Mathematical Problems of Computer Science 51, 98–106, 2019. UDC 519.6 On Homeomorphism Between Euclidean Subspace and Conformally Euclidean Manifold Ashot S. Gevorkyan1,2, Alek A. Aleksanyan1 and Suren B. Alaverdyan1 1Institute for Informatics and Automation Problems of NAS RA 2Institute of Chemical Physics after A. B. Nalbandyan of NAS RA e-mail: gashot@ipia.sci.am Abstract The article presents the proof of the homeomorphism between Euclidean subspace E6 of the classical three-body system and 6D Riemannian manifold M, which allows reducing the dynamical problem to the system of the 6th-order. Keywords: System of underdetermined algebraic equations, Orientated 3D Rie- mannian manifold, Topology of 3D manifolds. 1. Introduction As is known, the time evolution of the classical system is uniquely determined by the Hamil- ton equations and is usually reduced to a system of ordinary differential equations of the second order. Integrating this system of a differential equation means finding all possible functions of one variable “t” (time), which, when substituted into equations, turns them into an identity. In the case of dynamical systems, as a rule, the system of equations cannot be fully integrated, since the number of integrals of motion often is less than the number of degrees of freedom. In the series of works [1]–[5], using the example of the classical three-body problem, it was shown that the use of Riemannian geometry makes it possible to reveal new hid- den symmetries of a dynamical system, which makes the integration of the problem more completel. In this paper we examine the question of homeomorphism between 6D Euclidean subspace E6 and 6D manifold M. In particular, the question of the decomposition of a manifold in the form M :⇔ M(3) × S3Mi is proved, where M(3) denotes the sum of 84 oriented in 9D Euclidean space 3D manifolds and S3Mi is the group symmetry SO(3) at the given point Mi ∈ M(3). 98 A. Gevorkyan, A. Aleksanyan and S. Alaverdyan 99 2. On Homeomorphism Between the Euclidean Subspace and the Confor- mally Euclidean Manifold Proposition 1: Let E6 be a Euclidean subspace with metric γµν ({ρ}), on which an orthogonal coordinate system is given: ρ1, ..., ρ6 = {ρ} = ρ1, ρ6 ∈ E6, (1) and, respectively, M is a conformally Euclidean manifold, which is determined by the metric tensor gαβ({x}) and the local coordinate system {x}: gαβ({x}) = g({x̄})δαβ, {x} = x1, ..., x6, {x̄} = x1, x3, α, β = 1, 6, (2) where g({x̄}) > 0 is a smooth function belonging to the class C1(R6), then the Euclidean subspace E6 is homeomorphic to the manifold M. Proof. Let us consider a linear infinitesimal element ”ds” in both coordinate systems {ρ} ∈ E6 and {x} ∈ M. Equating them, we can write: (ds)2 = γαβ({ρ})dραdρβ = gµν ({x̄})dxµdxν , α, β, µ, ν = 1, 6, (3) from which one can obtain the following system of algebraic equations: γαβ({ρ})ρα,µρβ,ν = gµν ({x̄}) = g({x̄})δµν , (4) where it is necessary to prove that the coefficients ρα,µ({x}) = ∂ρα/∂xµ make sense of derivatives. In this regard, we must prove that the function ρα({x}) is twice differentiable and continuous in its domain of definition and, in