G Complete Riemannian Holonomy Metrics on 2 3 Deformations of Cones over S3 S3 1 × 0 2 Ya.V. Bazaikin, O.A. Bogoyavlenskaya n a February 1, 2013 J 1 3 Abstract G] CompleteRiemannianmetricswithholonomygroupG2areconstructed on themanifolds obtained by deformations of cones overS3×S3. D . h 1 Introduction t a m This article is a sequelto the works[1, 2, 3, 4] and is dedicated to studying the [ Riemannian manifolds with holonomy group G . Recently, this problem has 2 attracted considerable interest; moreover, though the most important problem 2 v is to study the compact manifolds admitting such metrics, still the studying of 9 non-compactmanifolds (for the mostpart,the vectorbundle spaces)with com- 7 pleteRiemannianG -holonomymetricsisquitelogical. Thisisexplainedbythe 2 3 fact that in the latter case one can succeed, as a rule, in setting a G -structure 6 2 explicitly and writing the equations which guarantee its being parallel. In ad- . 1 dition, if the symmetry group of the considered G -structure is large enough, 2 0 thentheproblemisreducedtoasystemofordinarydifferentialequations,which 3 allows either to find explicit solutions (in contrast to the compact case), or to 1 : study them qualitatively. The main idea of the article has been already used v in [1, 2, 3] for constructing complete Riemannian metrics with holonomy group i X Spin(7); it consists in the following: the standard conic metric is considered r over a Riemannian manifold with a special geometry. Then the deformation of a this metric depends on a certain number of functional parameters which allow defining explicitly a G (or Spin(7)) structure. In the presentwork we propose 2 (following [5]) to consider the space M = S3 S3 as such base of the cone. × Then the conic metric can be written as 3 3 ds¯2 =dt2+ A (t)2(η +η˜)2+ B (t)2(η η˜)2, i i i i i i − i=1 i=1 X X whereη ,η˜ isthestandardcoframeof1-forms,whereasthefunctionsA (t),B (t) i i i i define a deformationof the cone singularity. In the paper [5] a system of differ- ential equations is written down, which guarantees that the metric ds¯2 has the 1 holonomy group containing in G . In [5] a particular solution of this system is 2 found, which correspondsto a metric with the holonomy groupG on S3 R4. 2 × Let us indicate that in the papers [6, 7, 8, 9, 10] more general metrics on the cones overS3 S3 have also been studied; however,in the situation considered × by us noother exampleshavebeen discoveredbesides the example from[5]and the classicalexample from[11]. Inthe proposedworkwe continue tostudy this class of metrics, while setting A = A , B = B and considering boundary 2 3 2 3 conditions different from that from [5]. This yields the metrics with a different topological structure. Namely, we require that at the vertex of the cone only the function B turns to zero. This results in that the Riemannian metric ds¯2 1 is defined onH4 S3, where H is the space ofcanonicalcomplex linear bundle × overS2,whereH4 isitsfourthtensorpower. Notethatin[9]numericalinvesti- gationwasconducted(usingthedevelopmentofthesolutionofthebasicsystem into the Taylor series), which yielded some arguments in favor of existence of the metrics constructed by us. The main result of the paper is formulated in the following theorem: Theorem. Thereexistsaone-parametricfamilyofmutuallynon-homothetic complete Riemannian metrics of the form ds¯2 with holonomy groupG on H4 2 × S3, whereas the metrics can be parameterized by the set of initial data (A (0), 1 A (0), B (0), B (0))=(µ,λ,0,λ), where λ,µ>0 and µ2+λ2 =1. 