Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature
A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of th...
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nasplib_isofts_kiev_ua-123456789-1464432025-02-09T15:55:43Z Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature Herranz, F.J. Ballesteros, Á A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are explicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maximally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz system. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented. This work was partially supported by the Ministerio de Educaci´on y Ciencia (Spain, Project FIS2004-07913) and by the Junta de Castilla y Le´on (Spain, Projects BU04/03 and VA013C05). 2006 Article Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature / F.J. Herranz, Á. Ballesteros // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 43 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37J35; 22E60; 37J15; 70H06 https://nasplib.isofts.kiev.ua/handle/123456789/146443 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
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A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are explicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maximally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz system. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented. |
| format |
Article |
| author |
Herranz, F.J. Ballesteros, Á |
| spellingShingle |
Herranz, F.J. Ballesteros, Á Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Herranz, F.J. Ballesteros, Á |
| author_sort |
Herranz, F.J. |
| title |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_short |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_full |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_fullStr |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_full_unstemmed |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature |
| title_sort |
superintegrability on three-dimensional riemannian and relativistic spaces of constant curvature |
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Інститут математики НАН України |
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2006 |
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https://nasplib.isofts.kiev.ua/handle/123456789/146443 |
| citation_txt |
Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature / F.J. Herranz, Á. Ballesteros // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 43 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT herranzfj superintegrabilityonthreedimensionalriemannianandrelativisticspacesofconstantcurvature AT ballesterosa superintegrabilityonthreedimensionalriemannianandrelativisticspacesofconstantcurvature |
| first_indexed |
2025-11-27T17:05:54Z |
| last_indexed |
2025-11-27T17:05:54Z |
| _version_ |
1849964006312247296 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 010, 22 pages
Superintegrability on Three-Dimensional Riemannian
and Relativistic Spaces of Constant Curvature
Francisco José HERRANZ † and Ángel BALLESTEROS ‡
† Departamento de F́ısica, Escuela Politécnica Superior, Universidad de Burgos,
09001 Burgos, Spain
E-mail: fjherranz@ubu.es
‡ Departamento de F́ısica, Facultad de Ciencias, Universidad de Burgos,
09001 Burgos, Spain
E-mail: angelb@ubu.es
Received December 21, 2005, in final form January 20, 2006; Published online January 24, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper010/
Abstract. A family of classical superintegrable Hamiltonians, depending on an arbitrary
radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as
well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed.
Such systems admit three integrals of the motion (besides the Hamiltonian) which are ex-
plicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions
cover the six spaces in a unified way as these are parametrized by two contraction parameters
that govern the curvature and the signature of the metric on each space. Next two maxi-
mally superintegrable Hamiltonians are identified within the initial superintegrable family
by finding the remaining constant of the motion. The former potential is the superposition
of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers
(depending on each specific space), so that this generalizes the Smorodinsky–Winternitz sys-
tem. The latter one is a superposition of the Kepler–Coulomb potential with another two
oscillators or centrifugal barriers. As a byproduct, the Laplace–Runge–Lenz vector for these
spaces is deduced. Furthermore both potentials are analysed in detail for each particular
space. Some comments on their generalization to arbitrary dimension are also presented.
Key words: integrable systems; curvature; contraction; harmonic oscillator; Kepler–Cou-
lomb; hyperbolic; de Sitter
2000 Mathematics Subject Classification: 37J35; 22E60; 37J15; 70H06
1 Introduction
In [14] Evans obtained a classification of classical superintegrable systems [38] on the three-
dimensional (3D) Euclidean space E3. At this dimension he called minimally superintegrable
systems those endowed with three constants of the motion besides the Hamiltonian, that is,
they have one constant more than those necessary to ensure complete integrability, but one less
than the necessary number to determine maximal superintegrability. Amongst the resulting
potentials let us consider
U = F(r) +
β1
x2
+
β2
y2
+
β3
z2
, (1.1)
where F(r) is an arbitrary smooth function, the three βi are arbitrary real parameters, (x, y, z)
are Cartesian coordinates on E3, and r =
√
x2 + y2 + z2. Thus this potential is formed by
a central term with three centrifugal barriers. Next by analysing the radial function F(r) two
mailto:fjherranz@ubu.es
mailto:angelb@ubu.es
http://www.emis.de/journals/SIGMA/2006/Paper010/
2 F.J. Herranz and Á. Ballesteros
relevant and well known expressions arise conveying the additional constant of the motion. These
two cases then appear in the classification by Evans as maximally superintegrable systems as
they have the maximum number of functionally independent constants of the motion, four ones
plus the Hamiltonian. Explicitly, these are:
• The Smorodinsky–Winternitz (SW) potential [17] when F(r) = β0r
2:
USW = β0
(
x2 + y2 + z2
)
+
β1
x2
+
β2
y2
+
β3
z2
, (1.2)
which corresponds to the superposition of a harmonic oscillator with angular frequency√
β0 and the three centrifugal terms.
• And a generalized Kepler–Coulomb (GKC) potential when F(r) = −k/r:
UGKC = − k√
x2 + y2 + z2
+
β1
x2
+
β2
y2
, (1.3)
which is formed by the proper Kepler–Coulomb (KC) potential with parameter k together
with two of the famous centrifugal terms.
Superintegrable systems on E2 and E3 [14, 35] have also been implemented on the two classical
Riemannian spaces of constant curvature. In particular, some superintegrable systems on the 2D
and 3D spheres, S2 and S3, have been studied in [20], on the hyperbolic plane H2 in [29, 30], while
on H3 can be found in [21]. Moreover classifications of superintegrable systems on S2 and H2
have been carried out in [28, 31, 34, 39]. These results contain the corresponding (curved)
harmonic oscillator [26, 36] and KC potential [43], which in arbitrary dimension correspond, in
this order, to the following radial potential
F(r) =
β0 tan2 r, on SN ;
β0 r2, on EN ;
β0 tanh2 r, on HN .
F(r) =
−k/ tan r, on SN ;
−k/r, on EN ;
−k/ tanh r, on HN .
(1.4)
We recall that the SW system on SN and HN have been constructed in [9, 23] (curved harmonic
oscillator plus N terms) showing that this keeps maximal superintegrability for any value of the
curvature.
However, as far as we know, the construction of the GKC potential on SN and HN as well
as which are the corresponding SW and GKC systems on the relativistic spacetimes of constant
curvature is still lacking, that is, also covering the anti-de Sitter, Minkowskian and de Sitter
spacetimes. The aim of this paper is to present all of these Hamiltonians on these six 3D spaces
in a unified setting by making use of two explicit contraction parameters which determine the
curvature and the signature of the metric. In this sense, the results here presented can be
considered as the cornerstone for a further generalization of all of these systems to arbitrary
dimension. In this respect, we would like to mention that although very recently such potentials
have been deduced on the (1 + 1)D relativistic spacetimes [8, 12], this low dimension does not
show the guide for a direct generalization to ND.
The structure of this paper is as follows. The next section contains the necessary basics
on the Lie groups of isometries on the six spaces together with the two coordinate systems we
shall deal with throughout the paper: ambient (Weierstrass) coordinates in an auxiliary linear
space R4 and intrinsic geodesic polar (spherical) coordinates. The kinetic energy determining
the geodesic motion is then studied in Section 3 by starting from the metric. The generalization
of the Euclidean family (1.1) to these spaces is developed in Section 4 in such a manner that
general and global expressions for the Hamiltonian and its three integrals of motion are explicitly
given.
Superintegrability on 3D Spaces of Constant Curvature 3
The next two sections are devoted to the study of two maximal superintegrable Hamiltonians
arising in the above family by choosing in an adequate way the radial function F(r) (fulfil-
ling (1.4) for the Riemannian spaces) and finding at the same time the remaining constant of
the motion. In this way we obtain the generalization of the SW (1.2) and GKC (1.3) potentials
for any value of the curvature and signature of the metric. Furthermore a detail description of
such systems is performed on each particular space. We stress that, by following the geometrical
interpretation formerly introduced in [40, 41, 42] and generalized in [8, 9, 23], the SW potential is
interpreted as the superposition of a central harmonic oscillator with three non-central oscillators
or centrifugal barriers according to each specific space. Likewise, the GKC system can be seen
as the superposition of the KC potential with two oscillators or centrifugal barriers; in this case,
we moreover deduce the corresponding Laplace–Runge–Lenz vector. Finally, some remarks and
comments mainly concerning the pattern for the construction of such systems for arbitrary
dimension close the paper.