addition, satisfy the symmetry condition: ρα,µν ({x}) = ρα,νµ({x}), ∀ µ, ν = 1, 6, (5) (Schwartz’s theorem on the symmetry of second derivatives). Recall that the set of coefficients ρα,µ({x}) allows us to perform coordinate transformations {ρ} 7→ {x}, which we shall call direct transformations. γαβ({ρ})g−1({x̄}) = xµ, αxν, β δµν , (6) where xµ, α({ρ}) = ∂xµ/∂ρα and γαβ({ρ}) = γαᾱ({ρ})γββ̄({ρ})γᾱβ̄({ρ}). At first we consider the system of equations (4), which is related to direct coordinate transformations. It is easy to see that the system of algebraic equations (4) is underdeter- mined with respect to the variables ρα,µ({x}), since it consists of 21 equations, while the number of unknown variables is 36. Obviously, when these equations are compatible, then the system of equations (4) has an infinite number of real and complex solutions. Note that for the classical three-body problem, the real solutions of the system (4) are important, which form a 15-dimensional manifold. Since the system of equations (6) is still defined in a rather arbitrary way we can impose additional conditions on it in order to find the minimal dimension of the manifold allowing a separation of the base M(3) from the layer ∪iS3Mi . Similarly, from (3), one can obtain a system of algebraic equations defining inverse trans- formations: 100 On Homeomorphism Between Euclidean Subspace and Conformally Euclidean Manifold Let us make new notations: αµ = ρ1,µ, βµ = ρ2,µ, ζµ = ρ3,µ, uµ = ρ4,µ, vµ = ρ5,µ, wµ = ρ6,µ. (7) We also require that the following additional conditions be met: α4 = α5 = α6 = 0, β4 = β5 = β6 = 0, ζ4 = ζ5 = ζ6 = 0, u1 = u2 = u3 = 0, v1 = v2 = v3 = 0, w1 = w2 = w3 = 0. (8) Using (7) and conditions (8) from the equation (4) we can obtain two independent systems of algebraic equations: α21 + β 2 1 + γ 33ζ21 = ğ({ρ̄}), α1α2 + β1β2 + γ33ζ1ζ2 = 0, α22 + β 2 2 + γ 33ζ22 = ğ({ρ̄}), α1α3 + β1β3 + γ33ζ1ζ3 = 0, α23 + β 2 3 + γ 33ζ23 = ğ({ρ̄}), α2α3 + β2β3 + γ33ζ2ζ3 = 0, (9) (recall that at obtaining (9) it is assumed that γ11 = γ22 = 1) and, correspondingly: γ44u24 + γ 55v24 + γ 66w24 + 2(γ 45u4v4 + γ 46u4w4 + γ 56v4w4) = ğ({ρ̄}), γ44u25 + γ 55v25 + γ 66w25 + 2(γ 45u5v5 + γ 46u5w5 + γ 56v5w5) = ğ({ρ̄}), γ44u26 + γ 55v26 + γ 66w26 + 2(γ 45u6v6 + γ 46u6w6 + γ 56v6w6) = ğ({ρ̄}), a4u4 + a5v4 + a6w4 = 0, b4u5 + b5v5 + b6w5 = 0, c4u6 + c5v6 + c6w6 = 0. (10) In equations (10), the following notations are made: ai = γ i4u5 + γ i5v5 + γ i6w5, bj = γ j4u6 + γ j5v6 + γ j6w6, ck = γ k4u4 + γ k5v4 + γ k6w4, where i, j, k = 4, 6. It should be noted that the solutions of algebraic systems (9) and (10) form two different 3D manifolds S(3) and R(3), respectively. The manifold S(3) is in a one-to-one mapping on the one hand with the subspace E3 3 {ρ̄} (where E3 ⊂ E6 the internal space in the hyperspherical coordinate system), and on the other hand with the manifold M(3) (see Fig. 1). Note that this statement follows from the fact that all points of the manifold M(3) and the subspace E3, are pairwise connected through the corresponding derivatives (see (4)), which, as unknown variables, enter the algebraic equations (9), and, in addition, as shown