THE REAL JACOBI GROUP REVISITED

The real Jacobi group G1 (R), defined as the semidirect product of the group SL(2,R) with the Heisenberg group H1, is embedded in the 4× 4 matrix realisation of the group Sp(2,R). The left-invariant one-forms on G1 (R) and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates (x, y, θ, p, q, κ), and the left-invariant metric is obtained. The Heisenberg group H1 and Sp(2,R) are considered as subgroups of SL(2,R). An invariant metric in the variables (x, y, θ) on the Sasaki manifold SL(2,R) is presented. The well known Kähler balanced metric in the variables (x, y, p, q) of the four-dimensional Siegel-Jacobi upper half-plane X1 = GJ1 (R) SO(2)×R ≈ X1 × R 2 is written down as sum of the squares of four invariant oneforms, where the Siegel upper half-plane X1 is realized as C 3 τ := x+i y, y > 0. A fivedimensional manifold, called extended Siegel-Jacobi upper half-plane X̃1 = GJ1 (R) SO(2) ≈ X1×R, is considered, and its left-invariant metric in the variables (x, y, p, q, κ) is determined. As a bi-product, we directly prove that the Siegel-Jacobi upper half-plane X1 is not a naturally reductive space with respect to the balanced metric, but we underline that it is naturally reductive in the coordinates furnished by the FC-transform.


Introduction
The Jacobi group [39,55] of degree n is defined as the semi-direct product G J n = H n Sp(n, R) C , where Sp(n, R) C = Sp(n, C)∩U(n, n) and H n denotes the (2n+1)-dimensional Heisenberg group [16,17,108]. To the Jacobi group G J n it is associated a homogeneous manifold, called the Siegel-Jacobi ball D J n [16], whose points are in C n × D n , i.e., a partially-bounded space. D n denotes the Siegel (open) ball of degree n. The non-compact Hermitian symmetric space Sp(n, R) C / U(n) admits a matrix realization as a homogeneous bounded domain [63]: The Jacobi group is an interesting object in several branches of Mathematics, with important applications in Physics, see references in [13,16,21,28,29].
Our special interest to the Jacobi group comes from the fact that G J n is a coherent state (CS) group [79,80,84,85,86,87], i.e., a group which has orbits holomorphically embedded into a projective Hilbert space, for a precise definition see [12,Definition 1], [13], [22,Section 5.2.2] and [29,Remark 4.4]. To an element X in the Lie algebra g of G we associated a first order differential operator X on the homogenous space G/H, with polynomial holomorphic coefficients, see [23,24,25] for CS based on hermitian symmetric spaces, where the maximum degree of the polynomial is 2. In [12,26,27] we have advanced the hypothesis that for CS groups the coefficients in X are polynomial, and in [13] we have verified this for G J 1 .
It was proved in [16,17,21] that the Kähler two-form on D J n , invariant to the action of the Jacobi group G J n , has the expression −iω D J n (z, W ) = k 2 Tr(B ∧B) + µ Tr A tM ∧Ā , A = dz + dWη, It was emphasized [17] that the change of coordinates (z, W ) → (η, W ), called FC-transform, has the meaning of passing from un-normalized to normalized Perelomov CS vectors [92]. Also, the FC-transform (1.2) is a homogeneous Kähler diffeomorphism from D J n to C n × D n , in the meaning of the fundamental conjecture for homogeneous Kähler manifolds [53,61,103].
We reproduce a proposition which summarizes some of the geometric properties of the Jacobi group and the Siegel-Jacobi ball [21,22], see the definitions of the notions appearing in the enunciation below in [13,19,20,22] and also Appendix B for some notions on Berezin's quantization: (i) The Jacobi group G J n is a unimodular, non-reductive, algebraic group of Harish-Chandra type.
(ii) The Siegel-Jacobi domain D J n is a homogeneous, reductive, non-symmetric manifold associated to the Jacobi group G J n by the generalized Harish-Chandra embedding. (v) The manifold D J n is a quantizable manifold. (vi) The manifold D J n is projectively induced, and the Jacobi group G J n is a CS-type group. (vii) The Siegel-Jacobi ball D J n is not an Einstein manifold with respect to the balanced metric corresponding to the Kähler two-form (1.1), but it is one with respect to the Bergman metric corresponding to the Bergman Kähler two-form.
(viii) The scalar curvature is constant and negative.
The properties of geodesics on the Siegel-Jacobi disk D J 1 have been investigated in [13,19,20], while in [22] we have considered geodesics on the Siegel-Jacobi ball D J n . We have explicitly determined the equations of geodesics on D J n . We have proved that the FC-transform (1.2) is not a geodesic mapping on the non-symmetric space D J n , see definition in [82]. However, it was not yet anlayzed whether the Siegel-Jacobi ball is a naturally reductive space or not, even if its points are in C n × D n , both manifolds being naturally reductive, see Definition A.8 and Proposition A. 9. In fact, this problem was the initial point of the present investigation. The answer to this question has significance in our approach [19] to the geometry of the Siegel-Jacobi ball via CS in the meaning of Perelomov [92]. We have proved in [8] that for symmetric manifolds the FC-transform gives geodesics, but the Siegel-Jacobi ball is not a symmetric space. Similar properties are expected for naturally reductive spaces [9,10].
In the standard procedure of CS, see [16,92], the Kähler two-form on a homogenous manifold M is obtained from the Kähler potential f (z,z) = log K(z,z) via the recipe where K(z, z) := (ez, ez) is the scalar product of two CS at z ∈ M . In [21] we have underlined that the metric associated to the Kähler two-form (1.3) is a balanced metric, see more details in Appendix B.
The real Jacobi group of degree n is defined as G J n (R) := Sp(n, R) H n , where H n is the real (2n+1)-dimensional Heisenberg group. Sp(n, R) C and G J n are isomorphic to Sp(n, R) and G J n (R) respectively as real Lie groups, see [17,Proposition 2].
