We present a complete description of a class of linearizable planar geodesic
webs which contain a parallelizable 3-subweb.Comment: 7 page

Goldberg, Vladislav V.

Lychagin, Valentin V.

English

arXiv

We present a complete description of a class of linearizable planar
geodesic webs which contain a parallelizable 3-subweb

Munin - Open Research Archive

arXiv:0812.2547v1  [math.DG]  13 Dec 2008On a class of linearizable planar geodesic websVladislav V. Goldberg and Valentin V. LychaginAbstractWe present a complete description of a class of linearizable planargeodesic webs which contain a parallelizable 3-subweb.1 IntroductionThe paper is a continuation of [3]. In the paper [3] we considered some classicalproblems of the theory of planar webs. In particular, at the end of the paper weproved that a planar d-web is linearizable if and only if the web is geodesic andthe Liouville tensor of one of its 4-subwebs vanishes. In the current paper wedescribe all linearizable planar geodesic webs satisfying the following additionalcondition: the curvature K of one of its 3-subwebs vanishes.2 The ProblemBelow we give some (not all) definitions and notions which will be used in thepaper. For additional information a reader is advised to look into [3].We consider the plane M endowed with a torsion-free connection ∇ and ageodesic d-web in M , i.e., a d-web all leaves of all foliations of which are geodesicwith respect to the connection ∇. We have proved in [3] that there is a uniqueprojective structure associated with a planar 4-web in such a way that the 4-webis geodesic with respect to the structure.The flatness of the projective structure can be checked by the Liouville tensor(see [6], [5], [4]). This tensor can be constructed as follows (see, for example,[7]).Let ∇ be a representative of the canonical projective structure, and Ric bethe Ricci tensor of the connection ∇. Define a new tensor P asP(X, Y ) =23Ric(X, Y ) +13Ric(Y, X),where X and Y are arbitrary vector fields.The Liouville tensor L is defined as follows:L(X, Y, Z) = ∇X(P)(Y, Z) −∇Y (P)(X, Z)where X, Y and Z are arbitrary vector fields.1The tensor is skew-symmetric in X and Y , and therefore it belongs toL ∈ Ω1(R2) ⊗ Ω2(R2).It is known (see [6], [7], [5], [4]) that the Liouville tensor depends on the projectivestructure defined by ∇ and vanishes if and only if the projective structure is flat.For the case of the projective structure associated with a planar 4-web weshall call this tensor the Liouville tensor of the 4-web.Let us consider a 4-web with a 3-subweb given by a web function f (x, y)and a basic invariant a (see [3] for more details) and introduce the followingthree invariants:w =fyfx, α =aay − waxwa(1 − a), k = (log w)xy . (1)Then the Liouville tensor has the form [3]:L = (L1ω1 +L2wω2) ⊗ ω1 ∧ ω2,where L1 and L2 are relative differential invariants of order three.The explicit formulas for these invariants are3L1 = w(−(kw)x + αxx + ααx) + (αwxx + (α2 + 3αx)wx − 2αxy − 2ααy)+w−1(−αwxy − 2αywx + αw2x) + w−2αwxwy,3L2 = w2(−(kw−1)y + 2ααx) + w(2α2wx − 2αxy − ααy)+(−αwxy − 2αywx + αyy) + w−1(αwxwy − αywy).(2)As we said in Introduction, at the end of the paper [3] we proved that aplanar d-web is linearizable if and only if the web is geodesic and the Liouvilletensor of one of its 4-subwebs vanishes.In the current paper we consider a class of planar d-webs for which thecurvature K of one of its 3-subwebs vanishes.In order to prove the main theorem, we need the following lemma.Lemma 1 If K = 0, we can reduce w (see (1)) to one: w = 1.Proof. In fact, becauseK = −1fxfy(logfxfy)xy,it follows from K = 0 that (log w)xy = 0. Hence log w = u(x)+v(y), where u(x)and v(y) are arbitrary functions. It follows that w = a(x)b(y), where a(x) =eu(x) and b(y) = ev(y). Taking the gauge transformation x → X (x) , y → Y (y),with X ′ (x) = eu(x) and Y ′ (y) = e−v(y), we get that w = 1.We shall prove now the main theorem.2Theorem 2 A planar d-web, for which the curvature K of one of its 3-subwebsvanishes, is linearizable if and only if the web is geodesic, and the invariants αdefined by its 4-subwebs have one of the following forms:(i)α =℘′(2x + y + λ1, g2, g3) − ℘′(x + 2y + λ2, g2,−g3)℘(2x + y + λ1, g2, g3) − ℘(x + 2y + λ2, g2,−g3), (3)where ℘ is the Weierstrass function, g2 and g3 are invariants, and λ1 andλ2 are arbitrary constants.(ii)α = kek(x−y+C) + 1ek(x−y+C) − 1, (4)where k and C are arbitrary constants.(iii)α = −k tanx − y + C2, (5)where k and C are arbitrary constants.(iv)α =2x − y + C, (6)where C is an arbitrary constant.Here x, y are such coordinates that the 3-subweb is defined by the web func-tions x, y and x + y.Proof. By Theorem 9 of [3], the conditions of linearizability are L1 =0, L2 = 0. By (1) and Lemma 1, the condition K = 0 implies k = 0, w = 1.It follows that the conditions L1 = 0, L2 = 0 becomeαxx − 2αxy + ααx − 2ααy = 0,αyy − 2αxy + 2ααx − ααy = 0.