there exist also inverse coordinate transformations (see Appendix). Now we prove the continuity of these mappings. Recall that the unknowns in the equations (9), are in fact functions of coordinates {ρ̄}. Performing a shift of coordinates {ρ̄} → {ρ̄} + {δρ̄} in (9), we get the following system of equations: ᾱ21 + β̄ 2 1 + γ̄ 33ζ̄21 = ḡ({ρ̄}), ᾱ1ᾱ2 + β̄1β̄2 + γ̄33ζ̄1ζ̄2 = 0, ᾱ22 + β̄ 2 2 + γ̄ 33ζ̄22 = ḡ({ρ̄}), ᾱ1ᾱ3 + β̄1β̄3 + γ̄33ζ̄1ζ̄3 = 0, ᾱ23 + β̄ 2 3 + γ̄ 33ζ̄23 = ḡ({ρ̄}), ᾱ2ᾱ3 + β̄2β̄3 + γ̄33ζ̄2ζ̄3 = 0, (11) A. Gevorkyan, A. Aleksanyan and S. Alaverdyan 101 Fig. 1: In this diagram all spaces are homeomorphic to each other, i.e., E3 ' S(3) ' M(3). where ḡ({ρ̄}) = ğ ( {ρ̄} + {δρ̄} ) , {δρ̄} = (δρ1, δρ2, δρ3). Assuming that |δ{ρ̄}| ¿ 1, in the equations (11), we can expand the functions in a Taylor series on these small parameters and taking into account the system of equations (9), we get: δρi { 2(α1α1 i + β1β1 i + γ 33ζ1ζ1 i) + γ 33 , i ζ 2 1 − ğ, i({ρ̄}) } = O(|δ{ρ̄}|2), δρi { 2(α2α2 i + β2β2 i + γ 33ζ2ζ2 i) + γ 33 , i ζ 2 2 − ğ, i({ρ̄}) } = O(|δ{ρ̄}|2), δρi { 2(α3α3 i + β3β3 i + γ 33ζ3ζ3,i) + γ 33 , i ζ 2 3 − ğ, i({ρ̄}) } = O(|δ{x̄}|2), δρi { α1α2 i + α2α1 i + β1β2 i + β2β1 i + γ 33(ζ1ζ2 i + ζ2ζ1 i) + γ 33 , i ζ1ζ2 } = O(|δ{ρ̄}|2), δρi { α1α3 i + α3α1 i + β1β3 i + β3β1 i + γ 33(ζ1ζ3 i + ζ3ζ1 i) + γ 33 , i ζ1ζ3 } = O(|δ{ρ̄}|2), δρi { α2α3, i + α3α2 i + β2β3 i + β3β2 i + γ 33(ζ2ζ3 i + ζ3ζ2 i) + γ 33 , i ζ2ζ3 } = O(|δ{ρ̄}|2), (12) where i = 1, 3 and summation is performed by dummy indices. If we require that the expressions with the same increments be equal to zero, then from (12) one can obtain an underdetermined system of algebraic equations, i.e., 18 equations for finding 27 unknowns variables: 2(α1α1 i + β1β1 i + γ 33ζ1ζ1 i) + γ 33 , i ζ 2 1 − ğ, i({ρ̄}) = 0, 2(α2α2 i + β2β2 i + γ 33ζ2ζ2 i) + γ 33 , i ζ 2 2 − ğ, i({ρ̄}) = 0, 2(α3α3 i + β3β3 i + γ 33ζ3ζ3 i) + γ 33 , i ζ 2 3 − ğ, i({ρ̄}) = 0, α2α1 i + α1α2 i + β2β1 i + β1β2 i + γ 33(ζ2ζ1 i + ζ1ζ2 i) + γ 33 , i ζ1ζ2 = 0, α3α1 i + α1α3 i + β3β1 i + β1β3 i + γ 33(ζ3ζ1 i + ζ1ζ3 i) + γ 33 , i ζ1ζ3 = 0, α3α2 i + α2α3 i + β3β2 i + β2β3 i + γ 33(ζ3ζ2 i + ζ2ζ3 i) + γ 33 , i ζ2ζ3 = 0. (13) Recall that the set of coefficients {σ} = (σ1, ..., σ9) = [α = (α1, α2, α3), β = (β1, β2, β3), ζ = (ζ1, ζ2, ζ3)] belongs to the manifold S (3). Now, we can require that the second derivatives be symmetric σij = σji, where {σ} = [α, β, ζ] and i, j = 1, 3. This, as can be easily seen, allows us to reduce the number of unknown variables and make the system of equations definite, i.e., 18 equations for 18 unknown variables. The system of equations (13) can be written in canonical form: AX = B, A = (dµν ), µ, ν = 1, 18, (14) where A ∈ R18×18 is the basic matrix of the system, B ∈ R18 and X ∈ R18 are columns of free terms and system solutions, respectively. Note that, for an arbitrary point {ρ̄i} ∈ E3, 102 On Homeomorphism Between Euclidean Subspace and Conformally Euclidean Manifold Fig. 2: The form of an oriented manifold generated by a system of equations (9). Note that the calculations of the equations system (9) were performed taking into account the following