We have applied the partial Cayley transform from the Siegel-Jacobi ball to the Siegel-Jacobi upper half-plane and we have obtained the balanced metric on X J n , see [17,Proposition 3]. However, the mentioned procedure of obtaining the invariant metric on homogeneous Kähler manifolds works only for even dimensional CS manifolds. For example, starting from the six dimensional real Jacobi group G J 1 (R) = SL(2, R) H 1 , we have obtained the Kähler invariant two-form ω X J 1 (2.4) on the Siegel-Jacobi upper half-plane, a four dimensional homogeneous manifold attached to the Jacobi group, X J 1 = G J 1 (R) SO(2)×R ≈ X 1 × R 2 [13,14,18,19], obtained previously by Berndt [37,38], and Kähler [68,69].
In the present paper we determine the invariant metric on a five dimensional homogeneous manifold, here called the extended Siegel-Jacobi upper half-plane, denotedX J 1 = G J 1 (R) SO(2) ≈ X 1 × R 3 . It will be important to find applications in Physics of the invariant metric (5.25) on the five-dimensional manifoldX J 1 . In order to obtain invariant metric on odd dimensional manifolds, we are obliged to change our strategy applied previously to get the invariant metric on homogeneous Kähler manifolds. Instead of the mentioned first order differential operators on M = G/H with holomorphic polynomial coefficients X associated to X in the Lie algebra g of G [12,26], we have to use the fundamental vector field X * associated with X, see Appendix A. We have to abandon the approach in which the Jacobi algebra is defined as the semi-direct sum g J 1 := h 1 su(1, 1), where only the generators of su(1, 1) have a matrix realization, see [13] and the summary in Section 2.
The approach of mathematicians is to consider the real Jacobi group G J 1 (R) as subgroup of Sp(2, R). In the present paper we follow the notation in [39,55] for the real Jacobi group G J 1 (R), realized as submatrices of Sp(2, R) of the form is related to the Heisenberg group H 1 .
To get the invariant metric onX J 1 , we have determined the invariant one-forms λ 1 , . . . , λ 6 on G J 1 (R), the main tool of the present paper, see details on the method in Appendix D.1.1. Then we have determined the invariant vector fields L j verifying the relations λ i | L j = δ ij , i, j = 1, . . . , 6, such that L j are orthonormal with respect to the metric ds 2 G J 1 (R) in the S-variables (x, y, θ, p, q, κ), see [39, p. 10]. This is the idea of the method of the moving frame of E. Cartan [49,50,56] explained in Section 5.4.
The paper is laid out as follows. In Section 2 we recall how we have obtained the Kähler two-forms on the Siegel-Jacobi disk D J 1 and on the Siegel-Jacobi upper half-plane X J 1 , specifying the FC-transforms. Section 3 describes the real Heisenberg group H 1 embedded into Sp(2, R): invariant one-forms, invariant metrics in the variables (λ, µ, κ). Note that in the formula (3.4) the last parenthesis (dκ − µdλ + λdµ) 2 replaces (dκ) 2 on the Euclidean space R 3 (κ, λ, µ) and the idea of the paper is to see the effect of this substitution in the invariant metric of the five-dimensional manifoldX J 1 . Section 4 deals with the SL(2, R) group as subgroup of Sp(2, R) in the variables (x, y, θ), which describe the Iwasawa decomposition. SL(2, R) is treated as a Sasaki manifold, with the invariant metric written down as sum of squares of the invariant one-forms λ 1 , . . . , λ 3à la Milnor [83], while the metric on X 1 is just λ 2 1 + λ 2 2 . Invariant metrics on SL(2, R) in other coordinates previously obtained by other authors are mentioned in Comment 4.2. Details on the calculations referring to SL(2, R) are presented also in Appendix C.3. Section 5 presents the real Jacobi group G J 1 (R) in the EZ and S-coordinates [39]. The action of the reduced Jacobi group G J (R) 0 on the four-dimensional manifold X J 1 is recalled [13,14] and the fundamental vector fields on it are obtained. Also the action of G J 1 (R) on the 5-dimensional manifoldX J 1 , called extended Siegel-Jacobi upper half-plane, is established in Lemma 5.1. The well known Kählerian balanced metric on the Siegel-Jacobi upper-half plane is written down as sum of the square of four invariant one-forms in Section 5.4. For this we have obtained the invariant one-forms on G J 1 (R) in (5.16). In Comment 5.5 we discuss the connection of our previous papers [13,15,19] on G J 1 (R) with the papers of Berndt [37,38,39] and Kähler [68,69], developed by Yang [108,109,110,111,112] for G J n (R). We have also determined the Killing vector fields as fundamental vector fields on the Siegel-Jacobi upper half-plane with the balanced metric (5.21b). The same procedure is used to establish the invariant metric on the extended Siegel-Jacobi upper half-plane, which is not a Sasaki manifold. All the results concerning the invariant metrics on homogenous manifolds of dimensions 2-6 attached to the real Jacobi group of degree 1 are summarized in Theorem 5.7. As a consequence, we show by direct calculation that the Siegel-Jacobi upper half-plane is not a naturally reductive space with respect to the balanced metric, but it is one in the coordinates furnished by the FC-transform. In fact, this is the answer to the starting point of our investigation referring to the natural reductivity of X J 1 . We also calculate the g.o. vectors [77] on X J 1 applying the geodesic Lemma A. 19.
In four appendices we recall several basic mathematical concepts used in paper. Appendix A is devoted to naturally reductive spaces [51,71,88]. We have included the notions of Killing vectors, Riemannian homogeneous spaces [4], the list of 3 and 4-dimensional naturally reductive spaces [35,36,76,100], the famous BCV-spaces [41,48,104]. Appendix B recalls the notion of balanced metric in the context of Berezin quantization. The Killing vectors on S 2 , D 1 , R 2 are presented in Appendix C. Appendix D refers to notions on Sasaki manifolds [42,45,95].
The main results of this paper are stated in Lemma 5.1, Remark 4.3, Propositions 4.1-5.8, and Theorem 5.7.
Notation. We denote by R, C, Z, and N the field of real numbers, the field of complex numbers, the ring of integers, and the set of non-negative integers, respectively. We denote the imaginary unit √ −1 by i, and the Real and Imaginary part of a complex number by Re and respectively Im, i.e., we have for z ∈ C, z = Re z + i Im z, andz = cc(z) = Re z − i Im z. We denote by |M | or by det(M ) the determinant of the matrix M . M (n, m, F) denotes the set of n × m matrices with entries in the field F. We denote by M (n, F) the set M (n, n, F). If A ∈ M n (F), then A t (A † ) denotes the transpose (respectively, the Hermitian conjugate) of A.
