On symplectic invariants of planar 3-webs

The classical web geometry [1,3,4] studies invariants of foliation families with respect to pseudogroup of diffeomorphisms. Thus for the case of planar 3-webs the basic semi invariant is the Blaschke curvature, [2]. It is also curvature of the Chern connection [4] that are naturally associated with a planar 3-web. In the present paper we investigate invariants of planar 3-webs with respect to group of symplectic diffeomorphisms. We found the basic symplectic invariants of planar 3-webs that allow us to solve the symplectic equivalence problem for planar 3-webs in general position. The Lie-Tresse theorem, [5], gives the complete description of the field of rational symplectic differential invariants of planar 3-webs. We also give normal forms for homogeneous 3-webs, i.e. 3-webs having constant basic invariants. Анотація. В роботі вивчаються класи еквівалентності плоских 3-тканин відносно дії симплектичної псевдогрупи дифеоморфізмів площини. Знайдено поле раціональних диференціальних симплектичних інваріантів, яке далі використано для встановлення критеріїв симплектичної еквівалентності 3-тканин. 1. NORMALIZATION OF 3-WEBS Let D Ă R2 be a connected and simply connected domain in the plane equipped with the symplectic structure given by the differential 2-form Ω = dx^ dy in the standard coordinates on the plane. Recall that a 3-web in D is a family of three foliations being in general position. We will assume that these foliations are integral curves of differential 1-forms ωi, i = 1, 2, 3, and write W3 = xω1, ω2, ω3y ,

Proof. Since for every x P D the cotangent vectors ω 1 (x) and ω 2 (x) are pairwise linearly independent in the contangent space Tx D, one can find unique everywhere non-zero functions a 1 , a 2 , b P C 8 (D) such that ω 3 = a 1 ω 1 + a 2 ω 2 , ω 1^ω2 = b(x, y)dx^dy = bΩ.
Smoothness of b is evident, and its non-vanishing follows from linear independence of ω 1 and ω 2 . Also smoothness of a 1 and a 2 follows from Cramer's rule giving explicit formulae for the solution of a system of linear equations. Finally, since for i = 1, 2 and x P D the pair of cotangent vectors ω 3 (x) and ω i (x) is also linearly independent, the functions a 1 and a 2 are everywhere non-zero. Let ε = sign(a 1 a 2 b) P t´1, 1u be the sign of the product a 1 a 2 b. Then the functions εa 1 a 2 b , satisfy (1.1). In particular, f i =´a i f 3 , i = 1, 2.
Proof. This follows from uniqueness of "normalizing" functions f i from Lemma 1.1. The second condition in (1.2) requires that the numbering i Ñ ω i is defined up to even permutations and the first condition in (1.2) requires that rescaling Recall also that the first condition in (1.2) implies the existence of the Chern connection given by the differential 1-form γ such that The Chern form γ satisfies the following condition and its exterior differential dγ P Ω 2 (D) is an invariant of planar 3-webs with respect to the diffeomorphism group. In our case the normalization (1.2) and the above proposition shows that the Chern form γ is itself a symplectic invariant of 3-webs.
Let us write down γ in following form where he functions x i P C 8 (D) are barycentric coordinates of γ, i.e.
Then we have On symplectic invariants of planar 3-webs

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Using the second normalization (1.2) condition we can rewrite these relations in the following form

Theorem 2.3. The functions
are symplectic invariants of 3-webs.
Proof. The action of the cyclic group A 3 permute λ i while the action of group Z 2 change sign of λ i . Therefore, the invariants are the functions of λ 2 i , that are A 3 invariants. Hence they are generated by symmetric polynomials and the Vandermonde polynomial. Here J 1 , J 2 , J 3 correspond to the first elementary symmetric functions, and J w is the Vandermonde polynomial. □ Remark 2.4. Condition λ 1 +λ 2 +λ 3 = 0 implies that the following syzygies hold:

COORDINATES
Assume that differential 1-forms ω i have the following form in canonical coordinates (x, y), i.e. Ω = dx^dy, and Then Invariants J 1 and J 3 are differential invariants of the first order:

HOMOGENEOUS 3-WEBS
We say that a 3-web is homogenous if invariants J 1 , J 3 are constant. In this case syzygy relation (2.1) shows that all λ i are constant too.
We consider in series the following four cases: Case C 1 . Here we have and therefore for some smooth functions in the domain D.
Notice that the following symplectic transformations (x, y) Ñ ( x, y + ψ(x) ) transform the differential 1-form ω 2 to the form Therefore, the transformation associated with function ψ satisfying the following differential equation transforms the pair (ω 1 , ω 2 ) to the pair (dx, dy´λ 2 ydx).

Theorem 4.2. Any planar 3-web with invariants
can be transformed by a symplectomorphism to the 3-web formed by the family of curves x = const 1 , y = const 2 exp(αx), y = const 3 exp(αx) + 1.

Theorem 4.3.
Any planar 3-web with constant invariants J 1 , J 2 , J 3 , J w could be transformed by a symplectomorphism to the 3-web formed by the family of curves

GENERAL CASE
In addition to the above classification of homogeneous webs we consider also the singular case, when J 3 = 0.
Then strictly normalized triples (ω 1 , ω 2 , ω 3 ) and (r ω 1 , r ω 2 , r ω 3 ) define the same 3-web if and only if r ω i = ω σ(i) , for some even permutation σ P A 3 . Let's consider the following invariants of the A 3 action: Then, as above, we get the following syzygy: 2 3 , and the following proposition.
Let W I ω = Φ ω (W 3 ) be the image of the 3-web W 3 under this diffeomorphism. We call it invariantization of 3-web W 3 .
Summarize we get the following.