transformations γ33ζ1 → ζ1, γ33ζ2 → ζ2 and γ33ζ3 → ζ3. The first figure shows a general view of a manifold in three-dimensional space, which obviously is a sphere with topological features. The second figure shows the projection of a sphere onto a plane (α2, α3) in the form of a circle, from which one can see a cutting circle in the center. There are obviously six such circular cuts on a sphere. Fig. 3: As can be seen, this manifold also has a topology. Fig. 4: As can be seen, this manifold also has a topology. A. Gevorkyan, A. Aleksanyan and S. Alaverdyan 103 Fig. 5: As can be seen, this manifold also has a topology. the system of equations (9) generates sets of solutions {σ} = [α, β, ζ] that continuously fill a region of E3 space, forming 3D manifold S(3). As for the system of equations (14), it has a solution if the determinant of the basic matrix A is nonzero (see Appendix): det(dµν ) 6= 0, µ, ν = 1, 18. On the other hand, the algebraic system (14) does not have a solution when det(dµν ) = 0. In this case at each point {ρ̄i} there exists a countable set W of coefficients {σ} = [α, β, ζ] such that det(dµν ) = 0. It is easy to verify that the measure of this set in comparison with the measure of the S(3) for which det(dµν ) 6= 0, is equal to zero, i.e., W = {0}. In other words, for the case under consideration Schwartz’s theorem holds, and σς (where ς = 1, 9) and dµν (see (13)) have the sense of the first and second derivatives, respectively. The same is easily proved for inverse mappings. Let us consider the open set ∀ G = ∪αGα, consisting of the union of cards Gα arising at continuous mappings f : {ρ̄} 7→ {x̄} using algebraic equations (9). Proceeding from the foregoing, it is obvious that the maps can be chosen so that the immediate neighbors have intersections comprising at least one common point, that is a necessary condition for the continuity of the mappings. Using the above arguments, we assert that the atlas G can be widened up to G ∼= M(3). Now let us discuss the structure of the manifold M(3). It is easy to see that the in- dependent {σ} parameters form 9D space R9, in which the system of algebraic equations (9) generates 3D oriented manifolds. These manifolds can be summed up as sets using a certain order by gluing manifolds having common planes. As a result of this gluing, which similar to the the operation of connected sum of topological manifolds, the 3D manifold M(3) = ∪iM(3)i , is formed. The number of submanifolds M (3) i can easily be calculated by the formula Cmn = n! m!(n−m)! , where n and m denote the dimension of space R 9 and the di- mension of the manifold M(3)i immersed into R9, respectively. As the calculations show (see Fig. 2-5), the generated C39 = 84 topological manifolds can be grouped into four incongruent groups of manifolds. It is also necessary to note that all these varieties are oriented in a 9-dimensional space in the sense that they are well-defined 3D submanifolds. Thus, all the conditions of the theorem of a homeomorphism between metric spaces E3 and M(3) are satisfied, and therefore we can say that these spaces are homeomorphic or 104 On Homeomorphism Between