arXiv:1510.06327v1 [math.DS] 21 Oct 2015 h uvtr,ahroi ucinta ep l one or justifications. bounded unifying all different keeps need that for problems function is, Other harmonic which a potential, curvature, Kepler instanc the the for by spheres, preserved hyperbolic and properties spheres to problem body 1] hspriua oi smc eerhdtdyi h c the in h today researched to Lobache much mechanics Nikolai is celestial topic and of particular Bolyai This problem János [14]. 2-body of the work generalize chal the This to with new. 1830s not the is manifolds in general more to space clidean n antb rae lbly h aua generalization natural i however, The issue, globally. treated This be sense. cannot physical and any has couple equations a the just mention to [9], Schrödinger system, for Vlasov-Poisson PDE as the for such and pursued mechanics, [7], been classical [6], also of [5], have boundaries [4], manifolds various curvature, to constant tensions of spaces in problem h tepst xedcascleutoso mathematical of equations classical extend to attempts The nalteepolm ti motn oudrtn hte t whether understand to important is it problems these all In ETRFED HOG AIOD FCONSTANT OF MANIFOLDS THROUGH FIELDS VECTOR H OTNOSTASTO FHAMILTONIAN OF TRANSITION CONTINUOUS THE < κ U n hnapyorcnlsost h gravitational the o to are which conclusions cases, our 3-dimensional apply and t then answer 2- and the the that in prove We both positive, case. Euclidean the to corresponds osatGusa curvature Gaussian constant Abstract. κ κ , 0 lrnDau lmIrhm n edzjŚniatycki Je˛drzej and Ibrahim, Slim Diacu, Florin ∈ ,ps otnosytruhtevalue the through continuously pass ), R htdfiete aif h property the satisfy them define that , ic@vcc,irhm@vcc,[email protected] [email protected], [email protected], eakwehrHmloinvco ed endo pcsof spaces on defined fields vector Hamiltonian whether ask We aicIsiuefrteMteaia Sciences Mathematical the for Institute Pacific eateto ahmtc n Statistics and Mathematics of Department itra C aaa 8 2Y2 V8W Canada, BC, Victoria, ..Bx10 T CSC STN 1700 Box P.O. 1. nvriyo Victoria of University etme ,2018 6, September κ CURVATURE shrs for (spheres, Introduction and 1 > κ κ 0 = 0 n yeblcshrs for spheres, hyperbolic and , lim N fteptnilfunctions potential the if κ bd problem. -body → 0 U seuto 1,[] [13], [3], [1], equation ’s 1] 1] 1] u ex- But [12]. [11], [10], κ hsclinterest, physical f ftegravitational the of pcfi oec case each to specific s n osatvleof value constant any neto the of ontext hsqeto is question this o ,i utfidthrough justified is e, = egn rn started trend lenging . proi pc,[2], space, yperbolic iscoe,[] [7]. [6], closed, bits ,ee eodthe beyond even s, U 0 hsc rmEu- from physics where , sy h tried who vsky, eetninof extension he U 0 N -body N - 2 Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki
But a basic aspect of common interest to all attempts of extending the dynamics from Euclidean space to more general manifolds is that the classical equations are recovered when the manifolds are flattened. In the absence of this property, any extension is devoid of meaning. Of course, this attribute alone is not enough to warrant a specific generalization, since many extensions can manifest this quality, so a choice must be also based on other criteria. Nevertheless, checking the pres- ence of this feature is an indispensable first step towards justifying any extension of the classical equations to more general spaces. In this note we’ve set a modest goal in the basic direction mentioned above. We would like to see whether Hamiltonian vector fields defined on spheres and hyperbolic spheres tend to the classical Euclidean vector field as the curvature approaches zero, assuming that the classical potential is reached in the limit in a sense that will be made precise. As we will see, the answer to this problem is positive, which may be surprising, given the fact that, from the geometrical point of view, 2-spheres do not project isometrically on the Euclidean plane, and