Y)X for all vector ﬁelds X, Y on M
- 13. K¨ahler, Sasaki manifolds (6) Metric cone (1) A Sasakian structure may also be reinterpreted and characterized in terms of the metric cone as follows. The metric cone of a Riemannian manifold (M, g) is the manifold C(M) = (0, ∞) × M with the metric given by ¯g = dr2 + r2 g, where r is a coordinate on (0, ∞). M is a Sasaki manifold iff its metric cone C(M) is K¨ahler. The cone C(M) is equipped with an integrable complex structure J and a K¨ahler 2-form Ω, both of which are parallel with respect to the Levi-Civita connection ¯ of ¯g.
- 14. K¨ahler, Sasaki manifolds (7) Metric cone (2) M has odd dimension 2n + 1, where n + 1 is the complex dimension of the K¨ahler cone. The Sasakian manifold (M, g) is naturally isometrically embedded into the cone via the inclusion {r = 1} × M ⊂ C(M) and the K¨ahler structure of the cone (C(M), ¯g) induces an almost contact metric structure (φ, ξ, η, g) on M.
- 15. K¨ahler, Sasaki manifolds (8) Einstein manifolds An Einstein-Sasaki manifold is a Riemannian manifold (M, g) that is both Sasaki and Einstein, i.e. Ricg = λg for some real constant λ, where Ricg denotes the Ricci tensor of g. Einstein manifolds with λ = 0 are called Ricci-ﬂat manifolds. An Einstein-K¨ahler manifold is a Riemannian manifold (M, g) that is both K¨ahler and Einstein. The most important subclass of K¨ahler-Einstein manifolds are the Calabi-Yau manifolds (i. e. K¨ahler and Ricci-ﬂat). A Sasaki manifold M is Einstein iff the cone metric C(M) is K¨ahler Ricci-ﬂat. In particular the K¨ahler cone of a Sasaki-Einstein manifold has trivial canonical bundle and the restricted holonomy group of the cone is contained in SU(m), where m denotes the complex dimension of the K¨ahler cone.
- 16. K¨ahler, Sasaki manifolds (9) Progression from Einstein-K¨ahler to Einstein-Sasaki to Calabi-Yau manifolds (1) Suppose we have an Einstein-Sasaki metric gES on a manifold M2n+1 of odd dimension 2n + 1. An Einstein-Sasaki manifold can always be written as a ﬁbration over an Einstein-K¨ahler manifold M2n with the metric gEK twisted by the overall U(1) part of the connection ds2 ES = (dψn + 2A)2 + ds2 EK , where dA is given as the K¨ahler form of the Einstein-K¨ahler base. The metric of the cone manifold M2n+2 = C(M2n+1) is ds2 cone = dr2 + r2 ds2 ES = dr2 + r2 (dψn + 2A)2 + ds2 EK .
- 17. K¨ahler, Sasaki manifolds (10) Progression from Einstein-K¨ahler to Einstein-Sasaki to Calabi-Yau manifolds (2) The cone manifold is Calabi-Yau and its K¨ahler form is Ωcone = rdr ∧ (dψn + 2A) + r2 ΩEK , and the K¨ahler condition dΩcone = 0 implies dA = ΩEK , where ΩEK is K¨ahler form of the Einstein-K¨ahler base manifold M2n. The Sasakian 1-form of the Einstein-Sasaki metric is η = 2A + dψn , which is a special unit-norm Killing 1-form obeying for all vector ﬁelds X X η = 1 2 X−| dη , X (dη) = −2X∗ ∧ η .
- 18. K¨ahler, Sasaki manifolds (11) Hidden symmetries (1) The hidden symmetries of the Sasaki manifold M2n+1 are described by the special Killing (2k + 1)−forms Ψk = η ∧ (dη)k , k = 0, 1, · · · , n. Besides these Killing forms, there are n closed conformal Killing forms ( also called ∗-Killing forms) Φk = (dη)k , k = 1, · · · , n.