1 n denotes the identity matrix of degree n. We consider a complex separable Hilbert space H endowed with a scalar product which is antilinear in the first argument, (λx, y) =λ(x, y), x, y ∈ H, λ ∈ C \ 0. We denote by "d" the differential. We use Einstein convention that repeated indices are implicitly summed over. The set of vector fields (1-forms) are denoted by D 1 (D 1 ). If λ ∈ D 1 and L ∈ D 1 , then λ | L denotes their pairing. We use the symbol "Tr" to denote the trace of a matrix. If X i , i = 1, . . . , n are vectors in vector space V over the field F, then X 1 , X 2 , . . . , X n F denotes their span over F.

The starting point in the coherent states approach
We recall firstly our initial approach [11,13] to the Jacobi group G J 1 which we have followed in all our papers devoted to the Jacobi group, except [28] and [29]. The Lie algebra attached to G J 1 is where h 1 is an ideal in g J 1 , i.e., h 1 , g J 1 = h 1 , determined by the commutation relations a, a † = 1, (2.1a) The Heisenberg algebra is where a † (a) are the boson creation (respectively, annihilation) operators which verify the canonical commutation relations (2.1a). The Lie algebra of the group SU(1, 1) is where the generators K 0 , K + , K − verify the standard commutation relations (2.1b), and we have considered the matrix realization We have determined the invariant metric on the Siegel-Jacobi upper half-plane X J 1 from the metric on D J 1 and the FC-transforms, see [13,14,18,21] We have the homogeneous Kähler diffeomorphism FC : The Kähler two-form (2.3) is invariant to the action (g, α) × (η, w) = (η 1 , w 1 ) of G J 0 on C × D 1 : Using the partial Cayley transform we get the Kähler two-form We have the homogeneous Kähler diffeomorphism The situation is summarized in the commutative diagram of the table FC-transforms We recall that in Proposition 2.1 the parameters k and µ come from representation theory of the Jacobi group: k indexes the positive discrete series of SU(1, 1) (2k ∈ N), while µ > 0 indexes the representations of the Heisenberg group. Note that in the Berndt-Kähler approach the Kähler potential (5.24) is just "guessed", see Comment 5.5.
Here we just verify the invariance of the Kähler two-form (2.4) to the action (2.5), see also Lemma 5.1. We use equations (2.6) 6) and the particular case n = 1 of equations in [17, p. 17] where B is given in (2.4).
We determine the left-invariant vector fields L j such that λ i | L j = δ ij , i, j = 1, 2, 3, If we take in (4.11) the limit θ → 0, we project the invariant vector fields of SL(2, R) on the Siegel half-plane X 1 = (x, y) ∈ R 2 | y > 0 , and we recover the invariant vector fields which appear in Theorem A.10(2) equation (A.11) (4.12) Now we calculate the fundamental vector fields f * , g * , h * of manifold SL(2, R) attached to the base F , G, respectively H, invariant to the action (x, y, θ) · (x , y , θ ) = (x 1 , y 1 , z 1 ) given by the composition law M M = M 1 , applying (C.21), (4.4), (4.5): where we have denoted with a subindex 1 the fundamental vector fields (C.23) of SL(2, R), corresponding to the action of the group on the Siegel upper half-plane X 1 . In fact, F * 1 , G * 1 , H * 1 are F, G, H in the convention of Section 1. Evidently, the vector fields (4.13) verify the same commutation relations as F , G, H, with a minus sign.
Using (C.22) or directly with (4.13), we calculate the fundamental vector fields v * , h1 * , w * of SL(2, R) corresponding to Now we consider SL(2, R) as a contact manifold in the meaning of Definition D.3. Firstly we define an almost contact structure (Φ, ξ, η) as in Definition D.1. We take η = λ 3 and ξ = L 3 , verifying (D.3a). We have dη = dλ 3 = √ β dx∧dy y 2 , and the condition (D.6) (with n = 1) that η be a contact form is verified. The only nonzero component of the associated two formΦ in (D.7) It is convenient to work with the matrix With (η, ξ, Φ) chosen as λ 3 , L 3 , Φ , equation (D.3b) and the conditions of Theorem D.2 for an almost contact structure for the manifold SL(2, R), where Rank(Φ) = 2, are verified. We have and we can write the (1, 1)-tensor Φ (D.2) as We also observe that SL(2, R) is a homogeneous contact manifold in the sense of Definition D.5. Now we construct the 4-dimensional symplectization (C(SL(2, R)), ω,ḡ) of SL(2, R), wherē In order to see that the Riemann cone (C(SL(2, R)), ω,ḡ) of the manifold SL(2, R) is normal in the sense of Definition D.10, we calculate the components (D.17) of the (1, 2)-tensor N 1 (D.9), using equations (4.15) and (4.16). Because the tensor (D.10) is antisymmetric in the lower indexes i, j, we have to calculate only the 9 components N 1 i x,y , N 1 i x,θ , N 1 i y,θ , i = x, y, θ, which were found to be 0. In accord with Definition D.10, the Riemann cone (C(SL(2, R)), ω,ḡ) is Sasaki, and, in accord with Theorem D.12, it is a Kähler manifold.
It can be verified that the vector ξ = L 3 is a Killing vector for the metric (4.20), and SL(2, R) has a K-contact structure, in the sense of Definition D. 10. In fact, with Remark A.3, it is verified that ∂ ∂x and ∂ ∂θ are Killing vectors for the metric (4.21) below, because none of the coordinates x and θ appear explicitly in (4.20). For completness, if X = X 1 ∂ ∂x + X 2 ∂ ∂y + X 3 ∂ ∂θ then the equations (A.6) of the Killing vectors in the case of the homogeneous metric (4.21) are In fact, we have The matrix associated with the metric (4.20) is The invariant vector fields L 1 , L 2 , L 3 given by (4.11) are orthonormal with respect to the metric (4.20). The Killing vector fields associated to the metric (4.20), solutions of the equations (4.19), are given by (4.14). L 3 , λ 3 , Φ defines an almost contact structure on SL(2, R), where L 3 , λ 3 , Φ are given respectively by (4.11c), (4.10c), (4.16). λ 3 is the contact structure for SL(2, R), L 3 is the Reeb vector and the contact distribution D is given by (4.17). SL(2, R)(x, y, θ), X 1 , ds 2 D is a sub-Riemannian manifold and where ds 2 X 1 is the (Beltrami) Kähler metric (A.11), (A.17), on the Siegel upper half-plane x, y, ∈ R, y > 0. The invariant vector fields l 1 0 , l 2 0 given by (4.12) are orthonormal with respect to the metric (4.22).