2 Riemannian spaces and relativistic spacetimes
Let us consider a subset of real Lie algebras contained in the family of the Cayley–Klein orthogo-
nal algebras [4, 18]. These can also be obtained as the Z2⊗Z2 graded contractions of so(4) and
are denoted soκ1,κ2(4) where κ1 and κ2 are two real contraction parameters. The Lie brackets
of soκ1,κ2(4) in the basis spanned by {Jµν} where µ, ν = 0, 1, 2, 3 and µ < ν read [4]
[J12, J13] = κ2J23, [J12, J23] = −J13, [J13, J23] = J12,
[J12, J01] = J02, [J13, J01] = J03, [J23, J02] = J03,
[J12, J02] = −κ2J01, [J13, J03] = −κ2J01, [J23, J03] = −J02,
[J01, J02] = κ1J12, [J01, J03] = κ1J13, [J02, J03] = κ1κ2J23,
[J01, J23] = 0, [J02, J13] = 0, [J03, J12] = 0. (2.1)
There are two Casimir invariants
C1 = κ2J
2
01 + J2
02 + J2
03 + κ1J
2
12 + κ1J
2
13 + κ1κ2J
2
23,
C2 = κ2J01J23 − J02J13 + J03J12, (2.2)
where C1 is associated to the Killing–Cartan form.
Let us explain the geometrical role of the contraction parameters κ1 and κ2. The involutive
automorphisms defined by
Θ0 : Jij → Jij , J0i → −J0i, i = 1, 2, 3,
Θ01 : {J01, J23} → {J01, J23}, {J0j , J1j} → −{J0j , J1j}, j = 2, 3,
generate a Z2 ⊗ Z2-grading of soκ1,κ2(4) in such a manner that κ1 and κ2 are two graded
contraction parameters coming from the Z2-grading determined by Θ0 and Θ01, respectively.
By scaling the Lie generators each parameter κi can be reduced to either +1, 0 or −1; the
vanishment of κi is equivalent to apply an Inönü–Wigner contraction.
Furthermore, these automorphisms give rise to the following Cartan decompositions:
soκ1,κ2(4) = h0 ⊕ p0, h0 = 〈J12, J13, J23〉 = soκ2(3), p0 = 〈J01, J02, J03〉,
soκ1,κ2(4) = h01 ⊕ p01, h01 = 〈J01, J23〉 = soκ1(2)⊕ so(2), p01 = 〈J02, J03, J12, J13〉.
If H0 and H01 denote the Lie subgroups with Lie algebras h0 and h01, we obtain two families
of symmetric homogeneous spaces [22], namely the usual 3D space of points SOκ1,κ2(4)/H0
4 F.J. Herranz and Á. Ballesteros
and the 4D space of lines SOκ1,κ2(4)/H01, which have constant curvature equal to κ1 and κ2,
respectively.
We shall make use of the former space which has a metric with a signature governed by κ2
as diag(+1, κ2, κ2) and we denote it
S3
[κ1]κ2
= SOκ1,κ2(4)/SOκ2(3).
Thus when κ2 is positive we recover the three classical Riemannian spaces, while if this is
negative we find a Lorentzian metric. In this case, there is a kinematical interpretation for the
homogeneous spaces. Let P0, Pi, Ki and J (i = 1, 2) the usual generators of time translation,
space translations, boosts and spatial rotations, respectively. Under the following identification
P0 = J01, Pi = J0 i+1, Ki = J1 i+1, J = J23, i = 1, 2, (2.3)
the three algebras with κ2 = −1/c2 < 0 (c is the speed of light) are the Lie algebras of the groups
of motions of (2+1)D relativistic spacetime models. Thus the commutation relations (2.1) read
now
[J,Ki] = εijKj , [K1,K2] = − 1
c2
J, [P0,Ki] = −Pi, [Pi,Kj ] = − 1
c2
δijP0,
[J, Pi] = εijPj , [P1, P2] = −κ1
c2
J, [P0, Pi] = κ1Ki, [P0, Ji] = 0, (2.4)
where εij is a skew-symmetric tensor such that ε12 = 1. In this framework the curvature of the
spacetime can be written in terms of the (time) universe radius τ as κ1 = ±1/τ2 (which is also
proportional to the cosmological constant). The Casimir invariants (2.2), C1 and C2, correspond
to the energy and angular momentum of a particle in the free kinematics of the relativistic
spacetime:
C1 = − 1
c2
P 2
0 + P 2
1 + P 2
2 + κ1
(
K2
1 + K2
2
)
− κ1
c2
J2,
C2 = − 1
c2
P0J − P1K2 + P2K1. (2.5)
On the other hand, if κ2 = 0 we obtain a degenerate metric which corresponds to Newtonian
spacetimes. Since our aim is to construct superintegrable systems on these homogeneous spaces,
for which the kinetic energy is provided by the metric, we avoid the contraction κ2 = 0. The
resulting six particular spaces contained in the family S3
[κ1]κ2
are displayed in Table 1.
Table 1. 3D symmetric homogeneous spaces S3
[κ1]κ2
= SOκ1,κ2(4)/SOκ2(3) and their metric in geodesic polar
coordinates according to κ1 ∈ {+1, 0,−1} and κ2 ∈ {+1,−1}.
3D Riemannian spaces (2 + 1)D Relativistic spacetimes
• Spherical space S3 • Anti-de Sitter spacetime AdS2+1
S3
[+]+ = SO(4)/SO(3) S3
[+]− = SO(2, 2)/SO(2, 1)
ds2 = dr2 + sin2 r dθ2 + sin2 r sin2 θ dφ2 ds2 = dr2 − sin2 r dθ2 − sin2 r sinh2 θ dφ2
• Euclidean space E3 • Minkowskian spacetime M2+1
S3
[0]+ = ISO(3)/SO(3) S3
[0]− = ISO(2, 1)/SO(2, 1)
ds2 = dr2 + r2 dθ2 + r2 sin2 θ dφ2 ds2 = dr2 − r2 dθ2 − r2 sinh2 θ dφ2
• Hyperbolic space H3 • De Sitter spacetime dS2+1
S3
[−]+ = SO(3, 1)/SO(3) S3
[−]− = SO(3, 1)/SO(2, 1)
ds2 = dr2 + sinh2 r dθ2 + sinh2 r sin2 θ dφ2 ds2 = dr2 − sinh2 r dθ2 − sinh2 r sinh2 θ dφ2
Superintegrability on 3D Spaces of Constant Curvature 5
2.1 Vector model and ambient coordinates
The vector representation of soκ1,κ2(4) is given by the following 4× 4 real matrices [4]:
J01 =
· −κ1 · ·
1 · · ·
· · · ·
· · · ·
, J12 =
· · · ·
· · −κ2 ·
· 1 · ·
· · · ·
,
J02 =
· · −κ1κ2 ·
· · · ·
1 · · ·
· · · ·
, J13 =
· · · ·
· · · −κ2
· · · ·
· 1 · ·
,
J03 =
· · · −κ1κ2
· · · ·
· · · ·
1 · · ·
, J23 =
· · · ·
· · · ·
· · · −1
· · 1 ·
. (2.6)
Their exponential provides the corresponding one-parametric subgroups of SOκ1,κ2(4):
exJ01 =
Cκ1(x) −κ1 Sκ1(x) · ·
Sκ1(x) Cκ1(x) · ·
· · 1 ·
· · · 1
, exJ12 =
1 · · ·
· Cκ2(x) −κ2 Sκ2(x) ·
· Sκ2(x) Cκ2(x) ·
· · · 1
,
exJ02 =
Cκ1κ2(x) · −κ1κ2 Sκ1κ2(x) ·
· 1 · ·
Sκ1κ2(x) · Cκ1κ2(x) ·
· · · 1
, exJ13 =
1 · · ·
· Cκ2(x) · −κ2 Sκ2(x)
· · 1 ·
· Sκ2(x) · Cκ2(x)
,
exJ03 =
Cκ1κ2(x) · · −κ1κ2 Sκ1κ2(x)
· 1 · ·
· · 1 ·
Sκ1κ2(x) · · Cκ1κ2(x)
, exJ23 =
1 · · ·
· 1 · ·
· · cos x − sinx
· · sinx cos x
, (2.7)
where we have introduced the κ-dependent cosine and sine functions defined by [3, 5]
Cκ(x) =
∞∑
l=0
(−κ)l x2l
(2l)!
=
cos
√
κ x, κ > 0;
1, κ = 0;
cosh
√
−κ x, κ < 0,
Sκ(x) =
∞∑
l=0
(−κ)l x2l+1
(2l + 1)!
=
1√
κ
sin
√
κ x, κ > 0;
x, κ = 0;
1√
−κ
sinh
√
−κ x, κ < 0.
Notice that κ ∈ {κ1, κ1κ2, κ2}. The tangent is defined as Tκ(x) = Sκ(x)/ Cκ(x). Properties and
trigonometric relations for these κ-functions, which are necessary in the further computations,
can be found in [24, 25]; for instance,
C2
κ(x) + κ S2
κ(x) = 1,
d
dx
Cκ(x) = −κ Sκ(x),
d
dx
Sκ(x) = Cκ(x).