the same is true for 3-spheres relative to the Euclidean space. The subtle point of the results we obtain stays with how the limit is taken. To formalize this problem, let us consider a family of Hamiltonian vector fields, given by potential functions Uκ, κ ∈ R, defined on spheres of Gaussian curvature κ> 0, S2 2 2 2 −1 κ = {(x, y, z) | x + y + z = κ }, embedded in the Euclidean space R3, as well as hyperbolic spheres of curvature κ< 0, H2 2 2 2 −1 κ = {(x, y, z) | x + y − z = κ }, 2,1 embedded in the Minkowski space R , such that Uκ tends to a potential U0 of the Euclidean plane as κ → 0 while the Euclidean distances in R3 and the Minkowski distances in R2,1 are kept fixed when κ → 0. This way of approaching the limit is essential here, since allowing distances to vary with the curvature makes all the points of the sphere and the hyperbolic sphere run to infinity as κ → 0. We would like to know whether the equations of motion defined for κ =06 tend to the classical equations of motion defined in flat space, i.e. for κ = 0. We are also interested in answering the same question in the 3-dimensional case, in other S2 H2 words to decide what happens when the manifolds κ and κ are replaced by 3-spheres, S3 2 2 2 2 −1 κ = {(x,y,z,w) | x + y + z + w = κ } (embedded in R4), and hyperbolic 3-spheres, H3 2 2 2 2 −1 κ = {(x,y,z,w) | x + y + z − w = κ } (embedded in the Minkowski space R3,1), respectively. Of course, we could work in general for any finite space dimension, but the cases of main physical interest are The continuous transition of vector fields through manifolds of constant curvature 3 dimensions 2 and 3. Moreover, we would like to explicitly see what the equations of motion look like in these two particular cases, such that we can use them in the gravitational N-body problem. Another reason for including both the 2- and 3-dimensional case in this note, instead of treating only the latter, is that the former allows the reader to build intuition and make then an easy step to the higher dimension. To address the above issues for N point masses, let us initially consider the more general problem in which the bodies are interacting on a complete, connected, n- dimensional Riemannian manifold M (n being a fixed positive integer), under a law given by a potential function. So let N point particles (bodies) of masses m1,...,mN > 0 move on the manifold M. In some suitable coordinate system, the position and velocity of the body mr are described by the vectors r r r r xr =(x1,...,xn), x˙ r = (x ˙ 1,..., x˙ n), r = 1,N, respectively. We attach to every particle mr a metric tensor given by the matrix r −1 ij Gr = (gij), and its inverse, Gr = (gr ), at the point of the manifold M where the particle mr,r = 1,N, happens to occur at a given time instant. The law of motion is given by a sufficiently smooth potential function, U : MN \ ∆ → (0, ∞), U = U(x), where x =(x1,..., xN ) is the configuration of the particle system and ∆ represents the set of singular configurations, i.e. positions of the bodies for which the potential is not defined. Then U generates a Lagrangian function 1 N n (1) L(x, x˙ )= m gr x˙ rx˙ r − U(x), 2 r ij i j Xr=1 i,jX=1 which corresponds to a Hamiltonian vector field. In this setting, we can derive the equations of motion on the Riemannian manifold M in the result stated below, whose proof we also provide for the completeness of our presentation.
Lemma 1. Assume that the point masses m1,...,mN > 0 interact on the com- plete and connected n-dimensional Riemannian manifold M under the law im- posed by Lagrangian (1). Then the system of differential equations describing the motion of these particles has the form n ∂U n (2) m x¨r = − gsi − m Γs,rx˙ rx˙ r, s = 1,n, r = 1,N, r s r ∂xr r lj l j Xi=1 i l,jX=1 where 1 n ∂gr ∂gr ∂gr (3) Γs,r = gsi il + ij − lj , s = 1, n, lj 2 r ∂xr ∂xr ∂xr Xi=1 j l i 4 Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki are the Christoffel symbols corresponding to the particle mr, r = 1, N.