- 19. K¨ahler, Sasaki manifolds (12) Hidden symmetries (2) Moreover, in the case of holonomy SU(n + 1) , i.e. the cone M2n+2 = C(M2n+1) is K¨ahler and Ricci-ﬂat, or equivalently M2n+1 is Einstein-Sasaki, it follows that we have two additional Killing forms of degree n + 1 on the manifold M2n+1. These additional Killing forms are connected with the additional parallel forms of the Calabi-Yau cone manifold M2n+2 given by the complex volume form and its conjugate.
- 20. K¨ahler, Sasaki manifolds (13) Hidden symmetries (3) In order to extract the corresponding additional Killing forms on the Sasaki-Einstein space we make use of the fact that for any p -form ω on the space M2n+1 we can deﬁne an associated (p + 1) -form ωC on the cone C(M2n+1) ωC := rp dr ∧ ω + rp+1 p + 1 dω . The 1-1-correspondence between special Killing p -forms on M2n+1 and parallel (p + 1) -forms on the metric cone C(M2n+1) allows us to describe the additional Killing forms on Sasaki-Einstein spaces.
- 21. Killing forms on Kerr-NUT-(A)dS spaces (1) In the case of Kerr-NUT-(A)dS spacetime the Einstein-K¨ahler metric gEK and the K¨ahler potential A are gEK = ∆µdx2 µ Xµ + Xµ ∆µ n−1 j=0 σ(j) µ dψj 2 , Xµ = −4 n+1 i=1 (αi − xµ) − 2bµ , A = n−1 k=0 σ(k+1) dψk ,
- 22. Killing forms on Kerr-NUT-(A)dS spaces (2) with ∆µ = ν=µ (xν − xµ) , σ(k) µ = ν1<···<νk νi =µ xν1 . . . xνk , σ(k) = ν1<···<νk xν1 . . . xνk . Here, coordinates xµ (µ = 1, . . . , n) stands for the Wick rotated radial coordinate and longitudinal angles and the Killing coordinates ψk (k + 0, . . . , n − 1) denote time and azimuthal angles with Killing vectors ξ(k) = ∂ψk . αi (i = 1, . . . , n + 1) and bµ are constants related to the cosmological constant, angular momenta, mass and NUT parameters.
- 23. Killing forms on Kerr-NUT-(A)dS spaces (3) Write the metric gEK on the Einstein-K¨ahler manifold M2n in the form gEK = oˆµ oˆµ + ˜oˆµ˜oˆµ , and the K¨ahler 2-form Ω = dA = oˆµ ∧ ˜oˆµ . where oˆµ = ∆µ Xµ(xµ) dxµ , ˜oˆµ = Xµ(xµ) ∆µ n−1 j=0 σ(j) µ dψj .
- 24. Killing forms on Kerr-NUT-(A)dS spaces (4) We introduce the following complex vierbeins on Einstein- K¨ahler manifold M2n ζµ = oµ + i˜oˆµ , µ = 1, · · · , n . On the Calabi-Yau cone manifold M2n+2 the complex vierbeins are Λµ = rζµ for µ = 1, · · · , n and Λn+1 = dr r + iη . The standard complex volume form of the Calabi-Yau cone manifold M2n+2 is dV = Λ1 ∧ Λ2 ∧ · · · ∧ Λn+1 .
- 25. Killing forms on Kerr-NUT-(A)dS spaces (5) As real forms we obtain the real part respectively the imaginary part of the complex volume form. For example, writing Λj = λ2j−1 + iλ2j, j = 1, ..., n + 1, the real part of the complex volume is given by Re dV = [n+1 2 ] p=0 1≤i1<i2<...<in+1≤2n+2 (C) (−1)p λi1 ∧λi2 ∧...∧λin+1 (1) where the condition (C) means that in the second sum are taken only the indices i1, ..., in+1 such that i1 + ... + in+1 = (n + 1)2 + 2p and (ik , ik+1) = (2j − 1, 2j), for all k ∈ {1, ..., n} and j ∈ {1, ..., n + 1}.