Under the above matrix algebra and group representations it is verified that
XT Iκ + IκX = 0, X ∈ soκ1,κ2(4), Y T IκY = Iκ, Y ∈ SOκ1,κ2(4),
6 F.J. Herranz and Á. Ballesteros
(XT is the transpose matrix of X) with respect to the bilinear form
Iκ = diag(+1, κ1, κ1κ2, κ1κ2).
Therefore SOκ1,κ2(4) is a group of isometries of Iκ acting on a linear ambient space R4 =
(x0, x1, x2, x3) through matrix multiplication. The origin O in S3
[κ1]κ2
has ambient coordinates
O = (1, 0, 0, 0) and this point is invariant under the subgroup H0 = SOκ2(3) = 〈J12, J13, J23〉
(see (2.7)). The orbit of O corresponds to the homogeneous space S3
[κ1]κ2
which is contained in
the “sphere”
Σ ≡ x2
0 + κ1x
2
1 + κ1κ2x
2
2 + κ1κ2x
2
3 = 1, (2.8)
determined by Iκ. The ambient coordinates (x0, x1, x2, x3), subjected to (2.8), are also called
Weierstrass coordinates. The metric on S3
[κ1]κ2
follows from the flat ambient metric in R4 divided
by the curvature and restricted to Σ:
ds2 =
1
κ1
(
dx2
0 + κ1dx2
1 + κ1κ2dx2
2 + κ1κ2dx2
3
) ∣∣∣
Σ
. (2.9)
A differential realization of soκ1,κ2(4), fulfilling (2.1), as first-order vector fields in the ambient
coordinates is provided by the vector representation (2.6) and reads
J01 = κ1x1∂0 − x0∂1, J0j = κ1κ2xj∂0 − x0∂j ,
J23 = x3∂2 − x2∂3, J1j = κ2xj∂1 − x1∂j , (2.10)
where j = 2, 3 and ∂µ = ∂/∂xµ.
2.2 Geodesic polar coordinate system
Let us consider a point Q in S3
[κ1]κ2
with Weierstrass coordinates (x0, x1, x2, x3). This can be
parametrized in terms of three intrinsic quantities of the space itself in different ways. We shall
make use of the geodesic polar coordinates (r, θ, φ) which are defined through the following action
of the one-parametric subgroups (2.7) on the origin O = (1, 0, 0, 0):
Q(r, θ, φ) = exp{φJ23} exp{θJ12} exp{rJ01}O,
x0
x1
x2
x3
=
Cκ1(r)
Sκ1(r) Cκ2(θ)
Sκ1(r) Sκ2(θ) cos φ
Sκ1(r) Sκ2(θ) sinφ
. (2.11)
Let l1 be a (time-like) geodesic and l2, l3 two other (space-like) geodesics in S3
[κ1]κ2
orthogonal
at O in such a manner that each translation generator J0i moves the origin along li. Then the
(physical) geometrical meaning of the coordinates (r, θ, φ) is as follows.
• The radial coordinate r is the distance between Q and O measured along the (time-like)
geodesic l that joins both points. In the curved Riemannian spaces with κ1 = ±1/R2, r has
dimensions of length, [r] = [R]; notice however that the dimensionless coordinate r/R is
usually taken instead of r, and so the former is considered as an ordinary angle (see,
e.g., [27]). In the relativistic spacetimes with κ1 = ±1/τ2, r has dimensions of a time-like
length, that is, [r] = [τ ].
• The coordinate θ is an ordinary angle in the three Riemannian spaces (κ2 = +1) and this
parametrizes the orientation of l with respect to l1, whilst θ corresponds to a rapidity in
the spacetimes (κ2 = −1/c2) with dimensions [θ] = [c].
Superintegrability on 3D Spaces of Constant Curvature 7
• Finally, φ is an ordinary angle for the six spaces that determines the orientation of l with
respect to the reference flag spanned by l1 and l2, that is, the 2-plane l1l2.
In the Riemannian spaces (r, θ, φ) parametrize the complete space, while in the spacetimes
these only cover the time-like region (in ambient coordinates this is x2
2 + x2
3 ≤ x2
1) limited by
the light-cone on which θ → ∞. The flat contraction κ1 = 0 gives rise to the usual spherical
coordinates in the Euclidean space (κ2 = 1).
By introducing the parametrization (2.11) in the metric written in terms of ambient coordina-
tes (2.9) we obtain that
ds2 = dr2 + κ2 S2
κ1
(r)
(
dθ2 + S2
κ2
(θ)dφ2
)
, (2.12)
which is particularized in Table 1 to each space. From it we compute the Levi-Civita connec-
tion Γk
ij , the Riemann Ri
jkl and Ricci Rij tensors [13]. Their nonzero components are given by
Γθ
θr = Γφ
φr = 1/ Tκ1(r), Γφ
φθ = 1/ Tκ2(θ), Γr
θθ = −κ2 Sκ1(r) Cκ1(r),
Γr
φφ = −κ2 Sκ1(r) Cκ1(r) S2
κ2
(θ), Γθ
φφ = −Sκ2(θ) Cκ2(θ),
Rr
θrθ = Rφ
θφθ = κ1κ2 S2
κ1
(r), Rr
φrφ = Rθ
φθφ = κ1κ2 S2
κ1
(r) S2
κ2
(θ), Rθ
rθr = Rφ
rφr = κ1,
Rrr = 2κ1, Rθθ = 2κ1κ2 S2
κ1
(r), Rφφ = 2κ1κ2 S2
κ1
(r) S2
κ2
(θ). (2.13)
Therefore all the sectional curvatures turn out to be constant Kij = κ1, while the scalar curvature
reads K = 6κ1.
3 Geodesic motion
The metric (2.12) can be read as the kinetic energy of a particle written in terms of the velocities
(ṙ, θ̇, φ̇), that is, the Lagrangian of the geodesic motion on the space S3
[κ1]κ2
given by
T =
1
2
(
ṙ2 + κ2 S2
κ1
(r)
(
θ̇2 + S2
κ2
(θ)φ̇2
))
. (3.1)
Then the canonical momenta (pr, pθ, pφ) are obtained through p = ∂T /∂q̇ (q̇ = ṙ, θ̇, φ̇), namely,
pr = ṙ,
pθ = κ2 S2
κ1
(r)θ̇,
pφ = κ2 S2
κ1
(r) S2
κ2
(θ)φ̇, (3.2)
so that the free Hamiltonian in the geodesic polar phase space (q; p) = (r, θ, φ; pr, pθ, pφ) with
respect to the canonical Lie–Poisson bracket,
{f, g} =
3∑
i=1
(
∂f
∂qi
∂g
∂pi
− ∂g
∂qi
∂f
∂pi
)
, (3.3)
turns out to be
T =
1
2
(
p2
r +
p2
θ
κ2 S2
κ1
(r)
+
p2
φ
κ2 S2
κ1
(r) S2
κ2
(θ)
)
. (3.4)
Note that the connection (2.13) would allow one to write the geodesic equations whose solution
would correspond to the geodesic motion associated with T (see [8] for the 2D case).
8 F.J. Herranz and Á. Ballesteros
Now we proceed to deduce a phase space realization of the Lie generators of soκ1,κ2(4). In
Weierstrass coordinates xµ and momenta pµ this comes from the vector fields (2.10) through the
replacement ∂µ → −pµ:
J01 = x0p1 − κ1x1p0, J0j = x0pj − κ1κ2xjp0,
J23 = x2p3 − x3p2, J1j = x1pj − κ2xjp1. (3.5)
The metric (2.9) can also be understood as the kinetic energy in the ambient velocities ẋµ so
that the momenta pµ are (j = 2, 3):
p0 = ẋ0/κ1, p1 = ẋ1, pj = κ2ẋj . (3.6)
Next if we compute the velocities ẋi in the parametrization (2.11) and introduce the mo-
menta (3.2) and (3.6) we obtain the relationship between the ambient momenta and the geodesic
polar ones:
p0 = −Sκ1(r) pr,
p1 = Cκ1(r) Cκ2(θ) pr −
Sκ2(θ)
Sκ1(r)
pθ,
p2 = κ2 Cκ1(r) Sκ2(θ) cos φ pr +
Cκ2(θ) cos φ
Sκ1(r)
pθ −
sinφ
Sκ1(r) Sκ2(θ)
pφ,
p3 = κ2 Cκ1(r) Sκ2(θ) sinφ pr +
Cκ2(θ) sinφ
Sκ1(r)
pθ +
cos φ
Sκ1(r) Sκ2(θ)
pφ.