r r ij ji Proof. Recall first that gij = gji and gr = gr for i, j = 1, n. A simple computation leads to ∂L 1 n ∂gr ∂U ∂L n = m lj x˙ rx˙ r − , = m gr x˙ r, ∂xr 2 r ∂xr l j ∂xr ∂x˙ r r il l i l,jX=1 i i i Xl=1 d ∂L n n ∂gr = m gr x¨r + m ij x˙ rx˙ r. dt ∂x˙ r r il l r ∂xr l j i Xl=1 l,jX=1 l Then the Euler-Lagrange equations, d ∂L ∂L r = r , i = 1,n, r = 1,N, dt ∂x˙ i ∂xi which describe the motion of the N particles, take the form n n ∂gr 1 ∂gr ∂U m gr x¨r + m ij − lj x˙ rx˙ r = − , i = 1,n, r = 1,N. r il l r ∂xr 2 ∂xr l j ∂xr Xl=1 l,jX=1 l i i si Let us fix r in the above equations, multiply the ith equation by gr , and add all n equations thus obtained. The result is n n ∂gr 1 ∂gr n ∂U m gsigr x¨r + m gsi ij − lj x˙ rx˙ r = − gsi , r = 1,N. r r il l r r ∂xr 2 ∂xr l j r ∂xr i,lX=1 i,l,jX=1 l i Xi=1 i n si r Using the fact that i=1 gr gil = δsl, where δsl is the Kronecker delta, and the identity P n ∂gr 1 n ∂gr ∂gr ij x˙ rx˙ r = il + ij x˙ rx˙ r, ∂xr l j 2 ∂xr ∂xr l j j,lX=1 l j,lX=1 j l which is easy to prove by expanding the double sums, the above equations become those given in the statement of the lemma, a remark that completes the proof.
2. The 2-dimensional case In this section we will prove the continuity of Hamiltonian vector fields through S2 H2 R2 the value κ =0 of the curvature, i.e. when we move from κ to κ through . For this purpose, let us first define some trigonometric functions that unify circular and hyperbolic trigonometry, namely the κ-sine function, snκ, as κ−1/2 sin κ1/2s if κ> 0 snκs := s if κ =0 |κ|−1/2 sinh |κ|1/2s if κ< 0, The continuous transition of vector fields through manifolds of constant curvature 5 the κ-cosine function, csnκ, as cos κ1/2s if κ> 0 csnκs := 1 if κ =0 cosh |κ|1/2s if κ< 0, as well as the functions κ-tangent, tnκ, and κ-cotangent, ctnκ, as
snκs csnκs tnκs := and ctnκs := , csnκs snκs respectively. The following relationships, which will be useful in subsequent com- putations, follow from the above definitions: 2 2 κ snκs + csnκs =1, d d sn s = csn s and csn s = −κ sn s. ds κ κ ds κ κ Also notice that all the above unified trigonometric functions are continuous rel- ative to κ. We can now prove the following result. M2 Theorem 1. Consider the point masses (bodies) m1,...,mN on the manifold κ S2 H2 (representing κ or κ), κ =06 , whose positions are given by spherical coordinates (sr,ϕr),r = 1, N. Then the equations of motion for these bodies are
∂Uκ 2 s¨r = − +ϕ ˙ r snκsr csnκsr (4) ∂sr −2 ∂Uκ ϕ¨r = −snκ sr − 2s ˙rϕ˙ r ctnκsr, r = 1,N, ∂ϕr where M2 N Uκ : ( κ) \ ∆κ → (0, ∞), κ =06 , represent the potentials, with the sets ∆κ, κ ∈ R, corresponding to singular con- 3 figurations. Moreover, if limκ→0 Uκ = U0 while the Euclidean distances in R (and the Minkowski distances in R2,1) between the North Pole N = (0, 0, 0) and the bodies are kept fixed, where
2 N U0 : (R ) \ ∆0 → (0, ∞) is the potential in the Euclidean case and ∆0 is the corresponding set of singular configurations, then system (4) tends to the classical equations
