4 1. INTRODUCTION
1.6.3. Normal forms for Goursat distributions. Giaro, Kumpera, and
Ruiz discovered in [GKR] the first Goursat germ not covered by Cartan. In so
doing they initiated the study of singular Goursat germs. Kumpera and Ruiz
introduced special coordinates in dimension 2 + i with associated Goursat normal
forms depending on (i−2) real parameters which covered all rank 2 Goursat germs.
They and their followers calculated which parameters could be “killed”, which could
be reduced to 1 or −1 and which must be left continuous (moduli). The outcome
of these computations is that the set of equivalence classes of Goursat germs on
is finite for i ≤ 7. Consequently the set of equivalence classes of points
i ≤ 7 is finite. This number is 2, 5, 13, 34, 93 for i = 3, 4, 5, 6, 7. See
the works [GKR], [KR], [Ga], [Mor1], [Mor2] of Giaro, Kumpera, and Ruiz
(i = 3, 4), Gaspar (i = 5) and Mormul (i = 6, 7). Mormul discovered [Mor2],
[Mor3], [Mor7] the first moduli, which appears at i = 8. The length of these
computations increases exponentially with i. Beyond Cartan’s theorem, the only
results which are valid for all dimensions are Mormul’s classifications of codimension
one singularities in [Mor4] and his classification of the simplest codimension two
singularities in [Mor5].
1.7. Our approach
We reduce the problem of classifying points in the Monster to a well-studied
classification problem: that of germs of Legendrian curves.
1.7.1. Integral curves. By a curve in PiR2 we mean a map γ : (a, b) → PiR2.
A curve in PiR2 tangent to the 2-distribution ∆i is called integral curve. Integral
curves in a contact 3-manifold such as
are called Legendrian curves.
1.7.2. Equivalent curves. Two germs of curves γ : (R, 0) →
γ : (R, 0) →
˜) p are called equivalent if there exists a local symmetry Φ :
˜) p and a local diffeomorphism φ : (R, 0) → (R, 0) (a reparame-
terization of a curve) such that γ = Φ ◦ γ ◦ φ.
Remark. When i = 1 we will call this equivalence RL-contact equivalence
(“RL” is for Right-Left, see for example [AVG]).
1.7.3. Classification problems. Consider the following problems.
(1) Classify germs of integral curves in the Monster tower.
(2) Classify germs of Legendrian curves.
(3) Classify points in the Monster tower.
(4) Classify finite jets of Legendrian curves.
We will show (1) and (2) are equivalent problems, and we will show that (3)
and (4) are equivalent problems. The equivalence between (3) and (4) allows us to
translate well-known results regarding Legendrian germs into classification results
on points of the Monster which leads to a number of new classification results and
gives simple unified proofs of previously disparate results. For example, Mormul’s
classification [Mor4] of the codimension one singularities of Goursat 2-distributions
now becomes a corollary of the contact classification of the simplest (A-type) sin-
gularities of Legendrian germs, see section 4.4.1 and Theorem 4.16.