2 1 2 For t the metrics of this family are approximated arbitrary closely by → ∞ thedirectproductS1 C(S2 S3),whereC(S2 S3)istheconeovertheproduct × × × of spheres. Moreover, the sphere S2 arises as factorization of the diagonally embedded in S3 S3 three-dimensional sphere with respect to the circle action corresponding to×the vector field ξ1+ξ˜ 1 2 G -structure on the cone over S3 S3 2 × Consider the Lie group G=SU(2) with the standard bi-invariant metric X,Y = tr (XY), h i − where X,Y su(2). Let us consider three Killing vector fields on G: ∈ i 0 0 1 0 i ξ1 = , ξ2 = , ξ3 = . 0 i 1 0 i 0 (cid:18) − (cid:19) (cid:18) − (cid:19) (cid:18) (cid:19) It is not difficult to see that they satisfy the relations [ξi,ξi+1]=2ξi+2, where the indices i = 1,2,3 are reduced modulo 3. Let η ,η ,η be the dual 1 2 3 basis of 1-forms, that is, η (ξj)=δj. Then i i dη = 2η η . i i+1 i+2 − ∧ 2 Let M =G G, then on M there arise 6 Killing fields ξi, ξ˜i, i=1,2,3, which × are tangent to the first and second factor, respectively, and 6 dual 1-forms η , i η˜. Consider the cone M =R M with the metric i + × 3 3 ds¯2 =dt2+ A (t)2(η +η˜)2+ B (t)2(η η˜)2, i i i i i i − i=1 i=1 X X whereA (t) andB (t) aresome positivefunctions defining adeformationofthe i i standard conic metric. Introducing the orthonormal coframe e1 =A (η +η˜ ), e4 =B (η η˜ ), 1 1 1 1 1 1 − e2 =A (η +η˜ ), e5 =B (η η˜ ), 2 2 2 2 2 2 − e3 =A (η +η˜ ), e6 =B (η η˜ ), 3 3 3 3 3 3 − e7 =dt. we define the following 3-form: Ψ=e564+e527+e513+e621+e637+e432+e417, where eijk = ei ej ek. The form Ψ defines a G -structure on M, which is 2 ∧ ∧ parallel provided the following equations hold: dΨ=0,d Ψ=0. (1) ∗ In the present work we consider a particular case when A =A , B =B . 2 3 2 3 Lemma1. Equations(1)areequivalenttothefollowingsystemofordinary differential equations: dA1 = 1 A21 A21 dt 2 A22 − B22 dA2 = 1(cid:16)B22−A22+B(cid:17)12 A1 dt 2 B1B2 − A2 (2) dB1 = A22(cid:16)+B22−B12 (cid:17) dt A2B2 dB2 = 1 A22−B22+B12 + A1 dt 2 A2B1 B2 (cid:16) (cid:17) For t=0 we have a conic singularity of the space M which can be resolved by setting the initial values of the functions A ,B . At that, there appear, i i up to symmetry of system (2), two types of singularity resolution listed below (compare with [1, 2, 3]). Type 1. A (0) = 0,B(0) = 0. In this case, a collapse takes place of the i integralthree-dimensional sph6eres generated by the vector fields ξi+ξ˜i. These spheres are the orbits of a free action of G on M, defined by the relation h ∈ G : (g ,g ) (hg ,hg ). It can be demonstrated that in this case the metric 1 2 1 2 ds¯2 on M ca7→n be continued to a space , homeomorphic to S3 R4. Here we M × omit the details because this case is not investigated in the present article. 3 Type 2. B (0) = 0, B (0) = 0,A (0) = 0. Consider a free action of the 1 2 i 6 6 group U(1)=S1 on M: z 0 z 1 0 z U(1):(U,V) U, − V . ∈ 7→ 0 z−1 0 z (cid:18)(cid:18) (cid:19) (cid:18) (cid:19) (cid:19) It is clear that the orbits of this action coincide with the integral curves of the field ξ ξ˜. Thus, it is possible to continue the metric ds¯2 on [0, ) M by 1 1 − ∞ × contracting each orbit into a point for t=0. The diffeomorphism φ:M M :(U,V) (U,U 1V). − → 7→ transforms the above-consideredaction of U(1) into the action of the following form: z 0 z U(1):(U,V) U,V . ∈ 7→ 0 z−1 (cid:18)(cid:18) (cid:19) (cid:19) FactorizationwithrespecttotheU(1)actiononthefirstfactordefinestheHopf fibration G = S3 S2 = G/U(1). After contraction