Euclidean Subspace and Conformally Euclidean Manifold topologically equivalent, i.e., f : E3 7→ M(3). As for the system of algebraic equations (10), then at each point of the internal space Mi(x 1, x2, x3)i ∈ M(3), it generates 3D manifold R(3) that is a local analogue of the Euler angles and, consequently, ∪iS3Mi ' R(3). The layer R(3), continuously passing through all points of the basis M(3), fills the subspace E6. Finally, taking into account the aforesaid, we can conclude that the spaces E6 and M, are homeomorphic too. In addition, the manifold M can be represented in the form of decomposition M ∼= M(3) × S3Mi . Proposition 1 is proved. 3. Conclusion As A. Poincaré rightly pointed out, there is no finest geometry, there is a geometry convenient for solving a specific task. Usually, when studying complex dynamical systems, coordinate transformations are used to separate variables and reduce the original system. In particular, by coordinate transformations, the three-body problem, which is a system of 18th order, is reduced to the system of 8th order. However, as we have shown, it is possible to make the reduction of a dynamical system more complete if we use the curve (Riemannian) geometry. Note that in this case it becomes possible to reveal the hidden symmetries of internal motion and, accordingly, to obtain additional integrals of motion. For a three-body system, replacing the geometry allows us to reduce the problem to the 6th order system. The main difficulty arising at the solution of this problem is the generalization of the well-known Poincaré theorem on a homomorphism between the 3D sphere with unit radius and 3D compact. In this work, the possibility of such a generalization is strictly proved. 4. Appendix As mentioned (see (14)), the vector X consists of 18 independent components. Its transposed form looks like this: XT = ( α11, α12, α13, α22, α23, α33, β11, β12, β13, β22, β23, β33, ζ11, ζ12, ζ13, ζ22, ζ23, ζ33 ) . Taking into account the form of the vector X, we can write the explicit form of the basic matrix: A =   d11 · · · d181 · · · · · · · · · d118 · · · d1818   , (15) where the superscript indicates the column number, while the subscript indicates the line number. As for the explicit form of elements d νµ = dµν , where µ, ν = 1, 18, then we can find them by multiplying the basic matrix A with the vector X (see equation (14)) and comparing with the system of equations (13). In particular, it is easy to verify that these A. Gevorkyan, A. Aleksanyan and S. Alaverdyan 105 terms are equal: d 11 = d 2 2 = d 3 3 = 2d 2 10 = 2d 4 11 = 2d 5 12 = 2d 3 13 = 2d 5 14 = 2d 6 15 = 2α1, d 24 = d 4 5 = d 5 6 = 2d 1 10 = 2d 2 11 = 2d 3 12 = 2d 3 16 = 2d 5 17 = 2d 6 18 = 2α2, d 37 = d 5 8 = d 6 9 = 2d 1 13 = 2d 2 14 = 2d 3 15 = 2d 2 16 = 2d 4 17 = 2d 5 18 = 2α3, d 17 = d 2 8 = d 3 9 = 2d 8 10 = 2d 10 11 = 2d 11 12 = 2d 9 13 = 2d 11 14 = 2d 12 15 = 2β1, d 84 = d 10 5 = d 6 11 = 2d 7 10 = 2d 8 11 = 2d 9 12 = 2d 9 16 = 2d 11 17 = 2d 12 18 = 2β2, d 97 = d 11 6 = d 6 12 = 2d 7 13 = 2d 8 14 = 2d 9 15 = 2d 8 16 = 2d 10 17 = 2d 11 18 = 2β3, d 113 = d 2 14 = d 3 15 = 2d 17 10 = 2d 16 11 = 2d 17 12 = 2d 