The last assertion in Proposition (4.1) is well known, see [ We also give a direct proof of the some well-known facts, see (b2) in Theorem A.11. Remark 4.3. The Siegel upper half-plane X 1 admits a realization as noncompact Hermitian symmetric space X 1 is a symmetric, naturally reductive space.
Proof . We use the equivalence (C.14), but we look at the level of groups. We consider the case of Sp(n, R). The group Sp(n, K) is the group of matrices M ∈ M(2n, K), where K is R or C, for which i.e., we have (4.24) We can identify the complex linear group GL(n, C) with the subgroup of matrices of GL(2n, R) that commutes with J, i.e., a + ib ∈ GL(n, C) is identified with the real 2n × 2n matrix (4.25), see, e.g., [71, p. 115] It is easy to prove, see, e.g., [97,60], that if M ∈ Sp(n, R), then M is similar with M t and M −1 . If M ∈ Sp(n, R) is as in (4.24), then the matrices a, b, c, d ∈ M (n, R) in (4.24) verify the equivalent conditions Note that the inverse of the matrix (4.24) is given by The matrices from Sp(n, R) have the determinant 1.
In order to verify the condition (A.10), we take We mention that naturally reductive left-invariant metrics on SL(2, R) in the context of BCV-spaces have been investigated in [62]. 5 The Jacobi group G J 1 (R) embedded in Sp(2, R)

The composition law
The real Jacobi group G J 1 (R) is the semi-direct product of the real three dimensional Heisenberg group H 1 with SL(2, R). The Lie algebra of the Jacobi group G J 1 (R) is given by g J 1 (R) = P, Q, R, F, G, H R , where the first three generators P , Q, R of h 1 verify the commutation relations (3.2), the generators F , G, H of sl(2, R) verify the commutation relations (C.11) and the ideal h 1 in sp(2, R) is determined by the non-zero commutation relations where g 1 = aa + bc ab + bd ca + dc cb + dd , The inverse element of g ∈ G J 1 (R) is given by where Y was defined in (1.5).

The action
Let while (x 1 , y 1 ) are given by (4.7a).

Fundamental vector fields
In order to calculate the change of coordinates of a contravariant vector field under the change of variables (5.4) (x, y, ξ, η) → (x, y, p, q), where (p, q) = η y , ξ − η y x , we firstly observe that the Jacobian is ∂(x,y,ξ,η) ∂(x,y,p,q) = −y < 0, and we get easily In order to calculate the change of coordinates of a contravariant vector field under the change of variables (5.4) (x, y, p, q, κ) → (x, y, ξ, η, κ), we get easily With (5.9) and the action (5.5) on X J 1 , and then with (5.10) for the action (5.8) onX J 1 , we get Proposition 5.2. The fundamental vector fields expressed in coordinates (τ, z) of the Siegel-Jacobi upper half-plane X J 1 on which act the reduced Jacobi group G J (R) 0 by (5.5) are given by the holomorphic vector fields Then the real holomorphic fundamental vector fields corresponding to τ = x + iy, y > 0, z = ξ + iη in the variables (x, y, ξ, η) are If we express the fundamental vector fields in the variables (x, y, p, q) where ξ = px + q, η = py, we find Now we consider the action (5.8) of G J 1 (R) on the points (τ, z, κ) ofX J 1 . Instead of (5.11), we get the fundamental vector fields in the variables (τ, z, p, q, κ) Instead of (5.12), we get the fundamental vector fields inX J 1 in the variables (x, y, ξ, η, κ) Instead of (5.13), we ge the fundamental vector fields in the variables (x, y, p, q, κ)

Invariant metrics
We explain the method to get invariant metrics on G-homogeneous manifolds M from invariant metrics of G, see also Appendix D.1.1.
is the Lie algebra of G (respectively, H), then there exists a vector space m such that we have the vector space decomposition g = m + h, m ∩ h = ∅, and the tangent space at x, T x M , can be identified with m, where H = G x is the isotropy group at x, see Definition A.5 and Lemma A.6. Then let X i , i = 1, . . . , n be a basis of the Lie algebra g such that where dim m = m. The left-invariant one forms λ i on G are given by and the left-invariant vector fields L i on G are determined from the relations λ i | L j = δ ij , i, j = 1, . . . , n. Then the invariant metric on G is given by ds 2 . . , m are Killing vectors of the metric g M . Now we calculate the left-invariant one-forms on G J 1 (R) where g is as in (1.4) and g −1 as in (5.3). We find the left-invariant one-forms on G J 1 (R) In ( The invariant vector fields L F , L G , L H , L P , L Q , L R verify the commutations relations (C.11), (3.2) and (5.1) of the generators F , G, H, P , Q, R of the Lie algebra g J 1 (R). Besides the formulas for λ 1 , λ 2 , λ 3 defined in (4.10), we introduce the left-invariant one-forms: where λ P , λ Q , λ R are defined in (5.15). Note that the parameters γ, δ introduced in (5.17) will appear also in the invariant metrics (5.25) onX J 1 and (5.27) on G J 1 (R), while in the metric (5.22) on X J 1 appears only γ. The invariant metrics on D J n and X J n depend only of two parameters α, γ > 0 [16,17,21,109,111]. In fact, the first time they appear in the papers of Kähler [68,69] and Berndt [37,38], parameterizing the invariant metric on X J 1 . Also, besides the left-invariant vector fields L 1 , L 2 , L 3 defined in (4.11), we introduce the left invariant one forms where L P , L Q , L R are defined in (5.16). The vector fields L i , i = 1, . . . , 6 verify the commutations relations Similarly, we introduce Recalling also Proposition 2.1, where we have replaced u = pv + q, v = x + iy with z = pτ + q, respectively τ = x + iy, and k = 2c 1 , µ = c 2 2 , we have proved: Proposition 5.4. The balanced metric (5.21) on the Siegel-Jacobi upper half-plane X J 1 , leftinvariant to the action (5.5), (5.6), (5.7) of reduced group G J (R) 0 is The metric (5.21) is Kähler.