into a point of the orbits → ofthis actionfort=0inthe space[0, ) G,we obtainacylinder ofthe Hopf fibration, which can be easily seen to b∞e h×omeomorphic to a linear C-bundle H over S2, called a tautological bundle over S2. Since the action on the second factor is trivial, we conclude that the metric ds¯2 can be continued to the space H G. ×ConsidernowthecyclicsubgroupZ inU(1). ThegroupZ (followingU(1)) 4 4 acts on M; therefore,it is possible to expand this actionto the entire space M¯. Since this discrete action is in agreement with the orbits contraction for t =0, we obtain an actionof Z on H G. The factor-space (H G)/Z is naturally 4 4 × × diffeomorphic to = H4 G, where H4 is the fourth tensor power of the M × bundle H. Thus, in the considered case the metric ds¯2 can be continued to the manifold . M The next lemma is proven analogously to the Lemma 5 from [1]. Lemma 2. In order for the metric ds¯2 to be continued to a smooth metric on , it is necessary and sufficient that the following conditions hold: M (1) B (0)=0, B (0) =2; 1 | 1′ | (2) A (0)=B (0)=0,A (0)= A (0), 2 2 6 ′2 − ′2 (3) A (0)=0,A (0)=0; 1 6 ′1 (4) the functions A ,B are sign-definite on the interval (0, ). i i ∞ Remark. Adissimilaritywiththepaper[1],whichappearsinthecondition for the initial derivative of the function B in (1), is connected with normaliza- 1 tion: the length of the vector ξ ξ˜ equals 2, not one, as it was in Lemma 5 1 1 − from [1]. In[5]anexactsolutionofthefollowingformwasfoundforsystem(2)(other 4 solutions of the family, discovered in [5], are homothetic to this one): A1(r)= ((rr−93//44))((rr++93//44)), − A (r)=q1 (r+3/4)(r 9/4), 2 √3 − (3) B (r)=2r/3, 1 p B (r)= 1 (r 3/4)(r+9/4), 2 √3 − where r 9/4, and the variable r ips connected with t via the variables change ≥ dr dt= A (r),t|r=49 =0. 1 Metric (3) is a complete metric with holonomy group G on S3 R4. If we 2 × consider the case A =A =A =A and B =B =B =B, then the system 1 2 3 1 2 3 (2) can be integrated in elementary functions, and we obtain another complete metric with holonomy group G on S3 R4: 2 × dr2 r2 1 3 r2 3 ds¯2 = + 1 (η +η˜)2+ (η η˜)2. (4) 1 1 9 − r3 i i 3 i− i − r3 (cid:18) (cid:19)i=1 i=1 X X Metric (4) was constructed for the first time in [11], see also [12]. As far as we know, metrics (3) and (4) exhaust the list of known explicit solutions of the system (2) corresponding to complete Riemannian metrics with holonomy group G . 2 If we performa formalvariables changer r in the solution(4), then we →− get the following solution of (2): dr2 r2 1 3 r2 3 ds¯2 = + 1+ (η +η˜)2+ (η η˜)2. (5) 1+ 1 9 r3 i i 3 i− i r3 (cid:18) (cid:19)i=1 i=1 X X The solution (5) is defined for 0 < r < , but it does not yield any smooth ∞ Riemannian metric , because it has a singularity at r =0. 3 A family of new solutions Proceedinganalogouslyto[1],weconsiderthestandardEuclidianspaceR4 and set R(t) = (A (t),A (t),B (t),B (t)). Let V : R4 R4 be the function of the 1 2 1 2 → argument R, defined by the right-hand side of system (1) ( the function V is defined, ofcourse,only within the domainwhere A , B =0). Thus,system(1) i i 6 has the form: dR =V(R). dt Using the invariance of V with respect to homotheties R4, we perform substi- tution R(t)=f(t)S(t), where S(t) =1,f(t)= R(t), | | | | S(t)=(α (t),α (t),α (t),α (t)). 