- 26. Killing forms on Kerr-NUT-(A)dS spaces (6) The imaginary part of the complex volume is given by Im dV = [n 2 ] p=0 1≤i1<i2<...<in+1≤2n+2 (C ) (−1)p λi1 ∧λi2 ∧...∧λin+1 (2) where the condition (C ) in means that in the second sum are considered only the indices i1, i2, ..., in+1 such that i1 + ... + in+1 = (n + 1)2 + 2p + 1 and (ik , ik+1) = (2j − 1, 2j), for all k ∈ {1, ..., n} and j ∈ {1, ..., n + 1}.
- 27. Killing forms on Kerr-NUT-(A)dS spaces (7) Finally, the Einstein-Sasaki manifold M2n+1 is identiﬁed with the submanifold {r = 1} of the Calabi-Yau cone manifold M2n+2 = C(M2n+1) and the additional (n + 1)-Killing forms are accordingly acquired. The 1-1-correspondence between special Killing p -forms on M2n+1 and parallel (p + 1) -forms on the metric cone C(M2n+1) allows us to describe the additional Killing forms on Einstein-Sasaki spaces. In order to ﬁnd the additional Killing forms on the Einstein - K´ahler manifold M2n+1 we must identify the ωM form in the complex volume form of the Calabi-Yau cone.
- 28. Y(p,q) spaces (1) Inﬁnite family Y(p, q) of Einstein-Sasaki metrics on S2 × S3 provides supersymmetric backgrounds relevant to the AdS/CFT correspondence. The total space Y(p, q) of an S1-ﬁbration over S2 × S2 with relative prime winding numbers p and q is topologically S2 × S3. Explicit local metric of the 5-dimensional Y(p, q) manifold given by the line element ds2 ES = 1 − c y 6 (dθ2 + sin2 θ dφ2 ) + 1 w(y)q(y) dy2 + q(y) 9 (dψ − cos θ dφ)2 + w(y) dα + ac − 2y + c y2 6(a − y2) [dψ − cos θ dφ] 2 , where
- 29. Y(p,q) spaces (2) w(y) = 2(a − y2) 1 − cy , q(y) = a − 3y2 + 2cy3 a − y2 and a, c are constants. The constant c can be rescaled by a diffeomorphism and in what follows we assume c = 1. The coordinate change α = −1 6 β − 1 6 c ψ , ψ = ψ takes the line element to the following form ( with p(y) = w(y) q(y) ) ds2 ES = 1 − y 6 (dθ2 + sin2 θ dφ2 ) + 1 p(y) dy2 + p(y) 36 (dβ + cos θ dφ)2 + 1 9 [dψ − cos θ dφ + y(dβ + cos θ dφ)]2 ,
- 30. Y(p,q) spaces (3) One can write ds2 ES = dS2 EK + ( 1 3 dψ + σ)2 The Sasakian 1-form of the Y(p, q) space is η = 1 3 dψ + σ , with σ = 1 3 [− cos θ dφ + y(dβ + cos θ dφ)] . connected with local K¨ahler form ΩEK . This form of the metric with the 1-form η is the standard one for a locally Einstein-Sasaki metric with ∂ ∂ψ the Reeb vector ﬁeld.
- 31. Y(p,q) spaces (4) The local K¨ahler and holomorphic (2, 0) form for ds2 EK are ΩEK = 1 − y 6 sin θdθ ∧ dφ + 1 6 dy ∧ (dβ + cos θdφ) dVEK = 1 − y 6p(y) (dθ + i sin θdφ) ∧ dy + i p(y) 6 (dβ + cos θdφ)
- 32. Y(p,q) spaces (5) From the isometries SU(2) × U(1) × U(1) the momenta Pφ, Pψ, Pα and the Hamiltonian describing the geodesic motions are conserved. Pφ is the third component of the SU(2) angular momentum, while Pψ and Pα are associated with the U(1) factors. Additionally, the total SU(2) angular momentum given by J2 = P2 θ + 1 sin2 θ (Pφ + cos θPψ)2 + P2 Ψ , is also conserved.