If we denote c 1 4 = α, c 2 = γ, then the matrix attached to the left invariant metric (5.21b) on X J 1 reads The metric (5.21b) can be written as The vector fields L j 0 dual orthogonal to the invariant one-forms λ i , λ i | L j 0 = δ ij , i, j = 1, 2, 4, 5, with respect to the bases dx, dy, dp, dq and ∂ ∂x , ∂ ∂y , ∂ ∂p , ∂ ∂q are and We make a "historical" comment where c 1 = k 2 , c 2 = 2µ comparatively to our formula (2.4). Formula (5.24) is presented by Berndt as "communicated to the author by Kähler", where it is also given equation (5.21a), while (5.21b) has two printing errors. Later, in Section 36 of his last paper [68], reproduced also in [69], Kähler argues how to choose the potential as in (5.24); see also [68,Section 37,equation (9)], where c 1 = λ 2 , c 2 = iµπ, and the metric (8) differs from the metric (5.21) by a factor two, because the hermitian metric used by Kähler is ds 2 = 2g ij dz i dz j .
We also recall that in [110] Yang calculated the metric on X J n , invariant to the action of G J n (R) 0 . The equivalence of the metric of Yang with the metric obtained via CS on D J n and then transported to X J n via partial Cayley transform is underlined in [17]. In particular, the metric (5.21c) appears in [110, p. 99] for the particular values c 1 = 1, c 2 = 4. See also [109,111,112]. Now we shall establish a metric invariant to the action given in Lemma 5.1 of G J 1 (R) on the extended Siegel-Jacobi upper half-planeX J 1 . Because the manifoldX J 1 is 5-dimensional, we want to see if the extended Siegel-Jacobi upper half-plane is a Sasaki manifold, as in the case of SL(2, R) in Proposition 4.1. If we take as contact form η = λ 6 , then dη = −2 √ δdp ∧ dq, and η(∧η) 2 = 0. If we try to determine a contact distribution D = Ann(η), we get D = ∂ ∂p − q ∂ ∂θ , ∂ ∂q + p ∂ ∂θ . From (D.4), we find Φ κ λ = 0, and Φ κ p = qΦ λ κ , Φ λ q = −pΦ λ κ , where λ = x, y, p, q, κ. So Φ has Rank(Φ) < 4. In conclusion, (Φ, ξ, η) chosen as above can not be an almost contact structure for the extended Siegel-Jacobi upper half-planeX J 1 . We obtain Proposition 5.6. The metric on the extended Siegel-Jacobi upper half-planeX J 1 , in the partial S-coordinates (x, y, p, q, κ): is left-invariant with respect to the action given in Lemma 5.1 of the Jacobi group G J 1 (R). The matrix attached to metric (5.25) is , g pq = g pq − δpq, g pκ = δq, g qκ = −δp, g pp = g pp + δq 2 , g qq = g qq + δp 2 , g κκ = δ, (5.26) while g xx , g yy , g pp , g qq , g pq are given in the metric matrix (5.22) associated with the balanced metric (5.21b) on X J 1 . The metric (5.26) is orthonormal with respect to the invariant vector fields L i 0 , i = 1, 2, L i , i = 4, 5, 6. The fundamental vector fields with respect to the action (5.8) in the variables (x, y, p, q, κ) are given by (5.14).
We have where L 1 , . . . , L 3 are defined by is a naturally reductive manifold or not. The fact that X J 1 is not a naturally reductive 4dimensional manifold is well known, see Theorem A.12, but in Proposition 5.8 below we present a direct proof.
Proposition 5.8. The Siegel-Jacobi upper half-plane realized as homogenous Riemannian mani- is a reductive, non-symmetric manifold, not naturally reductive with respect to the balanced metric (5.21b).
The Siegel-Jacobi upper half-plane X J 1 is not a g.o. manifold with respect to the balanced metric.
When expressed in the variables that appear in the FC-transform given in Proposition 2.1, X J 1 is a naturally reductive space with the metric g X 1 × g R 2 , where g X 1 is given by (4.22) and g R 2 is the Euclidean metric (A.14). If then a geodesic vector of the homogeneous manifold X J 1 has one of the following expressions given in Table 1.
But [m, m] h, and X J 1 is not a symmetric manifold. We verify (A.10) written as Instead of (5.29) we take where L 1 , . . . , L 3 (L 4 , . . . , L 6 ) are defined in (4.11), (respectively (5.18)). We take and, with the commutation relations (5.19), we find Taking into account that the vector fields L 1 , . . . , L 6 are orthonormal with respect to the metric (5.27) on G J 1 as in Theorem 5.7, the condition (5.30) of the geodesic Lemma A.19 reads The condition (5.31) implies that the system of algebraic equations must have a solution for any a i , b i , c i , d i , i = 1, 2, 3, which is not possible, and X J 1 is not naturally reductive with respect to the balanced metric.
Due to Theorem A.20, the four-dimensional manifold X J 1 is not a g.o. manifold. We also recall that in [17,Propositions 3 and 4] it was proved that under the so called FCtransform, the manifold X J n is symplectomorph with X n × C n . The particular case of the Jacobi group of degree 1 was reproduced in Proposition 2.1 and, in particular, X J 1 is equivalent with the symmetric space X 1 × C, which is naturally reductive, as in Theorem A.12.
To find the geodesic vectors on the Siegel-Jacobi upper half-plane X J 1 , we look for the solution (5.28) that verifies the condition (A.22) of the geodesic lemma expressed in Proposition A. 19. Taking The solutions of the system (5.32) are written in Table 1.