∂U0 ∂U0 (5) x¨r = − , y¨r = − , r = 1,N. ∂xr ∂yr 6 Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki M2 Proof. In κ, the equations of motion (2) for the N-body system take the form
r 11 ∂Uκ 12 ∂Uκ 1,r r 2 1,r 1,r r r 1,r r 2 x¨1 = −gr r − gr r − Γ11 (x ˙ 1) − (Γ12 +Γ21 )x ˙ 1x˙ 2 − Γ22 (x ˙ 2) ∂x1 ∂x2 (6) r 21 ∂Uκ 22 ∂Uκ 2,r r 2 2,r 2,r r r 2,r r 2 x¨2 = −gr r − gr r − Γ11 (x ˙ 1) − (Γ12 +Γ21 )x ˙ 1x˙ 2 − Γ22 (x ˙ 2) , ∂x1 ∂x2 M2 r = 1,N. Consider now the (s,ϕ)-coordinates on κ, i.e. take in system (6) the variables r r sr := x1, ϕr := x2, r = 1,N, given in terms of extrinsic (x, y, z)-coordinates by −1/2 −1/2 xr = snκsr cos ϕr, yr = snκsr sin ϕr, zr = |κ| csnκsr −|κ| ,r = 1,N. Notice that for κ → 0, we have z → 0 and the remaining relations give the polar coordinates in R2. To make precise how we take the above limit, let us remark that while keeping the bodies fixed as κ → 0, the geodesic distances sr,r = 1,N, are not fixed. −1/2 Indeed, we can write sr = |κ| αr, where αr is the angle from the centre of the sphere that subtends the arc of geodesic length sr. The Euclidean/Minkowski distance corresponding to this arc is given by τr := 2snκ(αr/2), so we can conclude −1 that sr = 2 snκ (τr/2). The quantity τr is assumed fixed as κ varies, and sr → τr as κ → 0. Differentiating the expressions given the change of coordinates we obtain
dxr = csnκsr cos ϕr ds − snκsr sin ϕr dϕr,
dyr = csnκsr sin ϕr dsr + snκsr cos ϕr dϕr, 1/2 dzr = −σ|κ| snκsr dsr, where σ = 1 for κ > 0, but σ = −1 for κ < 0. Using these expression of the 2 2 2 differentials, we can compute the line element dxr + dyr + σdzr and obtain that the metric tensor and its inverse are given by matrices of the form
r 1 0 −1 ij 1 0 Gr =(gij)= 2 , Gr =(gr )= 1 , 0 2 0 snκsr snκsr respectively, and that they cover the entire spectrum of metrics for κ ∈ R. So we obtain that
r r r r 2 11 12 21 22 1 g11 =1, g12 = g21 =0, g22 = snκsr, gr =1, gr = gr =0, gr = 2 . snκsr Using the matrices G and G−1 as well as equations (3), we compute the Christoffel symbols and obtain 1,r 1,r 1,r 2,r 2,r Γ11 =Γ12 =Γ21 =Γ11 =Γ22 =0, 1,r 2,r 2,r Γ22 = −snκsr csnκsr, Γ12 =Γ21 = ctnκsr. The continuous transition of vector fields through manifolds of constant curvature 7
Using the above results and reintroducing the index r to point out the dependence of the equations of motion on the position of each body, system (6) becomes
∂Uκ 2 s¨r = − +ϕ ˙ r snκsr csnκsr (7) ∂sr −2 ∂Uκ ϕ¨r = −snκ sr − 2s ˙rϕ˙ r ctnκsr, r = 1,N. ∂ϕr Given the definitions of the unified trigonometric functions at κ = 0, the above system takes in Euclidean space the form
∂U0 2 s¨r = − + srϕ˙ r (8) ∂sr −2 ∂U0 −1 ϕ¨r = −sr − 2sr s˙rϕ˙ r, r = 1,N. ∂ϕr S2 H2 But, as noted above, the (s,ϕ)-coordinates of κ and κ (for κ =6 0), become polar coordinates in R2 (for κ =0), so if we write
xr = sr cos ϕr, yr = sr sin ϕr, r = 1,N, and perform the computations, system (8) takes the desired form (5). This remark completes the proof.