- 33. Y(p,q) spaces (6) Speciﬁc conserved quantities for Einstein-Sasaki spaces (1) First of all from the 1-form η Ψ = η ∧ dη = 1 9 [(1 − y) sin θ dθ ∧ dφ ∧ dψ + dy ∧ dβ ∧ dψ + cos θ dy ∧ dφ ∧ dψ − cos θ dy ∧ dβ ∧ dφ + (1 − y)y sin θ dβ ∧ dθ ∧ dφ] . is a special Killing form. Let us note also that Ψk = (dη)k , k = 1, 2 , are closed conformal Killing forms ( -Killing forms).
- 34. Y(p,q) spaces (7) Speciﬁc conserved quantities for Einstein-Sasaki spaces (2) On the Calabi-Yau manifold the K¨ahler form is Ωcone = rdr ∧ η + r2 ΩEK . and the holomorphic (3, 0) form is dVcone = eψ r2 dVEK ∧ [dr + ir ∧ η] = eψ r2 1 − y 6p(y) dθ + i sin θdφ ∧ dy + i p(y) 6 (dβ + cos θdφ) ∧ dr + i r 3 [ydβ + dψ − (1 − y) cos θdφ
- 35. Y(p,q) spaces (8) Speciﬁc conserved quantities for Einstein-Sasaki spaces (3) The additional Killing 3-forms of the Y(p, q) spaces are extracted from the volume form dVcone. Using the the 1-1-correspondence between special Killing p -forms on M2n+1 and parallel (p + 1) -forms on the metric cone C(M2n+1) for p = 2 we get the following additional Killing 2-forms of the Y(p, q) spaces written as real forms: Ξ = Re ωM = 1 − y 6 p(y) × cos ψ −dy ∧ dθ + p(y) 6 sin θ dβ ∧ dφ − sin ψ − sin θ dy ∧ dφ − p(y) 6 dβ ∧ dθ + p(y) 6 cos θ dθ ∧ dφ
- 36. Y(p,q) spaces (9) Speciﬁc conserved quantities for Einstein-Sasaki spaces (4) Υ = Im ωM = 1 − y 6 p(y) × sin ψ −dy ∧ dθ + p(y) 6 sin θ dβ ∧ dφ + cos ψ − sin θ dy ∧ dφ − p(y) 6 dβ ∧ dθ + p(y) 6 cos θ dθ ∧ dφ
- 37. Y(p,q) spaces (10) Speciﬁc conserved quantities for Einstein-Sasaki spaces (5) The St¨ackel-Killing tensors associated with the Killing forms Ψ , Ξ , Υ are constructed as usual. Together with the Killing vectors Pφ, Pψ, Pα and the total angular momentum J2 these St¨ackel-Killing tensors provide the superintegrability of the Y(p, q) geometries.
- 38. Killing forms on mixed 3-Sasakian manifolds (1) 3-Sasakian manifolds (1) A Riemannian manifold (M, g) of real dimension m is 3-Sasakian if the holonomy group of the metric cone (C(M) , ¯g) = (R+ × M , dr2 + r2 g) reduces to a subgroup of Sp m+1 4 . In particular, m = 4n + 3 , n ≥ 1 and (C(M) , ¯g) is hyperK¨ahler. A 3-Sasakian manifold admits three characteristic vector ﬁelds (ξ1 , ξ2 , ξ3), satisfying any of the corresponding conditions of the Sasakian structure, such that g(ξα , ξβ) = δαβ , and [ξα , ξβ] = 2 αβγξγ .