A Naturally reductive spaces A.1 Fundamental vector fields
A homogeneous space is a manifold M with a transitive action of a Lie group G. Equivalently, it is a manifold of the form G/H, where G is a Lie group and H is a closed subgroup of G, cf., e.g., [6, p. 67]. Let (M, g), (N, g ) be Riemannian manifolds. An isometry is a diffeomorphism f : M → N that preserves the metric, i.e., Let G be a Lie group of transformations acting on the manifold M , cf. [63, Chapter II, Section 3, p. 121]. In [63, p. 122] it is introduced the notion of vector field on M induced by the one parameter subgroup exp tX, t ∈ R, X ∈ g, denoted X + , where g is the Lie algebra of G. In [70, Section 5, p. 51], in the context of principal fibre bundle P (M, G) over M with structure group the Lie group G, it is introduced the same notion under the name fundamental vector field associate to X ∈ g, denoted X * , see also [70,Proposition 4.1,p. 42].
Let M = G/H be a homogeneous n-dimensional manifold and let us suppose that G acts transitively on the left on M , G×M → M : g·x = y, where y = (y 1 , . . . , y n ) t . Then g(t)·x = y(t), where g(t) = exp(tX), t ∈ R, generates a curve in M with y(0) = x andẏ(0) = X. The fundamental vector field attached to X ∈ g at x ∈ M is defined as We write the fundamental vector field attached to X ∈ g as then the associated fundamental vector fields verify the commutation relations Note that if the action of G on M is on the right as in [70, p. 51], then
Let us consider a n-dimensional Riemannian manifold (M, g) and a vector field with the contravariant components X i , i = 1, . . . , n: If ∇ denotes the covariant derivative, we have the standard formulas Lemma A.2. Let (M, g) be a n-dimensional Riemannian manifold with a Riemannian (metric) connection. The field X is a Killing vector field if and only if its covariant components X µ , µ = 1, . . . , n verify the Killing equations If the coordinate x α is not present in the expression of metric tensor g λχ , λ, χ = 1, . . . , n, then ∂ ∂x α is a Killing vector field for the metric g λχ . With (A.4e), the condition (A.2) of a vector field (A.3) to be a Killing vector field is that its contravariant components to verify the equations X µ ∂ µ g λχ + g µχ ∂ λ X µ + g λµ ∂ χ X µ = 0, λ, χ, µ = 1, . . . , dim M = n. (A.6) The system (A.6) of n(n + 1)/2 equations of a Killing vector field X 1 (x), . . . , X n (x) is overdetermined, and no-nonvanishing solution is guaranteed, in general. The set ι(M ) of all Killing vector fields on n-dimensional manifold M forms a Lie algebra of dimension not exceeding n(n+1) 2 and dim(ι(M )) = n(n+1) 2 is obtained only for spaces of constant curvature, see [70,Theorem 3.3,p. 238]. For example, maximal solution is obtained for the (pseudo)-Euclidean spaces E r,n−r , for the sphere S n or the real projective space RP n = S n /(±I), see, e.g., [70,Theorem 1,p. 308], [104, p. 251] and [57, Section 4.6.6, p. 83]. We have ι(E r,s ) = so(r, s) R r+s . The Euclidean group E n of R n has dimension n(n+1)/2, where n degrees of freedom correspond to translations, the other n(n − 1)/2 correspond to rotations, see also Proposition A.13 and Remark C.4 below.
The following remark is very important for the determination of Killing vector fields on Riemannian homogeneous manifolds, see, e.g., see [35, p. 4]
The set of elements G x of a given group G, acting on a set M as group of transformations that leaves the point x fixed, is called isotropy group, also called stationary group or stabilizer. If G is a Lie group and H is a closed subgroup, then the coset space G/H, in particular, H = G x is taken with the analytic structure given in [63,Theorem 4.2,p. 123]. For x ∈ G, the diffeomorphism of G/H into itself is τ (x) : yH → xyH. The natural representation of the isotropy group of a differentiable transformation group in the tangent space to the underling manifold is called isotropy representation. If G is the group of differentiable transformations on the manifold M and G x is the corresponding isotropy subgroup at the point x ∈ M , then the isotropy representation Is x : G x → GL(T x M ) associates to each h ∈ G x the differential Is x (h) := (dτ (h)) λ(H) of the transformation h at x, where λ : G → G/H is the canonical projection. The image of the isotropy representation, Is x (G x ), is called the linear isotropy group at x.
If G is a Lie group with a countable base acting transitively and smoothly on M , then the tangent space T x M can be naturally identified with the space g/g x , where g ⊃ g x are respectively the Lie algebras of the groups G ⊃ G x . The isotropy representation Is x is now identified with the representation G x → GL(g/g x ), induced by the restriction of the adjoint representation Ad G of G to G x . See details below in Lemma A.6. Lemma A.6. If a homogeneous space M is reductive, then T x M can be identified with m, while Is x can be identified with the representation h → (Ad G h)| m . In this case, the isotropy representation is faithful if G acts effectively.
So let us denote by x(s) the 1-parameter subgroup of G generated by X ∈ m and let x * (s) = λ(x(s)) be the image of x(s) by the projection λ of G onto G/H: Identifying X * with X ∈ m, we can write down is given by Explicitly, the Ad(H) invariance of the symmetric non-degenerate form B in (A.8) means, see, e.g., [71, p. 201] Usually it is asked that the group of isometries G acts effectively on M , cf. [51].
The canonical connection, see [71, p. 192], or canonical affine connection of second type, see [88], on the reductive space M = G/H verifying (A.7a), (A.7b), is the unique G-invariant affine connection on M such that for any vector field X ∈ m and any frame u at the point o, the curve (exp tX)u in the principal fibration of frames over M is horizontal. The canonical connection is complete and the set of its geodesics through o coincides with the set of curves of the type (exp tX)o, where X ∈ m, see also [71,Proposition 2.4 and Corollary 2.5,p. 192]. In a reductive space there is a unique G-invariant affine connection with zero torsion having the same geodesics as the canonical connection, cf. [71,Theorem 2.1,p. 197]. This connection is called in [71] natural torsion-free connection on M = G/H relative to the decomposition (A.7a), or canonical affine connection of the first kind in [88].

A.4 Naturally reductive spaces
is verified; (ii) the Levi-Civita connection of (M, g) and the natural torsion-free connection with respect to the decomposition (A.7a) are the same; (iii) ( * ) is true, i.e., every geodesic in M is the orbit of a one-parameter subgroup of I(M ) generated by some X ∈ m.