3. The 3-dimensional case M3 S3 H3 In this section we consider the motion in κ, which stands for κ or κ. Our main goal is to prove the following result.
Theorem 2. Consider the point masses (bodies) m1,...,mN on the manifold M3 S3 H3 κ (representing κ or κ), κ =6 0, whose positions are given in hyperspherical coordinates (sr,ϕr, θr),r = 1, N. Then the equations of motion for these bodies are given by the system
∂Uκ 2 ˙2 2 s¨r = − + (ϕ ˙ r + θr sin ϕr) snκsr csnκsr ∂sr −2 ∂Uκ 2 (9) ϕ¨r = −snκ sr + θr sin ϕr cos ϕr − 2s ˙rϕ˙ r ctnκsr ∂ϕr ¨ −2 −2 ∂Uκ ˙ ˙ θr = −snκ sr sin ϕr − 2s ˙rθr ctnκsr − 2ϕ ˙ rθr cot ϕr, r = 1,N, ∂θr where M3 N Uκ : ( κ) \ ∆κ → (0, ∞), κ =06 , represent the potentials, with the sets ∆κ, κ ∈ R, corresponding to singular con- 4 figurations. Moreover, if limκ→0 Uκ = U0 while the Euclidean distances in R (and 8 Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki the Minkowski distances in R3,1) between the North Pole N = (0, 0, 0, 0) and the bodies are kept fixed, where 3 N U0 : (R ) \ ∆0 → (0, ∞) is the potential in the Euclidean case and ∆0 is the corresponding set of singular configurations, then system (9) tends to the classical equations, ∂U0 ∂U0 ∂U0 (10) x¨r = − , y¨r = − , z¨r = − , r = 1,N. ∂xr ∂yr ∂zr M3 Proof. In κ, system (2) takes the form (11) r 11 ∂Uκ 12 ∂Uκ 13 ∂Uκ 1,r r 2 1,r r 2 1,r r 2 x¨1 = −gr r − gr r − gr r − Γ11 (x ˙ 1) − Γ22 (x ˙ 2) − Γ33 (x ˙ 3) ∂x1 ∂x2 ∂x3 1,r 1,r r r 1,r 1,r r r 1,r 1,r r r − (Γ12 +Γ21 )x ˙ 1x˙ 2 − (Γ13 +Γ31 )x ˙ 1x˙ 3 − (Γ23 +Γ32 )x ˙ 2x˙ 3 ∂U ∂U ∂U 2 2 2 x¨r = −g21 κ − g22 κ − g23 κ − Γ ,r(x ˙ r)2 − Γ ,r(x ˙ r)2 − Γ ,r(x ˙ r)2 2 r r r r r r 11 1 22 2 33 3 ∂x1 ∂x2 ∂x3 2,r 2,r r r 2,r 2 r r 2,r 2,r r r − (Γ12 +Γ21 )x ˙ 1x˙ 2 − (Γ13 +Γ31)x ˙ 1x˙ 3 − (Γ23 +Γ32 )x ˙ 2x˙ 3
∂U ∂U ∂U 3 3 3 x¨r = −g31 κ − g32 κ − g33 κ − Γ ,r(x ˙ r)2 − Γ ,r(x ˙ r)2 − Γ ,r(x ˙ r)2 3 r r r r r r 11 1 22 2 33 3 ∂x1 ∂x2 ∂x3 3,r 3,r r r 3,r 3,r r r 3,r 3,r r r − (Γ12 +Γ21 )x ˙ 1x˙ 2 − (Γ13 +Γ31 )x ˙ 1x˙ 3 − (Γ23 +Γ32 )x ˙ 2x˙ 3,r = 1,N. M3 Consider now the intrinsic (s,ϕ,θ)-coordinates on the manifolds κ, i.e. take r r r sr := x1, ϕr := x2, θr := x3, r = 1,N, given in terms of the extrinsic (x,y,z,w)-coordinates by
xr = snκsr sin ϕr sin θr
yr = snκsr sin ϕr cos θr (12) zr = snκsr cos ϕr −1/2 −1/2 wr = |κ| csnκsr −|κ| , r = 1,N. Notice that for κ → 0, we have w → 0 and the remaining relations give the spherical coordinates in R3. The remark we made in the proof of Theorem 1 (that the quantities sr,r = 1, N, vary as we keep the Euclidean/Minkowski distance fixed, but tend to that distance as κ → 0) applies here too without any change. Differentiating in (12), we obtain