1 2 3 4 5 Thus, our system is split into ”radial” and ”tangential” parts: dS =V(S) V(S),S S =W(S), (6) du −h i 1 df = V(S),S , f du h i (7) dt=fdu. The solutions of the autonomous system (6) on the three-dimensional sphere 4 S3 = (α ,α ,α ,α ) α2 =1 { 1 2 3 4 | i } i=1 X allowus obtainingthe solutionsof(2) byintegratingequations(7). The follow- ing lemma is obvious. Lemma 3. Systems (2) and (6) admit the following symmetries: (α ,α ,α ,α ) ( α ,α ,α ,α ), 1 2 3 4 1 4 3 2 7→ − ((α (u),α (u),α (u),α (u)) ( α ( u),α ( u),α ( u), α ( u)), 1 2 3 4 1 2 3 4 7→ − − − − − − ((α (u),α (u),α (u),α (u)) ( α ( u), α ( u),α ( u),α ( u)), 1 2 3 4 1 2 3 4 7→ − − − − − − ((α (u),α (u),α (u),α (u)) (α (u),α (u), α (u), α (u)), 1 2 3 4 1 2 3 4 7→ − − ((α (u),α (u),α (u),α (u)) (α (u), α (u), α (u),α (u)) 1 2 3 4 1 2 3 4 7→ − − By virtue of Lemma 2, to the regular metric on there may correspond M only a trajectory of system (6) coming out of the point S =(µ,λ,0,λ), where 0 2λ2+µ2 =1. Due to symmetries of Lemma 3, we can assume that λ,µ>0. Lemma 4. For any point S = (µ,λ,0,λ) , considered above , there exists 0 a unique smooth trajectory of system (6), coming out of the point S into the 0 domain α >0,α >α . 3 4 2 Proof. Let J = (µ,λ,0,λ)µ > 0,λ > 0,2λ2 +µ2 = 1 be an arc of the { | } circle on which the point S is selected. Let us denote by U an open disc in 0 R2 with coordinates x = α , y = α α of the radius ε with the center at 3 4 2 − zero. TheninaneighborhoodofthearcJ wecanconsiderthelocalcoordinates x,y,z =α . In these coordinates the field W has the following components: 1 W =W , W =W W , W =W , x 3 y 2 4 z 1 − where W (S)=V (S) V(S),S α , j =1,2,3,4, j j j −h i S =(α ,α ,α ,α )= 1 2 3 4 1 1 z, 2 2x2 y2 2z2 y ,x, 2 2x2 y2 2z2+y , 2 − − − − 2 − − − (cid:18) (cid:16)p (cid:17) (cid:16)p (cid:17)(cid:19) 6 whereas the formulae for V (S) are obtained by the corresponding coordinate i change. SinceatthepointsofJ theoriginalsystemhasasingularity,weconsider in the neighborhood J U a modified system of differential equations: × x xW d x y = xW . (5) y dv     z xW z     Clearly, the trajectories of system (5) coincide with the trajectories of system (2)uptotheparameterchangedu=xdv. ThevectorfieldxW issmoothinthe neighborhood J U; and a direct calculation shows that for sufficiently small × ε>0thestationarypointsofsystem(5)inJ U arepreciselythepointsofthe × interval J. Consider linearization of system (5) in a neighborhood of the point S : 0 dx =2x, dv dy = µx y, dv √2 2µ2 − dz =0. − dv The linearized system has three eigenvectors e =(3, µ ,0), e =(0,1,0), 1 √2 2µ2 2 − e =(0,0,1) with the eigenvalues 2, 1 and 0, respectively. 3 − Adirectcalculationshowsthatif(x,y,z) S =(0,0,µ),then (0,0,1), xW → 0 h xW i→ 0, i. e. the angle between the vector xW and the vector, which is tangent| to| the arc J, tends to π/2 when we approach the points of J. This allows re- constructing the ”phase portrait” of system (5) in the neighborhood of J U × analogouslyto the wayitis done inthe classicalcase. Namely,considerthe do- mainΓ in J U,bounded by the paraboliccylinders µx +3y+αx2 =0, × −√2 2µ2 µx +3y αx2 =0 and the plane x=δ, where α,δ −>0. These cylinders −√2 2µ2 − − at the fixed level z are parabolas, which are tangent along the vector e . It is 1 easy to calculate that at the points of the first parabolic cylinder d µx +3y+αx2 =5αx2+O(x2+y2) 0, dv − 2 2µ2 ! ≥ − p if we choose the constant α to be sufficiently large (whereas equality is reached only on J). Thus, the trajectories intersect the first parabolic cylinder, coming from outside of the domain Γ inside. It can be demonstrated analogously that the trajectories of system (5) intersect the second parabolic cylinder, bounding the domain Γ, also