It is not always easy to decide whether a given homogenous Riemannian space is naturally reductive [1]. The Riemannian manifold M = G/H might be naturally reductive although for any reductive decomposition g = h + m none of the statements in Proposition A.9 holds, because that might exist another appropriate subgroupG ⊂ I(M ) such that M =G/H and with respect to such decomposition the conditions of Proposition A.9 are satisfied, see, e.g., [35, p. 5]. In accord with [35, Proposition 2, p. 5], a necessary and sufficient condition that a complete and simply connected manifold be naturally reductive is that there exists a homogeneous structure T on M with T * v = 0, for all tangent vectors v of M .
Ambrose and Singer found the condition for a Riemannian manifold be locally homogeneous [4].

A.5 Naturally reductive spaces of dimension ≤ 4
The connected homogeneous Riemannian V n naturally reductive spaces of dimension n ≤ 6 are classified.
For two dimensional manifolds, because the homogeneous manifolds V 2 have constant curvature, they are locally symmetric spaces, see, e.g., in [100, Theorem 4.1, Section 4].
Theorem A. 10. The only homogenous structure on R 2 and S 2 is given by T = 0, cf. Up to an isomorphism, H 2 has only two homogenous structures, namely: 1. T = 0, corresponding to the symmetric case H 2 = SO 0 (1, 2)/U(1), where SO 0 (1, 2) = SL(2, R)/±I is the connected component of the identity of the Lorentz group, see also (C.14).
X, Y ∈ D 1 (M ), r > 0. This homogenous structure corresponds to the Lie algebra g with the product (y 1 , y 2 )(y 1 , y 2 ) = (y 1 y 1 , y 1 y 2 + y 2 ), i.e., the semi-direct product of the multiplicative group R + 0 and the additive group R.
The case n = 3 was considered by Kowalski [74]. The proof of Theorems A.11 and A.12 below is based on the Ambrose and Singer theorem in the formulation of [100, Section 2] and the classification of 3-dimensional unimodular Lie groups with left-invariant metrics of Milnor [83].
The following theorem is [100, Theorem 6.5, p. 63], [36,Theorem 2] or [1, Theorem 5.2]: Theorem A.11. A three-dimensional complete, simply connected naturally reductive Riemannian manifold (M, g) is either: (a) a symmetric space realized by the real forms: R 3 , S 3 or the Poincaré half-space H 3 , and S 2 × R, H 2 × R, or (b) a non-symmetric space isometric to one of the following Lie groups with a suitable leftinvariant metric: The Poincaré half-space H n is the set (x 1 , . . . , x n ) ∈ R n , x 1 > 0, with the metric proportional with (A.12) For the left invariant Riemannian metrics which appear in Theorem A.11, see [74,Theorem 2] and [91]. For H 1 = R 3 [x, y, z] a left-invariant metric is Note that in [35, Theorem 1, p. 6] appear only the non-symmetric naturally reductive spaces of dimensions 3: SU(2) ∼ = S 3 , SL(2, R) and Nil 3 . The metrics of these spaces are particular cases of the 7-families of BCV-spaces that appear in Theorem A.15, because the naturally reductive spaces are a particular class of homogenous spaces. The case of four-dimensional manifolds was treated by Kowalski  Theorem A.12. Let (M, g) be a four-dimensional simply connected naturally reductive Riemannian manifold. Then (M, g) is either symmetric or it is a Riemannian product of the naturally reductive spaces of dimension 3 of type (b) appearing in Theorem A.11 times R. In the last cases, (M, g) is not locally symmetric.
A.6 V 2 and V 3 spaces with transitive group The determination of the groups G 3 of isometries with three parameters of a two-dimensional space V 2 with positive definite metric was done by Bianchi [41]. In Proposition A.13 below we follow Vranceanu, see [104,Chapter V,Section 14,p. 288]. The generators of G 3 in [104, equation (90)] considered by Vranceanu, in our notation (C.37), verifies the commutation relations Below we also write down V 2 as a homogenous manifolds. E(2) is the group of rigid motions of the Euclidean 2-space, denoted M (2) in [102, p. 195], see also [102,Section 8.5].
1. If dim(I(V 3 )) = 6, then V 3 is of the type of the real space forms, i.e., the real Euclidean space E 3 , the sphere S 3 (κ), or the hyperbolic space H 3 (κ).
The above classification contains the eight model geometries of Thurston [99]: Cartan classified all 3-dimensional spaces V 3 with a 4-dimensional isometry group G 4 in [48], see also [41] and [104]. See also [91] for a modern presentation of Cartan approach.
The Bianchi-Cartan-Vranceanu (BCV) spaces are V 3 spaces with dim(I(V 3 )) = 4 together with E 3 and S 3 (κ), while the hyperbolic space H 3 (κ) appearing in Theorem A.11 -a symmetric naturally reductive -is missing in the list of BCV-spaces.
For κ, τ ∈ R, it is defined the open subset of R 3  15. All 3-dimensional homogenous spaces V 3 with isometry group G 4 are locally isomorphic with the BCV-spaces. The BCV family also includes two real space forms, with isometry group G 6 , see Proposition A.14. The full classification of these spaces is as follows: Here the Poincaré (Siegel) disc is verifying the commutation relations The dual 1-forms ω i , ω i | e j = δ ij , i, j = 1, 2, 3, to the orthonormal vector fields (A. 20) are and we write down (A.19) as Let D be a distribution generated by e 1 , e 2 . The intrinsic (extrinsic) ideal is given by J = ω 3 (respectively, I = ω 1 , ω 2 ).
If τ = 0, the distribution is step 2 everywhere and ω 3 is a contact form. If we consider the sub-Riemannian metric then the BCV-space is a sub-Riemannian manifold BCV, D, ds 2 D . Remark A. 16. Note that the BCV metrics appearing in Cases 1, 2, 5, 6, 7 are metrics on the corresponding naturally reductive spaces of Theorem A.11. Note that naturally reductive space H 3 in Theorem A.11, corresponding to the isometry group of dimension 6, is not a BCV space.
See [58] we get for (A.18) where ζ := x + iy. If v := α + iβ, E := α 2 + (β + 1) 2 , then the left invariant one-forms (A. 21) in the new variables are Instead of the family of metrics (A. 19), we get in Theorem A.15 The natural reductivity is a special case of spaces with a more general property than ( * ), see [77]: ( * * ) Each geodesic of (M, g) = G/H is an orbit of a one parameter group of isometries {exp tZ}, Z ∈ g.