dxr = csnκsr sin ϕr sin θr dsr + snκsr cos ϕr sin θr dϕr + snκsr sin ϕr cos θr dθr,
dyr = csnκsr sin ϕ_rcosθr dsr + snκsr cos ϕr cos θr dϕr − snκsr sin ϕr sin θr dθr,
dzr = csnκsr cos ϕr dsr − snκsr sin ϕr dϕr 1/2 dwr = −σ|κ| snκsr dsr. The continuous transition of vector fields through manifolds of constant curvature 9
2 2 2 2 Using these expressions we can compute the line element dxr + dyr + dzr + σdwr to obtain the metric tensor and its inverse in matrix form, 10 0 10 0 r 2 −1 ij 1 0 2 0 Gr =(gij)= 0 snκsr 0 , Gr =(gr )= snκsr . 2 2 1 0 0 sn sr sin ϕr 0 0 2 2 κ snκsr sin ϕr These matrices give all the metrics for κ ∈ R. We can thus write that r r r r r r r r 2 r 2 2 g11 =1, g12 = g13 = g21 = g23 = g31 = g32 =0, g22 = snκsr, g33 = snκsr sin ϕr,
11 12 13 21 23 31 32 22 1 33 1 gr =1, gr = gr = gr = gr = gr = gr =0, gr = 2 , gr = 2 2 . snκsr snκsr sin ϕr Then we can compute the Christoffel symbols and obtain 1,r 2,r 2,r 1,r 1,r 1,r 1,r 1,r 1,r Γ11 =Γ11 =Γ22 =Γ12 =Γ21 =Γ13 =Γ31 =Γ23 =Γ32 =0, 1,r 1,r 2 Γ22 = −snκsr csnκsr, Γ33 = −snκsr csnκsr sin ϕr, 2,r 2,r 2,r 2,r 3,r 3,r 3,r 3,r 3,r Γ13 =Γ31 =Γ23 =Γ32 =Γ11 =Γ22 =Γ33 =Γ12 =Γ21 =0, 2,r 2,r 2,r 3,r 3,r 3,r 3,r Γ33 = − sin ϕr cos ϕr, Γ12 =Γ21 =Γ13 =Γ31 = ctnκsr, Γ23 =Γ32 = cot ϕr. Then system (11) becomes ∂Uκ 2 ˙2 2 s¨r = − + (ϕ ˙ r + θr sin ϕr) snκsr csnκsr ∂sr −2 ∂Uκ 2 (13) ϕ¨r = −snκ sr + θr sin ϕr cos ϕr − 2s ˙rϕ˙ r ctnκsr ∂ϕr ¨ −2 −2 ∂Uκ ˙ ˙ θr = −snκ sr sin ϕr − 2s ˙rθr ctnκsr − 2ϕ ˙ rθr cot ϕr, r = 1,N. ∂θr Given the definitions of the unified trigonometric functions at κ = 0, the above system takes in Euclidean space the form ∂Uκ 2 ˙2 2 s¨r = − + sr(ϕ ˙ r + θr sin ϕr) ∂sr −2 ∂Uκ 2 −1 (14) ϕ¨r = −sr + θr sin ϕr cos ϕr − 2sr s˙rϕ˙ r ∂ϕr ¨ −2 −2 ∂Uκ −1 ˙ ˙ θr = −sr sin ϕr − 2sr s˙rθr − 2ϕ ˙ rθr cot ϕr, r = 1,N. ∂θr Since the relations in (12) provide us with the spherical coordinates in R3, for κ =0 we can write that
xr = sr sin ϕr sin θr, yr = sr sin ϕr cos θr, zr = sr cos ϕr, r = 1,N. Then straightforward computations make system (14) take the classical form (10). This remark completes the proof. 10 Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki
4. Application to the gravitational N-body problem In this section we will apply Theorem 2 to the gravitational N-body problem. As we mentioned in the introduction, the potential we choose to define on spheres and hyperbolic spheres, which has a long history, provides the natural extension of gravitation to spaces of constant curvature. The application of Theorem 1 would follow in the same way. So we consider the cotangent potential function given by