Hence the generators (3.5) in geodesic polar coordinates and momenta turn out to be
J01 = Cκ2(θ) pr −
Sκ2(θ)
Tκ1(r)
pθ,
J02 = κ2 Sκ2(θ) cos φ pr +
Cκ2(θ) cos φ
Tκ1(r)
pθ −
sinφ
Tκ1(r) Sκ2(θ)
pφ,
J03 = κ2 Sκ2(θ) sinφ pr +
Cκ2(θ) sinφ
Tκ1(r)
pθ +
cos φ
Tκ1(r) Sκ2(θ)
pφ,
J12 = cos φ pθ −
sinφ
Tκ2(θ)
pφ,
J13 = sinφ pθ +
cos φ
Tκ2(θ)
pφ,
J23 = pφ. (3.7)
By direct computations it can be proven the following statement.
Proposition 1. The generators (3.7) fulfil the commutation relations (2.1) with respect to the
Lie–Poisson bracket (3.3) and all of them Poisson commute with T (3.4).
In this respect, notice that, under (3.7), the kinetic energy is related with the Casimir C1 (2.2)
by 2κ2T = C1, while the second Casimir C2 vanishes.
The realization of the generators (3.7) is particularized for each specific space and Poisson–
Lie algebra contained in S3
[κ1]κ2
and soκ1,κ2(4) in Table 2. In order to present the simplest
expressions, hereafter we shall set in all the tables κ1 ∈ {+1, 0,−1} and κ2 ∈ {+1,−1}, which
corresponds to deal with units R = τ = c = 1.
Superintegrability on 3D Spaces of Constant Curvature 9
Table 2. Phase space realization of the generators of soκ1,κ2(4) in canonical geodesic polar coordinates and
momenta (r, θ, φ; pr, pθ, pφ) on each space S3
[κ1]κ2
with κ1 ∈ {+1, 0,−1} and κ2 ∈ {+1,−1}.
3D Riemannian spaces (2 + 1)D Relativistic spacetimes
• Spherical space S3
[+]+ ≡ S3: so(4) • Anti-de Sitter spacetime S3
[+]− ≡ AdS2+1: so(2, 2)
J01 = cos θ pr −
sin θ
tan r
pθ J01 = cosh θ pr −
sinh θ
tan r
pθ
J02 = sin θ cos φ pr +
cos θ cos φ
tan r
pθ −
sin φ pφ
tan r sin θ
J02 = − sinh θ cos φ pr +
cosh θ cos φ
tan r
pθ −
sin φ pφ
tan r sinh θ
J03 = sin θ sin φ pr +
cos θ sin φ
tan r
pθ +
cos φ pφ
tan r sin θ
J03 = − sinh θ sin φ pr +
cosh θ sin φ
tan r
pθ +
cos φ pφ
tan r sinh θ
J12 = cos φ pθ −
sin φ
tan θ
pφ J12 = cos φ pθ −
sin φ
tanh θ
pφ
J13 = sin φ pθ +
cos φ
tan θ
pφ J13 = sin φ pθ +
cos φ
tanh θ
pφ
J23 = pφ J23 = pφ
• Euclidean space S3
[0]+ ≡ E3: iso(3) • Minkowskian spacetime S3
[0]− ≡ M2+1: iso(2, 1)
J01 = cos θ pr −
sin θ
r
pθ J01 = cosh θ pr −
sinh θ
r
pθ
J02 = sin θ cos φ pr +
cos θ cos φ
r
pθ −
sin φ
r sin θ
pφ J02 = − sinh θ cos φ pr +
cosh θ cos φ
r
pθ −
sin φ
r sinh θ
pφ
J03 = sin θ sin φ pr +
cos θ sin φ
r
pθ +
cos φ
r sin θ
pφ J03 = − sinh θ sin φ pr +
cosh θ sin φ
r
pθ +
cos φ
r sinh θ
pφ
J12 = cos φ pθ −
sin φ
tan θ
pφ J12 = cos φ pθ −
sin φ
tanh θ
pφ
J13 = sin φ pθ +
cos φ
tan θ
pφ J13 = sin φ pθ +
cos φ
tanh θ
pφ
J23 = pφ J23 = pφ
• Hyperbolic space S3
[−]+ ≡ H3: so(3, 1) • De Sitter spacetime S3
[−]− ≡ dS2+1: so(3, 1)
J01 = cos θ pr −
sin θ
tanh r
pθ J01 = cosh θ pr −
sinh θ
tanh r
pθ
J02 = sin θ cos φ pr +
cos θ cos φ
tanh r
pθ −
sin φ pφ
tanh r sin θ
J02 = − sinh θ cos φ pr +
cosh θ cos φ
tanh r
pθ −
sin φ pφ
tanh r sinh θ
J03 = sin θ sin φ pr +
cos θ sin φ
tanh r
pθ +
cos φ pφ
tanh r sin θ
J03 = − sinh θ sin φ pr +
cosh θ sin φ
tanh r
pθ +
cos φ pφ
tanh r sinh θ
J12 = cos φ pθ −
sin φ
tan θ
pφ J12 = cos φ pθ −
sin φ
tanh θ
pφ
J13 = sin φ pθ +
cos φ
tan θ
pφ J13 = sin φ pθ +
cos φ
tanh θ
pφ
J23 = pφ J23 = pφ
4 Superintegrable potentials
Now if we look for superintegrable potentials U(q) = U(r, θ, φ) which generalize the Euclidean
one (1.1) to the space S3
[κ1]κ2
we find
U = F ′(x0) +
β1
x2
1
+
β2
x2
2
+
β3
x2
3
= F(r) +
1
S2
κ1
(r)
(
β1
C2
κ2
(θ)
+
β2
S2
κ2
(θ) cos2 φ
+
β3
S2
κ2
(θ) sin2 φ
)
, (4.1)
where F ′( Cκ1(r)) ≡ F(r) is an arbitrary smooth function and βi are arbitrary real constants. As
in E3, the three βi-terms can be interpreted on the six spaces in a common way as “centrifugal
barriers”; for some particular curved spaces these may admit an alternative interpretation as
non-central harmonic oscillators. These facts will be explained in detail in the next section.
10 F.J. Herranz and Á. Ballesteros
The resulting Hamiltonian H = T + U , with kinetic energy (3.4) and potential (4.1), has
three integrals of the motion quadratic in the momenta which are associated with the (Lorentz)
rotation generators (j = 2, 3):
I1j = J2
1j + 2β1κ
2
2
x2
j
x2
1
+ 2βjκ2
x2
1
x2
j
, I23 = J2
23 + 2β2κ2
x2
3
x2
2
+ 2β3κ2
x2
2
x2
3
, (4.2)
which in the geodesic polar phase space explicitly read
I12 =
(
cos φ pθ −
sinφ
Tκ2(θ)
pφ
)2
+ 2β1κ
2
2 T2
κ2
(θ) cos2 φ +
2β2κ2
T2
κ2
(θ) cos2 φ
,
I13 =
(
sinφ pθ +
cos φ
Tκ2(θ)
pφ
)2
+ 2β1κ
2
2 T2
κ2
(θ) sin2 φ +
2β3κ2
T2
κ2
(θ) sin2 φ
,
I23 = p2
φ + 2β2κ2 tan2 φ +
2β3κ2
tan2 φ
. (4.3)
These constants of the motion do not Poisson commute with each other. In order to find
quantities in involution we define another integral from the above set:
I123 = I12 + I13 + κ2I23 + 2κ2(β1 + κ2β2 + κ2β3)
= p2
θ +
p2
φ
S2
κ2
(θ)
+
2β1κ2
C2
κ2
(θ)
+
2β2κ2
S2
κ2
(θ) cos2 φ
+
2β3κ2
S2
κ2
(θ) sin2 φ
, (4.4)
which is related with the Casimir of the rotation subalgebra h0 = soκ2(3).
Superintegrability of H is then characterized by:
Proposition 2. (i) The three functions {I12, I123,H} are mutually in involution. The same
holds for the set {I23, I123,H}.
(ii) The four functions {I12, I23, I123,H} are functionally independent, thus H is a superinte-
grable Hamiltonian.
These results, which can be checked directly, are displayed in Table 3 for each particular space
arising within S3
[κ1]κ2
. Notice that the integrals {I12, I23, I123} do depend on κ2 and (θ, φ; pθ, pφ)
but neither on the curvature κ1 nor on (r, pr), so these are the same for each set of three spaces
with the same signature.
A straightforward consequence of the complete integrability determined by {I23, I123,H} is
that H is separable and we obtain three equations, each of them depending on a canonical pair
(qi, pi):
I23(φ, pφ) = p2
φ + 2β2κ2 tan2 φ +
2β3κ2
tan2 φ
,
I123(θ, pθ) = p2
θ +
2β1κ2
C2
κ2
(θ)
+
1
S2
κ2
(θ)
(I23 + 2κ2(β2 + β3)) ,
H(r, pr) =
1
2
p2
r + F(r) +
1
2κ2 S2
κ1
(r)
I123, (4.5)
and H is so reduced to a 1D radial system.