- 39. Killing forms on mixed 3-Sasakian manifolds (2) 3-Sasakian manifolds (2) Let (M, g) be a 3-Sasakian manifold and (ϕα, ξα, ηα), α ∈ {1, 2, 3} , be its 3-Sasakian structure. then ηα(ξβ) = δαβ , ϕα(ξβ) = − αβγξγ , ϕαϕβ − ξα ⊗ ηβ = − αβγϕγ − δαβI . Theorem Every 3-Sasakian manifold (M , g) of dimension 4n + 3 is Einstein with Einstein constant λ = 4n + 2.
- 40. Killing forms on mixed 3-Sasakian manifolds (3) A mixed 3-structure on a smooth manifold M is a triple of structures (ϕα, ξα, ηα), α ∈ {1, 2, 3} , which are almost paracontact structures for α = 1, 2 and almost contact structure for α = 3 , satisfying the following compatibility conditions ηα(ξβ) = 0, ϕα(ξβ) = τβξγ, ϕβ(ξα) = −ταξγ, ηα ◦ ϕβ = −ηβ ◦ ϕα = τγηγ , ϕαϕβ − ταηβ ⊗ ξα = −ϕβϕα + τβηα ⊗ ξβ = τγϕγ , where (α, β, γ) is an even permutation of (1, 2, 3) and τ1 = τ2 = −τ3 = −1.
- 41. Killing forms on mixed 3-Sasakian manifolds (4) If a manifold M with a mixed 3-structure (ϕα, ξα, ηα)α=1,3 admits a semi-Riemannian metric g such that: g(ϕαX, ϕαY) = τα[g(X, Y) − εαηα(X)ηα(Y)], for all X, Y ∈ Γ(TM) and α = 1, 2, 3 , where εα = g(ξα, ξα) = ±1 then we say that M has a metric mixed 3-structure and g is called a compatible metric.
- 42. Killing forms on mixed 3-Sasakian manifolds (5) A manifold M endowed with a (positive/negative) metric mixed 3-structure (ϕα, ξα, ηα, g) is said to be a (positive/negative) mixed 3-Sasakian structure if (ϕ3, ξ3, η3, g) is a Sasakian structure, while both structures (ϕ1, ξ1, η1, g) and (ϕ2, ξ2, η2, g) are para-Sasakian, i.e. ( X ϕα)Y = τα[g(X, Y)ξα − αηα(Y)X] for all vector ﬁelds X, Y on M and α = 1, 2, 3. Theorem Any (4n + 3)−dimensional manifold endowed with a mixed 3-Sasakian structure is an Einstein space with Einstein constant λ = (4n + 2)θ, with θ = 1, according as the metric mixed 3-structure is positive or negative, respectively.
- 43. Killing forms on mixed 3-Sasakian manifolds (6) In (mixed) 3-Sasakian case any linear combination of the forms Ψk1,k2,k3 deﬁned by Ψk1,k2,k3 = k1 k1 + k2 + k3 [η1 ∧ (dη1)k1−1 ] ∧ (dη2)k2 ∧ (dη3)k3 + k2 k1 + k2 + k3 (dη1)k1 ∧ [η2 ∧ (dη2)k2−1 ] ∧ (dη3)k3 + k3 k1 + k2 + k3 (dη1)k1 ∧ (dη2)k2 ∧ [η3 ∧ (dη3)k3−1 ] for arbitrary positive integers k1, k2, k3 , is a special Killing form on M.
- 44. Outlook Complete integrability of geodesic equations Separability of Hamilton-Jacobi, Klein-Gordon, Dirac equations Hidden symmetries of other spacetime structures
- 45. References M. Visinescu, Mod. Phys. Lett. A 27, 1250217 (2012) M. Visinescu, G. E. Vˆılcu, SIGMA 8, 058 (2012) M. Visinescu, Mod. Phys. Lett. A 26, 2719 (2011) M. Visinescu, SIGMA 7, 037 (2011) M. Visinescu, J. Phys.: Conf. Series 411, 012030 (2013)