passing from outside of the domain inside. Then for each value z = z there exists a trajectory, which ends on the planar wall of the 0 domain at the point (δ,y,z ), and which comes out of a point on the axis J, if 0 wechooseδsufficientlysmallandαsufficientlylarge(thisfollowsfromthatsuch trajectory cannot deviate substantially along J, since the angle that it forms with J convergesto π/2). Hence, if we fix the point S =(0,0,µ) on the arc J, 0 thenunderdiminishingofδ andincreasingofαwecanfindatrajectory,coming outexponentiallywiththeorderofe2v fromthepointS intothedomainx>0. 0 7 Analogously, there will exists a trajectory, coming out of S from the opposite 0 side, i.e. from the side of the domain x < 0. Since the order of convergence of x to zero equals e 2v, then, with respect to the parameter u, there will take − place”comingout”fromthe pointS infinite time. Byanalogousreasoningwe 0 demonstrate the uniqueness of each of the trajectories. Let us note now that under the transition from the parameter u to the parameter v there occurs the reversal of the trajectories orientation in the domain x < 0. It means that for each point S there exists a unique trajectory which comes out of the point S 0 0 in finite time and enters the domain x > 0. Moreover, the trajectory, coming outofS ,willbe tangenttothevectore ,i.e. forsmalluwewillhaveα >α . 0 1 4 2 The Lemma is proved. Lemma 5. The stationary solutions of system (6) on S3 are exhausted by the following list of zeros of the vector field W, up to the symmetries of Lemma 3: 1 1 √3 √3 √3 √2 √3 , , , , 0, , , . 2√2 2√2 2√2 2√2! √10 √5 √10! Proof. At the points, where the vector field W turns to zero, the field V(S) is parallel to S(u); hence, the stationary solutions of system (2) satisfy the following system of equations 1 α21 α21 =βα , 2 α22 − α24 1 21(cid:16)α24−αα3α22+4α(cid:17)23 − αα12 =βα2, α22(cid:16)+αα2α24−4α23 =βα3, (cid:17) 21 α22−αα2α24+3α23 + αα14 =βα4, α2(cid:16)+α2+α2+α2(cid:17)=1, 1 2 3 4 where β = V(S),S R. Solution of the system is subdivided into two cases: h i ∈ if α = 0, then we easily obtain the second point from the conditions of the 1 Lemma. If α =0, then eliminating β, we express α ,α in terms of α ,α : 1 1 3 2 4 6 4 α2α2 α2 α2 2 α2 = 2 4 ,α2 =3 4− 2 , 1 3α2+α2 3 α2+α2 2 4 (cid:0) 2 4(cid:1) after which, we obtain the relation 4 α2 α2 2 = α2+α2 2, 4− 2 4 2 (cid:0) (cid:1) (cid:0) (cid:1) fromwhichweimmediatelyobtaintheremainingpoints. TheLemmaisproved. A pointS S3,where the fieldW is not defined, willbe calledconditionally ∈ stationary,if there exists a real-analyticcurveγ(u) onS3, u ( ε,ε), γ(0)=S ∈ − , such that the fields V, W are defined at all points γ(u), u ( ε,ε), u = 0, ∈ − 6 are continuously extendable to the entire curve γ(u), and lim W(γ(u))=0. u 0 → 8 Lemma 6. System (6) does not have any conditionally stationary solutions on S3. Proof. Let a point S = (α ,α ,α ,α ), 4 α2 = 1 be conditionally 1 2 3 4 i=1 i stationary, i.e. there exists a curve γ(u), u ( ε,ε) with the above-mentioned ∈ −P properties. Obviously, this is only possible in the case when at least one of the conditions holds: α (0)=0, α (0)=0 or α (0)=0. 2 3 4 1)Firstconsiderthecasewhenalltherelationsholdsimultaneously: α (0)= 2 α (0)=α (0)=0, α (0)= 1. Let us set for i=2,3,4 3 4 1 ± α (u)=c uki(1+o(1)), u 0, i i → where c = 0, k > 0. Note that if α (u) = α (u)+cuk, where c = 0, then V i i 2 4 1 6 6 cannot be continuously extended along γ(u) up to u = 0. It follows from the real analyticity that α (u)=α (u) and, in particular, k =k . Then 2 4 2 4 1 V2 = ± u−k2(1+o(1)), 2c 2 which is a contradiction with the existence of the limit of V(S) as u 0. → 2) Suppose that two out of three functions α ,α ,α turn to zero at u=0. 