Therefore there remains one constant of the motion to obtain maximal superintegrability
so that we shall say that H is a quasi-maximally superintegrable Hamiltonian. In the next
sections we study two specific choices for the arbitrary radial function F(r) that lead to an
additional integral thus providing maximally superintegrable potentials. The resulting systems
are generalizations of the (curved) harmonic oscillator and KC potentials with additional terms
(dependending on the βi).
Superintegrability on 3D Spaces of Constant Curvature 11
Table 3. Superintegrable Hamiltonian H = T + U and its three constants of the motion {I12, I23, I123} for the
six spaces S3
[κ1]κ2
with κ1 ∈ {+1, 0,−1} and κ2 ∈ {+1,−1}.
3D Riemannian spaces
• Spherical space S3
[+]+ ≡ S3
H =
1
2
(
p2
r +
p2
θ
sin2 r
+
p2
φ
sin2 r sin2 θ
)
+ F(r) +
1
sin2 r
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
+
β3
sin2 θ sin2 φ
)
• Euclidean space S3
[0]+ ≡ E3
H =
1
2
(
p2
r +
p2
θ
r2
+
p2
φ
r2 sin2 θ
)
+ F(r) +
1
r2
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
+
β3
sin2 θ sin2 φ
)
• Hyperbolic space S3
[−]+ ≡ H3
H =
1
2
(
p2
r +
p2
θ
sinh2 r
+
p2
φ
sinh2 r sin2 θ
)
+ F(r) +
1
sinh2 r
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
+
β3
sin2 θ sin2 φ
)
I12 =
(
cos φ pθ −
sin φ
tan θ
pφ
)2
+ 2β1 tan2 θ cos2 φ +
2β2
tan2 θ cos2 φ
I23 = p2
φ + 2β2 tan2 φ +
2β3
tan2 φ
I123 = p2
θ +
p2
φ
sin2 θ
+
2β1
cos2 θ
+
2β2
sin2 θ cos2 φ
+
2β3
sin2 θ sin2 φ
(2 + 1)D Relativistic spacetimes
• Anti-de Sitter spacetime S3
[+]− ≡ AdS2+1
H =
1
2
(
p2
r −
p2
θ
sin2 r
−
p2
φ
sin2 r sinh2 θ
)
+ F(r) +
1
sin2 r
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
+
β3
sinh2 θ sin2 φ
)
• Minkowskian spacetime S3
[0]− ≡ M2+1
H =
1
2
(
p2
r −
p2
θ
r2
−
p2
φ
r2 sinh2 θ
)
+ F(r) +
1
r2
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
+
β3
sinh2 θ sin2 φ
)
• De Sitter spacetime S3
[−]− ≡ dS2+1
H =
1
2
(
p2
r −
p2
θ
sinh2 r
−
p2
φ
sinh2 r sinh2 θ
)
+ F(r) +
1
sinh2 r
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
+
β3
sinh2 θ sin2 φ
)
I12 =
(
cos φ pθ −
sin φ
tanh θ
pφ
)2
+ 2β1 tanh2 θ cos2 φ− 2β2
tanh2 θ cos2 φ
I23 = p2
φ − 2β2 tan2 φ− 2β3
tan2 φ
I123 = p2
θ +
p2
φ
sinh2 θ
− 2β1
cosh2 θ
− 2β2
sinh2 θ cos2 φ
− 2β3
sinh2 θ sin2 φ
5 Harmonic oscillator potential
If we like to extend the (curved) harmonic oscillator potential (1.4) to our six spaces, we have
to consider the following choice for the arbitrary function appearing in (4.1):
F ′(x0) = β0
(
1− x2
0
κ1x2
0
)
= β0
(
x2
1 + κ2x
2
2 + κ2x
2
3
x2
0
)
, F(r) = β0 T2
κ1
(r), (5.1)
where β0 is an arbitrary real parameter. When the complete Hamiltonian is considered we
obtain the generalization of 3D SW system (1.2), HSW = T +USW, to the space S3
[κ1]κ2
, namely
USW = β0 T2
κ1
(r) +
1
S2
κ1
(r)
(
β1
C2
κ2
(θ)
+
β2
S2
κ2
(θ) cos2 φ
+
β3
S2
κ2
(θ) sin2 φ
)
. (5.2)
12 F.J. Herranz and Á. Ballesteros
As we already mentioned in the introduction, the proper SW Hamiltonian arises in the (flat)
Euclidean space [15, 16, 17, 19], here written in polar coordinates, which is formed by an isotropic
harmonic oscillator with angular frequency ω =
√
β0 together with three centrifugal barriers
associated with the βi’s. Different constructions of the SW system on the (curved) spherical
and hyperbolic spaces can be found in [9, 20, 23, 28, 30, 31, 39]. More recently such a potential
has also been deduced and analysed in the (1+1)D relativistic spacetimes of constant curvature
in [8, 12] as well as in 2D spaces of variable curvature in [8].
In our case, any of the translation generators J0i (3.7) provides a constant of the motion
quadratic in the momenta in the form (j = 2, 3):
I01 = J2
01 + 2β0
x2
1
x2
0
+ 2β1
x2
1
x2
0
,
I0j = J2
0j + 2β0κ
2
2
x2
j
x2
0
+ 2βjκ2
x2
0
x2
j
, (5.3)
that is,
I01 = J2
01 + 2β0 T2
κ1
(r) C2
κ2
(θ) +
2β1
T2
κ1
(r) C2
κ2
(θ)
,
I02 = J2
02 + 2β0κ
2
2 T2
κ1
(r) S2
κ2
(θ) cos2 φ +
2β2κ2
T2
κ1
(r) S2
κ2
(θ) cos2 φ
,
I03 = J2
03 + 2β0κ
2
2 T2
κ1
(r) S2
κ2
(θ) sin2 φ +
2β3κ2
T2
κ1
(r) S2
κ2
(θ) sin2 φ
. (5.4)
Obviously, the seven integrals of the motion {I01, I02, I03, I12, I23, I123,HSW} cannot be func-
tionally independent. One constraint for them is given by
2κ2HSW = κ2I01 + I02 + I03 + κ1I123,
which reminds the aforementioned relation for the geodesic motion 2κ2T = C1. Note also that
{I01, I23} = {I02, I13} = {I03, I12} = 0.
The final result concerning the superintegrability of HSW is established by:
Proposition 3. (i) Each function I0i (5.4) (i = 1, 2, 3) Poisson commutes with HSW.
(ii) The five functions {I0i, I12, I23, I123,HSW}, where i is fixed, are functionally independent,
thus HSW is a maximally superintegrable Hamiltonian.
The Hamiltonian HSW and the additional constant of the motion I01 (that ensures maximal
superintegrability) are presented for each particular space contained in S3
[κ1]κ2
in Table 4.
5.1 Description of the SW potential
The 2D version of the potential USW (5.2) on S2 has been interpreted in [40, 41, 42] as a super-
position of three spherical oscillators; the interpretation for arbitrary dimension on SN and HN
has been presented in [9, 23]. Furthermore, a detail description on this potential on AdS1+1,
M1+1 and dS1+1 was recently performed in [8]. In what follows we analyse the (physical) geo-
metrical role of the 3D potential (5.2) on each particular space S3
[κ1]κ2
thus generalizing all of
the mentioned 2D results.
Superintegrability on 3D Spaces of Constant Curvature 13
Table 4. Maximally superintegrable Smorodinsky–Winternitz Hamiltonian HSW = T + USW and the additional
constant of the motion I01 to the set {I12, I23, I123} for the six spaces S3
[κ1]κ2
with the same conventions given in
Table 3.