B Balanced metrics and Berezin quantization
In our approach to Berezin quantization on Kähler manifold M of complex dimension n, see, e.g., [21], we considered the Kähler two-form We have considered homogenous Kähler manifolds M = G/H, where the G-invariant Kähler two-form is deduced from a Kähler potential f We have applied Berezin recipe to quantization [31,32,33,34], where the Kähler potential is obtained from the scalar product of two Perelomov CS-vectors e z , z ∈ M [92] f (z,z) = ln K M (z,z), K M (z,z) = (ez, ez), i.e., (1.3).
This choice of f corresponds to the situation where the so called -function, see [46,93,94], is constant. The corresponding G-invariant metric is called balanced metric. This denomination was firstly used in [52] for compact manifolds, then it was used in [5] for noncompact manifolds and also in [81] in the context of Berezin quantization on homogeneous bounded domain, and we have used it in the case of the partially bounded domain D J n -the Siegel-Jacobi ball [21]. We recall that in [46,93,94] Berezin's quantization on homogenous Kähler manifolds via CS was globalized and extended to non-homogeneous manifolds in the context of geometric (pre-)quantization [73,107]. To the Kähler manifold (M, ω), it is also attached the triple σ = (L, h, ∇), where L is a holomorphic (prequantum) line bundle on M , h is the Hermitian metric on L and ∇ is a connection compatible with metric and the Kähler structure [30]. The connection ∇ has the expression ∇ = ∂ + ∂ lnĥ +∂. The manifold is called quantizable if the curvature of the connection F (X, Y ] has the property that F = −iω M , or ∂∂ logĥ = iω M , whereĥ is a local representative of h, takenĥ(z) = K −1 M (z,z). Then ω M is integral, i.e., the first Chern class is given by and we have (1.3).
C Killing vectors on S 2 , D 1 and R 2
We find for D 1 Remark C.2. The Killing vectors on the Siegel disk D 1 corresponding to the metric (C.9) are (C.10) The Killing vectors (C.10) on the Siegel disk D 1 verify the commutation relations C.3 Fundamental vector fields as Killing vector fields on D 1 and X 1 We recall some general facts about Hermitian symmetric spaces, see, e.g., [23,105,106]. Let • X n = G n /K: Hermitian symmetric space of noncompact type.
• X c : compact dual form of X n , X c = G c /K.
• G n : largest connected group of isometries of X n , a centerless semisimple Lie group.
• G c : compact real form of G n .
• G c = G c n = G c c = G: complexification of G c and G n . • K: maximal compact subgroup of G n .
• g n , g, g c , k: Lie algebras of G n , G, G c , K respectively.
• g n = k + m n , sum of +1 and −1 eigenspaces of the Cartan involution σ. To the complex Lie algebra A 1 = sl(2, C) are associated the compact real form sl(2, C) c = su(2) and the non-compact real forms su(1, 1) and sl(2, R), see [63, pp. 186, 446], [23,105,106], and we have We have also the isomorphisms between the compact real forms (1), and the non-compact real forms We have also the relations We calculate the fundamental vector fields for the real noncompact group SU(1, 1). Let us denote the elements of the Lie algebra su(1, 1) as Note the commutation relations If we make the notation G i = 2G i , i = 1, 2, 3, then the commutation relations (C.17) became We obtain, see also [102, p. 294], We get If we introduce w = ξ − iη, we write (C.20) as where X 1 , Y 1 , Z 1 are the Killing vector fields of the Siegel disk D 1 calculated in (C.10).
We also have the relations, see also [102, p. 353] e tF = 1 t 0 1 , e tG = 1 0 t 1 , e tH = e t 0 0 e −t , (C.21) e t(F +G) = cosh t sinh t sinh t cosh t , e t(F −G) = cos t sin t − sin t cos t , If we put τ = x + iy, we find the fundamental vector fields on the homogenous manifold X 1 , see Theorem A.10(1) and (4.23) In the convention of Section 1, the vector fields F * then, with formula (C.25), Ad(g)X = gXg −1 , g ∈ G, X ∈ g, (C. 25) we find easily We find out that in the base (C.12) and det(Ad) = 1, Now let us consider an element X ∈ sl(2, R) Then we find With (C.28) we find in the base H, F , G the expression of ad(X) for X given by (C.27) and Tr ad = 0.

C.4 Killing vectors on R 2
The Perelomov's coherent state vectors (Glauber's coherent states) for the oscillator group are, see, e.g., [12], e z := e za † e 0 , and the scalar product is (ez, ez ) = e zz . (C.35) The scalar product (C.35) of Glauber coherent states on C implies the metric on R 2 (A.14) ds 2 R 2 = dx 2 1 + dx 2 2 , where we have considered z = x 1 + ix 2 . Let as consider a vector field on R 2 We formulate a remark, see also in [57, Section 4.6.7, p. 83]: Remark C.4. The Killing vectors on R 2 associated with the metric (A.14) are Let G be a Lie group with Lie algebra g, which has the generators X 1 , . . . , X n verifying the commutation relations (A.1). To X ∈ g we associate the left-invariant vectorX on G such that X e = X, see [63, p. 99].
Let ω 1 , . . . , ω n be the 1-forms on G determined by the equations ω i |X j = δ ij , i, j = 1, . . . , n. Then we have the Maurer-Cartan equations, see, e.g., [63, Proposition 7.2, p. 137]: where c i jk are the structure constants (A.1). If G is embedded in GL(n) by a matrix valued map g = (g) ij , i, j = 1, . . . , n, then let λ (L) denote a left-invariant one-form (vector field) on G and ρ (R) a right-invariant one-form (respectively, vector field) on G. We have the relations
M 2n+1 from Theorem D.6 is said to be a contact (Riemannian) manifold associated with η. Let ω := d r 2 η .
Lemma D. 13. The components of the tensor (D.9) are given by Proof . In the calculation below we use the expressions With (D.11), we get for A, . . . , E the expressions  Note that formula given in [95, pp. 7-10] is wrong. The same wrong formula appears also in [98, equation (3.7)]. The Heisenberg group H 1 is a Sasaki manifold [44].