2 3 4 Consider the arising cases. The case of α (0)=α (0)=0, α (0)=0. If, in addition, α (0)=0, then 2 3 4 1 6 6 α (0) V = 1 u 2k2(1+o(1)), 1 2c2 − 2 which leads to a contradiction. If α (u)=c uk1(1+o(1)), c =0, k >0, then 1 1 1 1 6 k k (from the continuity of V ) and 1 2 1 ≥ α (0) V = 4 u k3(1+o(1)), 2 − c 3 which is again a contradiction. The case of α (0)= 0, α (0)= α (0)= 0 is symmetric to the previous one 2 3 4 6 and can be excluded analogously. The case of α (0)=α (0)=0, α (0)=0. This case is excluded, because 2 4 3 6 α (0)2 V3 = 3 u−k2−k4(1+o(1)). − c c 2 4 3) Suppose that only one of the functions α ,α ,α turns to zero at u=0. 2 3 4 Thecaseofα (0)=0,α (0),α (0)=0. ThecontinuityofV andV implies 2 3 4 1 3 6 in this case that α (0)=0 andα (0)= α (0)=0, while k k . In this case 1 3 4 1 2 ± 6 ≥ lim V(γ(u)) = (0,1,0,0) and lim W(γ(u)) = (0,1,0,0) = 0, which is a u 0 u 0 → → 6 contradiction. The case of α (0)=0, α (0),α (0)=0 is excluded in analogous fashion. 4 2 3 6 The case of α (0) = 0, α (0),α (0) = 0. The continuity of V immediately 3 2 4 6 yieldsthatα (0)= α (0). Thenlim V (γ(u))=2andlim W (γ(u))= 2 4 u 0 3 u 0 3 ± → → 2=0, again a contradiction. The Lemma is proved. 6 9 Ametricds¯2iscalledasymptoticallylocallyconic,ifthereexistthefunctions A˜ (t), B˜ (t), linear with respect to t up to a shift, such that i i A B i i 1 0, 1 0, t − A˜ → − B˜ → →∞ (cid:12) i(cid:12) (cid:12) i(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) The metric, defined(cid:12)by the (cid:12)function(cid:12)s A˜ (t)(cid:12), B˜ (t), is called locally conic. The (cid:12) (cid:12) (cid:12) i (cid:12) i following lemma is proved in [1]. Lemma 7. To the stationary solutions of system (6) there correspond the locally conic metrics on M, whereas to the trajectories of system (6), asymp- totically converging to stationary solutions, there correspond the asymptotically locally conic metrics on M. The following lemma follows directly from the analysis of systems (2) and (6). Lemma 8. If S = (α ,α ,α ,α ) is a solution of system (6), then there 1 2 3 4 take place the following relations: 1) d 2A A B B (B2 A2) =0, dt 1 2 2− 1 2 − 2 2) d (cid:0) α1α2α4 = (cid:1) α1α3 , du(cid:18)2α4α2α1−α3(α24−α22)(cid:19) 2α4α2α1−α3(α24−α22) 3) d lnα3(α24−α22) = 2α4α2α1−α3(α24−α22), 4) ddduu(cid:18)lnαα24 =α4ααα2224αα−13αα224(cid:19), α2 =2αα4α42,(α24−α22) 5) d α3 = 3 2 + α3 2 α3 α =0,α =α . du α4 2α4 √3 α4 √3 − α4 1 2 4 (cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17) Remark. Thus, the function F(t)=2A A B B (B2 A2) is anintegral 1 2 2− 1 2− 2 of system (2). Lemma 9. The trajectory of system (6), defined by the initial point S = 0 (µ,λ,0,λ), λ,µ>0, 2λ2+µ2 =1, converges, as u , to the stationary point →∞ S = 0, √3 ,√2, √3 . ∞ √10 √5 √10 (cid:16) (cid:17) Proof. Let us introduce notations for the following points in S3: O =(0,0,1,0), A=(0,0,0,1), B =(1,0,0,0), C =(0, 1 ,0, 1 ). √2 √2 Consider the domain Π S3, defined by the inequalities: ⊂ Π:α α 0,α 0,α 0. 4 2 1 3 ≥ ≥ ≥ ≥ Itisnotdifficulttoverifythatthe domainΠisthesphericalpyramid(OABC). The boundaries of the domain are the following sets: Π =(OAB)= α =0,α 0,α 0,α 0 , 1 2 4 1 3 { ≥ ≥ ≥ } Π =(OBC)= α =α ,α 0,α 0,α 0 , 2 4 2 2 1 3 { ≥ ≥ ≥ } Π =(OAC)= α α 0,α =0,α 0 , 3 4 2 1 3 { ≥ ≥ ≥ } Π =(ABC)= α α 0,α 0,α =0 . 4 4 2 1 3 { ≥ ≥ ≥ } 10