3D Riemannian spaces
• Spherical space S3
HSW =
1
2
(
p2
r +
p2
θ
sin2 r
+
p2
φ
sin2 r sin2 θ
)
+ β0 tan2 r +
1
sin2 r
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
+
β3
sin2 θ sin2 φ
)
I01 =
(
cos θ pr −
sin θ
tan r
pθ
)2
+ 2β0 tan2 r cos2 θ +
2β1
tan2 r cos2 θ
• Euclidean space E3
HSW =
1
2
(
p2
r +
p2
θ
r2
+
p2
φ
r2 sin2 θ
)
+ β0 r2 +
1
r2
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
+
β3
sin2 θ sin2 φ
)
I01 =
(
cos θ pr −
sin θ
r
pθ
)2
+ 2β0 r2 cos2 θ +
2β1
r2 cos2 θ
• Hyperbolic space H3
HSW =
1
2
(
p2
r +
p2
θ
sinh2 r
+
p2
φ
sinh2 r sin2 θ
)
+ β0 tanh2 r +
1
sinh2 r
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
+
β3
sin2 θ sin2 φ
)
I01 =
(
cos θ pr −
sin θ
tanh r
pθ
)2
+ 2β0 tanh2 r cos2 θ +
2β1
tanh2 r cos2 θ
(2 + 1)D Relativistic spacetimes
• Anti-de Sitter spacetime AdS2+1
HSW =
1
2
(
p2
r −
p2
θ
sin2 r
−
p2
φ
sin2 r sinh2 θ
)
+ β0 tan2 r +
1
sin2 r
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
+
β3
sinh2 θ sin2 φ
)
I01 =
(
cosh θ pr −
sinh θ
tan r
pθ
)2
+ 2β0 tan2 r cosh2 θ +
2β1
tan2 r cosh2 θ
• Minkowskian spacetime M2+1
HSW =
1
2
(
p2
r −
p2
θ
r2
−
p2
φ
r2 sinh2 θ
)
+ β0 r2 +
1
r2
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
+
β3
sinh2 θ sin2 φ
)
I01 =
(
cosh θ pr −
sinh θ
r
pθ
)2
+ 2β0 r2 cosh2 θ +
2β1
r2 cosh2 θ
• De Sitter spacetime dS2+1
HSW =
1
2
(
p2
r −
p2
θ
sinh2 r
−
p2
φ
sinh2 r sinh2 θ
)
+ β0 tanh2 r +
1
sinh2 r
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
+
β3
sinh2 θ sin2 φ
)
I01 =
(
cosh θ pr −
sinh θ
tanh r
pθ
)2
+ 2β0 tanh2 r cosh2 θ +
2β1
tanh2 r cosh2 θ
Consider the (time-like) geodesic l1 and the two (space-like) geodesics l2, l3 in S3
[κ1]κ2
orthog-
onal at the origin O and the generic point Q(r, θ, φ) as given in Subsection 2.2. Next let Qij
(i, j = 1, 2, 3; i < j) be the intersection point of the reference flag spanned by li and lj (the
2-plane lilj) with its orthogonal geodesic through Q. Hence we introduce the (time-like) geodesic
distance x = QQ23 and the two (space-like) distances y = QQ13, z = QQ12. Finally, let Q1
be the intersection point of l1 with its orthogonal (space-like) geodesic l′1 through Q for which
h = QQ1 is the (space-like) distance measured along l′1. Now by applying trigonometry [24] on
the orthogonal triangles OQQ1 (with inner angle θ), OQQ23 (with external angle θ), Q13QQ1
(with external angle φ) and Q12QQ1 (with inner angle φ), we find that
OQQ1 : Sκ1κ2(h) = Sκ1(r) Sκ2(θ),
OQQ23 : Sκ1(x) = Sκ1(r) Cκ2(θ),
14 F.J. Herranz and Á. Ballesteros
Q13QQ1 : Sκ1κ2(y) = Sκ1κ2(h) cos φ,
Q12QQ1 : Sκ1κ2(z) = Sκ1κ2(h) sinφ.
Hence the ambient coordinates xi (2.11) can be expressed as
x1 = Sκ1(r) Cκ2(θ) = Sκ1(x),
x2 = Sκ1(r) Sκ2(θ) cos φ = Sκ1κ2(y),
x3 = Sκ1(r) Sκ2(θ) sinφ = Sκ1κ2(z), (5.5)
so that the SW potential (5.2) can be rewritten as
USW = β0 T2
κ1
(r) +
β1
S2
κ1
(x)
+
β2
S2
κ1κ2
(y)
+
β3
S2
κ1κ2
(z)
, (5.6)
which allows for a unified interpretation on the six spaces:
• The β0-term is a central harmonic oscillator, that is, the Higgs oscillator [26] with center
at the origin O.
• The three βi-terms (i = 1, 2, 3) are “centrifugal barriers”.
Furthermore, the βi-potentials can be interpreted as non-central oscillators in some particular
spaces that we proceed to describe by considering the simplest values for κi ∈ {±1}.
5.1.1 Spherical space S3
Let Oi be the points placed along the basic geodesics li (i = 1, 2, 3) which are a quadrant apart
from the origin O, that is, each two points taken from the set {O,Oi} are mutually separated
a distance π
2 (if κ1 = 1/R2, a quadrant is π/(2
√
κ1) = Rπ/2). In fact, each Oi is the intersection
point between the geodesic li and the axis xi of the ambient space. If we denote by ri the distance
between Q and Oi measured along the geodesic joining both points then
r1 + x = r2 + y = r3 + z =
π
2
,
which means that each set of three points {O1QQ23}, {O2QQ13} and {O3QQ12} lie on the same
geodesic. Thus
x1 = sinx = cos r1, x2 = sin y = cos r2, x3 = sin z = cos r3,
so that the SW potential (5.2) on S3 can be expressed in two manners
USW = β0 tan2 r +
β1
sin2 x
+
β2
sin2 y
+
β3
sin2 z
(5.7)
= β0 tan2 r +
3∑
i=1
(
βi tan2 ri + βi
)
, (5.8)
which show a superposition of the central spherical oscillator with center at O either with three
spherical centrifugal barriers, or with three spherical oscillators with centers placed at Oi [9, 23].
5.1.2 Hyperbolic space H3
The analogous points to the previous “centers” Oi would be beyond the “actual” hyperbolic
space and so placed in the exterior (“ideal”) region of H3. The SW potential can only written
in the form (5.6):
USW = β0 tanh2 r +
β1
sinh2 x
+
β2
sinh2 y
+
β3
sinh2 z
, (5.9)
giving rise to the superposition of a central hyperbolic oscillator with three hyperbolic centrifugal
barriers [23].
Superintegrability on 3D Spaces of Constant Curvature 15
5.1.3 Euclidean space E3
The contraction κ1 → 0 (R → ∞) of the SW potential on S3 and H3 can be applied on both
expressions (5.7) and (5.9) reducing to
USW = β0r
2 +
β1
x2
+
β2
y2
+
β3
z2
, (5.10)
which is just the proper SW potential (1.2) formed by the flat harmonic oscillator with three
centrifugal barriers; in this case (x, y, z) are Cartesian coordinates on E3 and r2 = x2 + y2 + z2.
This contraction cannot be performed on S3 when the potential is written in the form (5.8);
notice that if κ1 → 0 the points Oi →∞.
5.1.4 Anti-de Sitter spacetime AdS2+1
We consider the intersection point O1 between the time-like geodesic l1 and the axis x1 of the
ambient space, which is at a time-like distance π
2 from the origin O [8]. If r1 denotes the time-like
distance QO1, then r1 + x = π
2 . Therefore the SW potential becomes
USW = β0 tan2 r +
β1
sin2 x
+
β2
sinh2 y
+
β3
sinh2 z
(5.11)
= β0 tan2 r + β1 tan2 r1 +
β2
sinh2 y
+
β3
sinh2 z
+ β1. (5.12)
The former expression corresponds to the superposition of a time-like (spherical) oscillator cen-
tered at O with a time-like (spherical) centrifugal potential and two space-like (hyperbolic)
ones. Under the latter form, the time-like centrifugal term is transformed into another spherical
oscillator now with center at O1.
5.1.5 De Sitter spacetime dS2+1
Recall that AdS2+1 and dS2+1 are related through an interchange between time-like lines and
space-like ones; the former are compact (circular) on AdS2+1 and non-compact (hyperbolic) on
dS2+1, while the latter are non-compact on AdS2+1 but compact on dS2+1.
So, we consider the intersection point Oj (j = 2, 3) between the space-like geodesic lj and
the axis xj which is at a space-like distance π
2 from O, so that rj is the space-like distance QOj
verifying r2 + y = r3 + z = π
2 [8]. Hence the SW potential can be rewritten as
USW = β0 tanh2 r +
β1
sinh2 x
+
β2
sin2 y
+
β3
sin2 z
(5.13)
= β0 tanh2 r +
β1
sinh2 x
+ β2 tan2 r2 + β3 tan2 r3 + β2 + β3. (5.14)
In this way, we find the superposition of a central time-like (hyperbolic) oscillator with a time-
like (hyperbolic) centrifugal barrier, and either with two other space-like (spherical) centrifugal
barriers or with two space-like (spherical) oscillators centered at Oj .