(7)Conditions (7) can be written in the form(∂x − 2∂y)(αx +12α2) = 0,(∂y − 2∂x)(αy −12α2) = 0.(8)Therefore, relations (8) implyαx +12α2 = A(2x + y),αy −12α2 = B(x + 2y)(9)3for some functions A and B.Differentiating the first equation of (9) with respect to y and the second onewith respect to x, we get the following compatibility conditions for (9):ααy + ααx = A′ − B′,which by (9) is equivalent to(A + B)α = A′ − B′. (10)We assume that A + B 6= 0. (The case A + B = 0 will be consideredseparately.) Then equation (10) impliesα =A′ − B′A + B. (11)Next, we substitute α from (11) into equations (7). As a result, we obtainthat(2A′′ − B′′)(A + B) − (A′ − B′)(2A′ + B′) + 12 (A′ − B′2= A(A + B)2,(A′′ − 2B′′)(A + B) − (A′ − B′)(A′ + 2B′) − 12 (A′ − B′2= B(A + B)2.(12)Adding and subtracting equations (12), we find that(A′′ − B′′)(A + B) − (A′2 − B′2) =(A + B)33(13)andA′′ + B′′ = A2 − B2. (14)Therefore,A′′ − A2 = c,(B′′ + B2 = −c,(15)for a constant c ∈ R .Multiplying equations (15) by A′ and B′, respectively, we getA′A′′ − A′2 = cA′;B′B′′ + B′2 = −cB′,and(12A′2 −13A′3 − cA)′= 0;(12B′2 +13B′3 + cB)′= 0,4respectively.This means that12A′2 −13A′3 − cA = a(s) (16)and12B′2 +13B′3 + cB = b(t), (17)where s = x + 2y and t = 2x + y.Now equations (16), (17) and (13) giveb = a = const . ∈ R (18)Remind that solutions of the equationy′2 = 4y3 − g2y − g3 (19)have the formy = ℘(x + λ, g2, g3), (20)where ℘ is the Weierstrass function, g2 and g3 are invariants, and λ is an arbi-trary constant.By (18), equations (16) and (17) can be written asA′2 =23A3 + 2cA + 2a,B′2 = −23B3 − 2cB − 2a.(21)Taking A = β℘ and B = γ℘, substituting them into (21) and comparing theresult with (19), we find thatβ = 6, γ = −6; g2 = −c3, g3 = −a18,i.e., g2 and g3 are the same for both equations (21).By (20), the solutions of (21) areA = 6℘(t + λ1, g2, g3),B = −6℘(t + λ2, g2,−g3),(22)where g2 and g3 are arbitrary constants.Equations (22) can be now written asA = 6℘(2x + y + λ1, g2, g3),B = −6℘(x + 2y + λ2, g2,−g3).(23)Finally, equations (11) and (23) give the following expression (3) for theinvariant α:α =℘′(2x + y + λ1, g2, g3) − ℘′(x + 2y + λ2, g2,−g3)℘(2x + y + λ1, g2, g3) − ℘(x + 2y + λ2, g2,−g3).5Consider now the cases for which A + B = 0, i.e., the casesA = v, B = −v, v ∈ R.Then system (9) has the formαx +12α2 = v,αy −12α2 = −v(24)and is consistent.It follows from (24) that αx + αy = 0. The solution of this equation isα = α(x − y). As a result, we can write two equations (24) as one equationα′ +12α2 = v. (25)Three cases are possible:(ii) v = 12k2, k 6= 0. Then the solution of (25) has the form (4).(iii) v = − 12k2, k 6= 0. Then the solution of (25) has the form (5).(ii) v = 0. Then the solution of (25) has the form (6).Corollary 3 If for a geodesic d web the basic invariants are solutions of theEuler equation and one of its 3-subwebs is parallelizable, then this web is lin-earizable.References[1] Goldberg, V. V., Lychagin, V. V., Abelian equations and rank problems forplanar webs (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 2007, no. 10, 40–76. English translation in Russian Math. (Iz. VUZ) 51, no. 11, 39–75 (2007).MR12381928 (2008k:53029)[2] Goldberg, V. V., Lychagin, V. V., Geodesic webs on a two-dimensional man-ifold and Euler equations, Acta Appl. Math. (2009) (to appear); see alsoarXiv: 0810.5392, pp. 1–15 (2009)[3] Goldberg, V. V., Lychagin, V. V., On rank problems for planar webs andprojective structures, in The Abel Symposium 2008, Springer (2009) (to ap-pear); see also arXiv: 0812.0125v2, pp. 1–31 (2009)1In the bibliography we will use the following abbreviations for the review journals: JFMfor Jahrbuch für die Fortschritte der Mathematik, MR for Mathematical Reviews, and Zbl forZentralblatt für Mathematik.6[4] Kruglikov, B., Point classification of 2nd order ODEs: Tresse classificationrevisited and beyond, arXiv: 0809.4653, pp. 1–22 (2008)[5] Lie, S., Klassification und Integration von gewöhnlichen Differentialgleichun-gen zwischen x, y, die eine Gruppe von Transformation gestatten. III, Archivfür Math. og Naturvidenskab 8 (Kristiania, 1883), 371–458 (JFM 15, p. 751);see also Gesammelte Abhandlungen. Bd. 5 (1924), paper XIV, 362–427. JFM50, p. 2.[6] Liouville, R., Mémoire sur les invariants de certaines équationsdifférentielles et sur leurs applications, Journal de l’École Polytechnique 59,7–76 (1889). JFM 21, p. 317[7] Nomizu, K., Sasaki, T., Affine Differential Geometry, Cambridge Tractsin Mathematics, 111, Cambridge University Press, Cambridge (1994).MR1311248 (96e:53014); Zbl 834:530027

On a class of linearizable planar geodesic webs

We present a complete description of a class of linearizable planar
geodesic webs which contain a parallelizable 3-subweb

On a class of linearizable planar geodesic webs

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