15 13 = 2d 17 14 = 2d 18 15 = 2γ 33ζ1, d144 = d 16 5 = d 6 17 = 2d 13 10 = 2d 14 11 = 2d 15 12 = 2d 13 13 = 2d 14 14 = 2d 15 15 = 2γ 33ζ2, d157 = d 17 8 = d 9 18 = 2d 14 16 = 2d 16 17 = 2d 17 18 = 2d 15 16 = 2d 17 17 = 2d 18 18 = 2γ 33ζ3. (16) All elements of the matrix (15) missing in (16) are identically zero. As is known, the algebraic system (13) or (14) does not have a solution in the case when the determinant of the matrix is zero det(A) = det(dµν ) = 0. A class consisting of sets of coefficients {σ} for which the determinant is zero, can be countable and the measure, respectively, will be equal to zero W = {0}. References [1] E. A. Ayryan, A. S. Gevorkyan and L. A. Sevastyanova, “On the motion of a three body system on hypersurface of proper energy”, Physics of Particles and Nuclei Letters, vol.10, no. 7, pp. 1-8, 2013. [2] A. S. Gevorkyan, “On reduction of the general three-body Newtonian problem and the curved geometry”, Journal of Physics: Conference Series, 496, 012030, 2014. [3] A. S. Gevorkyan, “On the motion of classical three-body system with consideration of quantum fluctuations”, Physics of Atomic Nuclei, vol. 80, no. 2, pp. 358-365, 2017. [4] A. S. Gevorkyan, “Fundamental irreversibility and times arrow of the classical three- body problem. New approaches and ideas in the study of dynamical systems”. arXiv:1706.09827v2[math-ph] 13 Dec 2017. [5] A. S. Gevorkyan, “Is the Hamiltonian mechanics and in general classical mechanics reversible?”, Book of abstracts, International Conference Dedicated to the 120th An- niversary of Emil Artin, Yerevan, Armenia, May 27-June 2, pp. 58-59, 2018. Submitted 04.12.2018, accepted 23.04.2019. 1 0 6 On Homeomorphism Between Euclidean Subspace and Conformally Euclidean Manifold ¾íÏÉÇ¹Û³Ý »Ýóï³ñ³ÍáõÃÛ³Ý ¨ ÏáÝýáñÙ-¿íÏÉÇ¹Û³Ý µ³½- Ù³Ó¨áõÃÛ³Ý ÙÇç¨ ÑáÙÇáÙáñýǽÙÇ í»ñ³µ»ñÛ³É 1 ÐÐ ¶²² ÆÝýáñÙ³ïÇϳÛÇ ¨ ³íïáÙ³ï³óÙ³Ý åñáµÉ»ÙÝ»ñÇ ÇÝëïÇïáõï 2 ÐÐ ¶²² ². ´. ܳɵ³Ý¹Û³ÝÇ ³Ýí³Ý ùÇÙÇ³Ï³Ý ýǽÇϳÛÇ ÇÝëïÇïáõï e-mail: gashot@ipia.sci.am ²Ù÷á÷áõÙ Ðá¹í³ÍáõÙ µ»ñí³Í ¿ ¹³ë³Ï³Ý »ñ»ù Ù³ñÙÝÇ ¿íÏÉÇ¹Û³Ý »Ýóï³ñ³ÍáõÃÛ³Ý E6 ¨ 6 D èÇÙ³ÝÇ µ³½Ù³Ó¨áõÃÛ³Ý M ÙÇç¨ ÑáÙÇáÙáñýǽÙÇ ³å³óáõÛóÁ, áñÁ ÃáõÛÉ ¿ ï³ÉÇë èÇÙ³ÝÇ µ³½Ù³Ó¨áõÃÛáõÝ, áõÕÕáñ¹í³Í ïáåáÉá·Ç³Ï³Ý µ³½Ù³Ó¨áõÃÛáõÝ, ÑáÙÇáÙáñýǽ٠µ³½Ù³Ó¨áõÃÛáõÝÝ»ñÇ ÙÇç¨: Î ãîìåîìîðôèçìå ìåæäó åâêëèäîâûì ïîäïðîñòðàíñòâîì è êîíôîðìíî-åâêëèäîâûì ìíîãîîáðàçèåì Àøîò Ñ. Ãåâîðêÿí 1;2, Àëåê À. Àëåêñàíÿí 1 è Ñóðåí Á. Àëàâåðäÿí1 1Èíñòèòóò ïðîáëåì èíôîðìàòèêè è àâòîìàòèçàöèè ÍÀÍ ÐÀ 2Èíñòèòóò õèìè÷åñêîé ôèçèêè èìåíè À. Á. Íàëáàíäÿíà ÍÀÍ ÐÀ e-mail: gashot@ipia.sci.am Àííîòàöèÿ  ñòàòüå ïðåäñòàâëåíî äîêàçàòåëüñòâî ãîìåîìîðôèçìà ìåæäó åâêëèäîâûì ïîäïðîñòðàíñòâîì E6 êëàññè÷åñêîé ñèñòåìû òðåõ òåë è 6 D ðèìàíîâûì ìíîãîîáðàçèåì M, ÷òî ïîçâîëÿåò ñâåñòè äèíàìè÷åñêóþ çàäà÷ó ê ñèñòåìå 6-ãî ïîðÿäêà. Êëþ÷åâûå ñëîâà: ñèñòåìà íåäîîïðåäåëåííûõ àëãåáðàè÷åñêèõ óðàâíåíèé, ðèìàíîâî ìíîãîîáðàçèå, îðèåíòèðîâàííîå òîïîëîãè÷åñêîå ìíîãîîáðàçèå, ãîìåîìîðôèçì ìåæäó ìíîãîîáðàçèÿìè. ²ßáï ê. ¶¨áñ·Û³Ý1;2, ²É»ք ². ²É»ùë³ÝÛ³Ý1 ¨ êáõñ»Ý ´. ²É³í»ñ¹Û³Ý1 ¹ÇݳÙÇÏ ËݹÇñÁ ѳݷ»óÝ»É 6-ñ¹ ϳñ·Ç ѳٳϳñ·Ç: ´³Ý³ÉÇ µ³é»ñ՝ ãÑëï³Ï»óí³Í ѳÝñ³Ñ³ßí³Ï³Ý ѳí³ë³ñáõÙÝ»ñÇ Ñ³Ù³Ï³ñ·, 08 08_Gevorgyan_98--106 8_Gevorgyan_Article 8_Gevorgyan_ABSTRACT