5.1.6 Minkowskian spacetime M2+1
Finally, the contraction κ1 → 0 (τ →∞) of (5.11) and (5.13) gives
USW = β0r
2 +
β1
x2
+
β2
y2
+
β3
z2
, (5.15)
16 F.J. Herranz and Á. Ballesteros
which is formed by a time-like harmonic oscillator β0r
2, one time-like centrifugal barrier β1/x2
together with two space-like ones β2/y2, β3/z2. The coordinates (x, y, z) are the usual time and
space ones such that r2 = x2 − y2 − z2. On the contrary, the expressions (5.12) and (5.14) are
not well defined when κ1 → 0 since the points O1 and Oj go to infinity.
6 Kepler–Coulomb potential
The generalization of the KC potential (1.4) to the space S3
[κ1]κ2
is achieved by choosing
F ′(x0) = −k
x0√
(1− x2
0)/κ1
= −k
x0√
x2
1 + κ2x2
2 + κ2x2
3
, F(r) = − k
Tκ1(r)
, (6.1)
where k is an arbitrary real parameter. As it already happens in E3 [14], it is not possible
to add the three potential terms depending on the βi’s keeping at the same time maximal
superintegrability; so that, at least, one of them must vanish. Consequently, we find, in principle,
three possible generalizations of the Euclidean potential (1.3) to S3
[κ1]κ2
:
UGKC
1 = − k
Tκ1(r)
+
1
S2
κ1
(r) S2
κ2
(θ)
(
β2
cos2 φ
+
β3
sin2 φ
)
,
UGKC
2 = − k
Tκ1(r)
+
1
S2
κ1
(r)
(
β1
C2
κ2
(θ)
+
β3
S2
κ2
(θ) sin2 φ
)
,
UGKC
3 = − k
Tκ1(r)
+
1
S2
κ1
(r)
(
β1
C2
κ2
(θ)
+
β2
S2
κ2
(θ) cos2 φ
)
. (6.2)
Thus each potential UGKC
i contains the proper KC k-term [11, 12, 28, 31, 32, 33, 37, 39, 43]
together with two additional βi-terms, which can further be interpreted as centrifugal barriers
or non-central oscillators; for each of them there is an additional constant of the motion given
by (i = 1, 2, 3):
Li =
3∑
l=1;l 6=i
J0lJli + k
κ2xi√
x2
1 + κ2x2
2 + κ2x2
3
− 2κ2
3∑
l=1;l 6=i
βl
x0xi
x2
l
, (6.3)
where Jli = −Jil for i < l. In terms of the geodesic polar phase space these integrals read
L1 = −J02J12 − J03J13 + k κ2 Cκ2(θ)−
2κ2 Cκ2(θ)
Tκ1(r) S2
κ2
(θ)
(
β2
cos2 φ
+
β3
sin2 φ
)
,
L2 = J01J12 − J03J23 + k κ2 Sκ2(θ) cos φ− 2κ2 cos φ
Tκ1(r)
(
β1 Sκ2(θ)
C2
κ2
(θ)
+
β3
Sκ2(θ) sin2 φ
)
,
L3 = J01J13 + J02J23 + k κ2 Sκ2(θ) sinφ− 2κ2 sin φ
Tκ1(r)
(
β1 Sκ2(θ)
C2
κ2
(θ)
+
β2
Sκ2(θ) cos2 φ
)
. (6.4)
The superintegrability of each Hamiltonian HGKC
i = T +UGKC
i (i = 1, 2, 3) is determined by:
Proposition 4. (i) The function Li (6.4) Poisson commutes with HGKC
i .
(ii) The five functions {Li, I12, I23, I123,HGKC
i } are functionally independent, thus HGKC
i is
a maximally superintegrable Hamiltonian.
Superintegrability on 3D Spaces of Constant Curvature 17
6.1 The Laplace–Runge–Lenz vector
When another βj is taken equal to zero in a given potential UGKC
i (j 6= i), the function Lj is also
a constant of the motion. Therefore when all the βj = 0, the three functions (6.4) are constants
of the motion for the GKC potential which reduces in this case to the proper KC system. This
is summed up in the following statements.
Proposition 5. Let one βj = 0 in the Hamiltonian HGKC
i = T + UGKC
i (i = 1, 2, 3) with j 6= i.
(i) The two functions Li, Lj Poisson commute with HGKC
i .
(ii) The functions {I12, I23, I123,HGKC
i } together with either Li or Lj are functionally inde-
pendent.
Proposition 6. Let the three βi = 0, then:
(i) The three GKC potentials reduce to its common k-term, UGKC
i ≡ UKC = −k/ Tκ1(r),
which is the (curved) KC potential on S3
[κ1]κ2
.
(ii) The three functions
L1 = −J02J12 − J03J13 + k κ2 Cκ2(θ),
L2 = J01J12 − J03J23 + k κ2 Sκ2(θ) cos φ,
L3 = J01J13 + J02J23 + k κ2 Sκ2(θ) sinφ, (6.5)
Poisson commutes with HKC = T + UKC and these are the components of the Laplace–Runge–
Lenz vector on S3
[κ1]κ2
.
(iii) The functions {I12, I23, I123,HGKC
i } together with any of the components Li are func-
tionally independent.
On the other hand, equivalence amongst the Hamiltonians HGKC
i comes from their interpre-
tation on S3
[κ1]κ2
that we proceed to study. We shall show that the three potentials (6.2) are
equivalent on the Riemannian spaces (take i = 3), meanwhile we can distinguish two different
potentials on the spacetimes (take i = 1, 3). Thus we display in Table 5 the corresponding
non-equivalent GKC potentials together with the additional constant of the motion (6.4).
6.2 Description of the GKC potential
In Subsection 5.1 we have interpreted each of the βi-terms appearing within the SW potential
either as a non-central oscillator or as a centrifugal barrier according to the particular space
under consideration. This, in turn, means that each potential (6.2) is a superposition of the KC
potential with either oscillators or centrifugal barriers. The latter interpretation arises directly
by introducing the distances (x, y, z) (5.5) and this holds simultaneously for the six spaces:
UGKC
1 = − k
Tκ1(r)
+
β2
S2
κ1κ2
(y)
+
β3
S2
κ1κ2
(z)
,
UGKC
2 = − k
Tκ1(r)
+
β1
S2
κ1
(x)
+
β3
S2
κ1κ2
(z)
,
UGKC
3 = − k
Tκ1(r)
+
β1
S2
κ1
(x)
+
β2
S2
κ1κ2
(y)
. (6.6)
These expressions clearly show that some HGKC
i are equivalent according to the value of κ2,
that is, the signature of the metric, so that we analyze the two possibilities separately.
18 F.J. Herranz and Á. Ballesteros
Table 5. Maximally superintegrable generalized Kepler–Coulomb potential UGKC
i , such that HGKC
i = T +UGKC
i ,
and the additional constant of the motion Li to the set {I12, I23, I123} for S3
[κ1]κ2
with the same conventions given
in Table 3 (i = 3 for the Riemannian spaces and i = 3, 1 for the spacetimes).
3D Riemannian spaces
• Spherical space S3
UGKC
3 = − k
tan r
+
1
sin2 r
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
)
L3 = J01J13 + J02J23 + k sin θ sin φ− 2 sin φ
tan r
(
β1 sin θ
cos2 θ
+
β2
sin θ cos2 φ
)
• Euclidean space E3
UGKC
3 = −k
r
+
1
r2
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
)
L3 = J01J13 + J02J23 + k sin θ sin φ− 2 sin φ
r
(
β1 sin θ
cos2 θ
+
β2
sin θ cos2 φ
)
• Hyperbolic space H3
UGKC
3 = − k
tanh r
+
1
sinh2 r
(
β1
cos2 θ
+
β2
sin2 θ cos2 φ
)
L3 = J01J13 + J02J23 + k sin θ sin φ− 2 sin φ
tanh r
(
β1 sin θ
cos2 θ
+
β2
sin θ cos2 φ
)
(2 + 1)D Relativistic spacetimes
• Anti-de Sitter spacetime AdS2+1
UGKC
3 = − k
tan r
+
1
sin2 r
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
)
L3 = J01J13 + J02J23 − k sinh θ sin φ +
2 sin φ
tan r
(
β1 sinh θ
cosh2 θ
+
β2
sinh θ cos2 φ
)
UGKC
1 = − k
tan r
+
1
sin2 r sinh2 θ
(
β2
cos2 φ
+
β3
sin2 φ
)
L1 = −J02J12 − J03J13 − k cosh θ +
2 cosh θ
tan r sinh2 θ
(
β2
cos2 φ
+
β3
sin2 φ
)
• Minkowskian spacetime M2+1
UGKC
3 = −k
r
+
1
r2
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
)
L3 = J01J13 + J02J23 − k sinh θ sin φ +
2 sin φ
r
(
β1 sinh θ
cosh2 θ
+
β2
sinh θ cos2 φ
)
UGKC
1 = −k
r
+
1
r2 sinh2 θ
(
β2
cos2 φ
+
β3
sin2 φ
)
L1 = −J02J12 − J03J13 − k cosh θ +
2 cosh θ
r sinh2 θ
(
β2
cos2 φ
+
β3
sin2 φ
)
• De Sitter spacetime dS2+1
UGKC
3 = − k
tanh r
+
1
sinh2 r
(
β1
cosh2 θ
+
β2
sinh2 θ cos2 φ
)
L3 = J01J13 + J02J23 − k sinh θ sin φ +
2 sin φ
tanh r
(
β1 sinh θ
cosh2 θ
+
β2
sinh θ cos2 φ
)
UGKC
1 = − k
tanh r
+
1
sinh2 r sinh2 θ
(
β2
cos2 φ
+
β3
sin2 φ
)
L1 = −J02J12 − J03J13 − k cosh θ +
2 cosh θ
tanh r sinh2 θ
(
β2
cos2 φ
+
β3
sin2 φ
)
6.2.1 Riemannian spaces
When κ2 = +1 the three distances (x, y, z) are completely equivalent, and their “label” in
the trigonometric functions is always κ1 (recall that in this case both θ and φ are ordinary
Superintegrability on 3D Spaces of Constant Curvature 19
angles). Hence the three Hamiltonians HGKC
i are also equivalent and we only consider a unique
potential, say UGKC
3 with constant of the motion L3. On the spherical space S3 both β1, β2
terms can alternatively be expressed as non-central oscillators as commented in Subsection 5.1.1,
meanwhile on E3 and H3 these only can be interpreted as centrifugal barriers. In this way we
find the following expressions for each space:
S3 : UGKC
3 =− k
tan r
+
β1
sin2 x
+
β2
sin2 y
= − k
tan r
+ β1tan2 r1 + β2tan2 r2 + β1 + β2;
E3 : UGKC
3 =− k
r
+
β1
x2
+
β2
y2
;
H3 : UGKC
3 =− k
tanh r
+
β1
sinh2 x
+
β2
sinh2 y
.