(15) Uκ(q)= − mimjctnκ(dκ(mi, mj)), κ =06 , 1≤Xi d0(mi, mj)= |qi − qj|. Notice that, for now, the position where the origin of the coordinate system is placed is irrelevant. Indeed, dκ(mi, mj) does not depend on this choice. However, the choice of the origin will become relevant later, when we take the limit κ → 0. Straightforward computations that use the definitions of snκ, csnκ, and ctnκ transform the potential (15) into m m |κ|1/2κqij (16) U (q)= − i j , κ |(κq2)(κq2) − (κqij)2|1/2 1≤Xi On the manifolds of constant curvature κ, Uκ becomes 2 mimj(2 − κq ) (19) U (q)= − ij , κ q (4 − κq2 )1/2 1≤Xi Acknowledgment. The authors are indebted to NSERC of Canada for partial financial support through its Discovery Grants programme. References [1] V. Banica, The nonlinear Schrödinger equation on the hyperbolic space, Comm. Partial Differential Equations 32, 10 (2007), 1643–1677. [2] W. Bolyai and J. Bolyai, Geometrische Untersuchungen, Hrsg. P. Stäckel, Teubner, Leipzig- Berlin, 1913. [3] N. Burq, P. Gérard, and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), 187–223. [4] F. Diacu, On the singularities of the curved N-body problem, Trans. Amer. Math. Soc. 363, 4 (2011), 2249–2264. [5] F. Diacu, Polygonal homographic orbits of the curved 3-body problem, Trans. Amer. Math. Soc. 364 (2012), 2783–2802. [6] F. Diacu, Relative equilibria of the curved N-body problem, Atlantis Studies in Dynamical Systems, vol. 1, Atlantis Press, Amsterdam, 2012. [7] F. Diacu, Relative equilibria of the 3-dimensional curved n-body problem, Memoirs Amer. Math. Soc. 228, 1071 (2013). [8] F. Diacu, The classical N-body problem in the context of curved space, arXiv:1405.0453. 12 Florin Diacu, Slim Ibrahim, and Je˛drzej Śniatycki [9] F. Diacu, S. Ibrahim, C. Lind, and S. Shen, The Vlasov-Poisson System for Stellar Dynamics in Spaces of Constant Curvature, arXiv:1506.07090, 34 p. [10] F. Diacu and S. Kordlou, Rotopulsators of the curved N-body problem, J. Differential Equations 255 (2013) 2709–2750. [11] F. Diacu, E. Pérez-Chavela, and J. Guadalupe Reyes Victoria, An intrinsic approach in the curved n-body problem, J. Differential Equations, 252 (2012), 4529–4562. [12] F. Diacu and B. Thorn, Rectangular orbits of the curved 4-body problem, Proc. Amer. Math. Soc. 143 (2015), 1583–1593. [13] A.D. Ionescu and G. Staffilani, Semilinear Schrödonger flows on hyperbolic spaces: scat- tering in H1, Math. Ann. 345 (2009), 133-158. [14] N. I. Lobachevsky, The new foundations of geometry with full theory of parallels [in Rus- sian], 1835-1838, In Collected Works, V. 2, GITTL, Moscow, 1949, p. 159.