Singular Legendre curves and Goursat systems

Goo ISHIKAWA Department of Mathematics, Hokkaido University, JAPAN. ishikawa@math.sci.hokudai.ac.jp

The classification problem of Goursat systems is reduced to that of Legendre curves [1][4][2]. Conversely, the classification of Goursat systems gives an insight to that of each individual Legendre curve. We describe their correspondence from general viewpoint using the results in [3].

The present work is a part of joint projects with P. Mormul and M. Zhitomirskii. Let E be a germ of differential system of rank 2 on a manifold N of dimension n

(n ≥ 2) at x_{0} ∈ N. This means that E ⊆ T N is a subbundle of rank 2. We call E a Goursat system if E^{2} := E + [E, E] is of rank 3, namely, of corank n − 3, and, setting E =: E_{n−2}, E^{2} =: E_{n−3}, if E_{i−1} := (E_{i})^{2} is of corank i − 1, for i = n − 2, n − 3, …, 2, 1. Fact([3]) For any n ≥ 3, there exist an n-manifold M^{n} and a Goursat system E_{n−2} (“Monster Goursat”) on M such that any Goursat germ (N^{n}, x_{0}, E) is isomorphic to (M, y_{0}, E) for some y_{0} ∈ M. The systems (M^{n}, E_{n−2}) are obtained by iterative prolongations from M^{2} := R^{2}, E_{0} := T R^{2}. Note that M^{3} = P T^{∗}R^{2}, the projective cotangent bundle of the plane with the canonical contact structure E_{1}.

Let n ≥ 3. Then there exists the intrinsic projection π_{n,3} : M^{n} → M^{3}. The projection π ∘ γ of an integral curve-germ γ : (R, 0) → M^{n} to E_{n−2} (γ̇(t) ∈ E_{n−2}, (t ∈ (R, 0))) is a Legendre curve in P T^{∗}R^{2}. Recall that a curve-germ c : (R, 0) → P T^{∗}R^{2} is called a Legendre curve if the velocity vector ċ(t) ∈ E_{1}, for t ∈ (R, 0). Here we do not assume c is an immersion. A Legendre curve is called of finite type if it is of finite type as a space curve-germ, i.e. if it is determined by its finite jet up to diffeomorphisms. If c : (R, 0) → M^{3} is a Legendre curve of finite type, there exists a unique integral lifting c̃: (R, 0) → M^{n} to E_{n−2} with π ∘ c̃ = c for all n ≥ 3.

Now we introduce an important invariant of Goursat systems: Let (N^{n}, x_{0}, E) be a germ of Goursat system of rank 2. Then there exists intrinsically the projection π : (N^{n}, x_{0}) → M^{3} making E is the prolongation of the contact E_{1}. Then we define intrinsically

C_{E} = \{π ∘ γ | \begin{matrix}γ : (R, 0) → M^{n} integral to E_{n−2}, \\ γ(0) = x_{0}, π ∘ γ is of finite type \end{matrix} \} / ∼,

where ∼ means the contactomorphism equivalence: For two Legendre curves c, c′ : (R, 0) → P T^{∗}R^{2}, we set c ∼ c′ if if there exist a diffeomorphism σ : (R, 0) → (R, 0) and a contactomorphism τ : (P T^{∗}R^{2}, c(0)) → (P T^{∗}R^{2}, c′(0)) such that τ ∘ c = c′ ∘ σ. Theorem 1: (Legendre invariants of Goursat systems). Let (N^{n}, x_{0}, E) and (N′^{n}, x′_{0}, E′) be germs of Goursat systems of rank two. Then the following conditions are equivalent to each other: (i) (N, x_{0}, E) ≅ (N′, x′_{0}, E′). (ii) C_{E} = C_{E′}. (iii) C_{E} ∩ C_{E′} ≠ ∅.

Let c : (R, 0) → P T^{∗}R^{2} be a Legendre curve of finite type. Then we define G_{c} to be the set of isomorphism classes of germs of Goursat systems E satisfying [c] ∈ C_{E}. Actually G_{c} is realized, up to isomorphisms, by the prolongation-deprolongation infinite sequence of Goursat systems which is obtained by lifting of c. Theorem 2: (Goursat-Legendre duality.) Let c : (R, 0) → P T^{∗}R^{2} be a Legendre curve of finite type. Then the contactomorphism class [c] of c is recovered from the sequence G_{c}: In fact we have ⋂_{[E]∈G_{c}} C_{E} = {[c]}. Conversely, let E be a germ of Goursat system of rank two. Assume E is not the contact system. Then the isomorphism class [E of E is recovered from C_{E}: In fact we have sup{n | #π_{n} (⋃_{[c]∈C_{E}} G_{c}) = 1} = d(E), where d(E) denote the dimension of the base manifold of E, and π_{n} means the operation of taking isomorphism classes of Goursat systems over n-dimensional manifold. Moreover we have π_{d(E)} (⋃_{[c]∈C_{E}} G_{c}) = {[E]}.

References

[1] V.I. Arnold, First steps of local contact algebra, Canad. J. Math. 51–6 (1999), 1123–1134.

[2] G. Ishikawa, Classifying singular Legendre curves by contactomorphisms, to appear in Journal of Geometry and Physics (in Press).

[3] R. Montgomery, M. Zhitomirskii, Geometric approach to Goursat flags, Ann. Inst. H. Poincaré, 18–4 (2001), 459–493.

[4] M. Zhitomirskii, Germs of integral curves in contact 3-space, plane and space curves, Isaac Newton Inst. Preprint NI00043-SGT, (2000).