When all the βi = 0 we obtain the components of the Laplace–Runge–Lenz vector (6.5) for the
three Riemannian spaces:
L1 = −J02J12 − J03J13 + k cos θ,
L2 = J01J12 − J03J23 + k sin θ cos φ,
L3 = J01J13 + J02J23 + k sin θ sinφ,
where the difference for each particular space comes from the translations J0i (3.7) that do
depend on the curvature κ1.
6.2.2 Relativistic spacetimes
On the contrary, if κ2 = −1 (in units c = 1), only the two space-like distances y and z are equi-
valent while x is a time-like distance (φ is also an angle for the three spacetimes but θ is
a rapidity). Thus UGKC
2 ' UGKC
3 containing a time-like centrifugal barrier and another space-like
one, while UGKC
1 defines a different potential with two space-like centrifugal barriers for the three
spacetimes. By taking into account the results given in Subsection 5.1, these potentials show
different superpositions of the KC potential with non-central harmonic oscillators and centrifugal
barriers on AdS2+1 and dS2+1. The explicit expressions on each spacetime turn out to be
AdS2+1 : UGKC
3 =− k
tan r
+
β1
sin2 x
+
β2
sinh2 y
= − k
tan r
+ β1tan2 r1 +
β2
sinh2 y
+ β1,
UGKC
1 =− k
tan r
+
β2
sinh2 y
+
β3
sinh2 z
;
M2+1 : UGKC
3 =− k
r
+
β1
x2
+
β2
y2
,
UGKC
1 =− k
r
+
β2
y2
+
β3
z2
;
dS2+1 : UGKC
3 =− k
tanh r
+
β1
sinh2 x
+
β2
sin2 y
= − k
tanh r
+
β1
sinh2 x
+ β2tan2 r2 + β2,
UGKC
1 =− k
tanh r
+
β2
sin2 y
+
β3
sin2 z
=− k
tanh r
+ β2tan2 r2 + β3tan2 r3 + β2 + β3.
The components of the Laplace–Runge–Lenz vector (6.5) (for βi = 0) written in terms of the
kinematical generators (2.3) are
L1 = −P1K1 − P2K2 − k cosh θ, L2 = P0K1 − P2J − k sinh θ cos φ,
L3 = P0K2 + P1J − k sinh θ sinφ.
20 F.J. Herranz and Á. Ballesteros
7 Concluding remarks
We have achieved the generalization of the 3D Euclidean superintegrable family (1.1) as well
as the maximally superintegrable SW (1.2) and GKC (1.3) potentials to the space S3
[κ1]κ2
by
applying a unified approach which makes use of a built-in scheme of contractions. Furthermore
the results so obtained have been described and interpreted on each particular space and have
also been displayed along the paper in tabular form. Thus we have explicitly shown that (maxi-
mal) superintegrability is preserved for any value of the curvature and for either a Riemannian
or Lorentzian metric. Notice that on the complex sphere (see e.g. [31]) and on the ambient
space R4 these two maximally superintegrable Hamiltonians read
HSW =
3∑
µ=0
(
1
2
p2
µ +
βµ
x2
µ
)
− β0, HGKC
3 =
1
2
3∑
µ=0
p2
µ −
k x0√
x2
1 + x2
2 + x2
3
+
β1
x2
1
+
β2
x2
2
,
where
3∑
µ=0
x2
µ = 1. Therefore the Hamiltonians here studied can be regarded as different real
forms coming from known complex superintegrable systems through graded contractions, that
is, by introducing the parameters κ1 and κ2.
As far as the superintegrable potential U (4.1) is concerned, we recall that in this 3D case, we
have one constant of the motion (besides de Hamiltonian) more than the two ones that ensure its
complete integrability, but one integral less than the four ones that determine maximal superin-
tegrability. By taking into account the former point of view one may claim that U is minimally
(or weak) superintegrable, while from the latter, U would be quasi-maximally superintegrable.
Our opinion is that when the corresponding Hamiltonian H = T + U is constructed on the ND
spaces SN
[κ1]κ2
, each of the N(N − 1) generators Jij (i, j = 1, . . . , N ; i < j) of the (Lorentz)
rotation subalgebra soκ2(N) would provide a constant of the motion Iij of the type (4.2). Next,
by following [9, 23], two subsets of N − 1 constants of the motion, Q(l) and Q(l), should be
deduced from the initial set of N(N − 1) integrals as:
Q(l) =
l∑
i,j=1
Iij , Q(l) =
N∑
i,j=N−l+1
Iij , l = 2, . . . , N, (7.7)
where Q(N) ≡ Q(N). In this way the complete integrability of H would be characterized by either
the N constants of the motion {Q(l),H} or by {Q(l),H}. The quasi-maximal superintegrability
would be provided by the 2N − 2 functions
{Q(2), Q(3), . . . , Q(N) ≡ Q(N), . . . , Q(3), Q(2),H}.
The corresponding SW potential on SN
[κ1]κ2
would be obtained by taking the same F(r) as
in (5.1) and the remaining constant of the motion would come from one of the translation
generators J0i in the form I0i (5.3). Likewise, a set of N GKC potentials could be constructed
by starting from the radial function (6.1) and then taking N − 1 centrifugal terms for each of
them as in (6.2); in this case the additional constant of the motion Li would be of the form (6.3).
We stress that this scheme of the possible ND generalization of all the 3D results here
presented (currently in progress) relies on the fact that the potential U can be endowed with
a coalgebra symmetry [10]. This indeed allowed us to obtain the integrals (7.7) for the ND SW
system on the three Riemannian spaces in [9, 23] by starting from the quantum deformation of
the Euclidean SW system introduced in [1, 2]. Furthermore, quantum deformations have been
shown [6, 7] to give rise to Riemannian and relativistic spaces of non-constant curvature on
which SW- and KC-type potentials can be considered [8].
Superintegrability on 3D Spaces of Constant Curvature 21
Acknowledgements
This work was partially supported by the Ministerio de Educación y Ciencia (Spain, Project
FIS2004-07913) and by the Junta de Castilla y León (Spain, Projects BU04/03 and VA013C05).
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1 Introduction
2 Riemannian spaces and relativistic spacetimes
2.1 Vector model and ambient coordinates
2.2 Geodesic polar coordinate system
3 Geodesic motion
4 Superintegrable potentials
5 Harmonic oscillator potential
5.1 Description of the SW potential
5.1.1 Spherical space S3
5.1.2 Hyperbolic space H3
5.1.3 Euclidean space E3
5.1.4 Anti-de Sitter spacetime AdS2+1
5.1.5 De Sitter spacetime dS2+1
5.1.6 Minkowskian spacetime M2+1
6 Kepler-Coulomb potential
6.1 The Laplace-Runge-Lenz vector
6.2 Description of the GKC potential
6.2.1 Riemannian spaces
6.2.2 Relativistic spacetimes
7 Concluding remarks
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