Contact Geometry of second order I - Dept. Math, Hokkaido Univ ...

When f = f 1 (r) (see [Y4] §2), there exists a canonical coordinate system (xi, z, pi, pij)
(1 ≦ i ≦ j ≦ n) of L(J) such that
R = { ∂2z ∂xi∂xα
(2) f2 (r) ⊂ S2 (V ∗ ) (r ≧ 2)
= 0 (1 ≦ i ≦ r, r + 1 ≦ α ≦ n)}.
(f 2 (r)) ⊥ = 〈{ei ⊚ ej | 1 ≦ i ≦ j ≦ r}〉 = S 2 (Vr),
When f = f 2 (r) (see [Y4] §3), there exists a canonical coordinate system (xi, z, pi, pij)
(1 ≦ i ≦ j ≦ n) of L(J) such that
R = { ∂2 z
∂xi∂xj
(3) f 3 (r) ⊂ S 2 (V ∗ ) (r ≦ n − 2)
= 0 (1 ≦ i ≦ j ≦ r)}.
(f 3 (r)) ⊥ = 〈{ei ⊚ ea | 1 ≦ i ≦ r, 1 ≦ a ≦ n}〉 = Vr ⊗S V,
When f = f 3 (r) (see [Y4] §4), there exists a canonical coordinate system (xi, z, pi, pij)
(1 ≦ i ≦ j ≦ n) of L(J) such that
R = { ∂2z = 0 (1 ≦ i ≦ r, 1 ≦ a ≦ n)}.
∂xi∂xa
Here {e1, . . . , en} is a basis of V , Vr = 〈{e1, . . . , er}〉 and Vs = 〈{er+1, . . . , en}〉 .
We need Reduction Theorems to explain why second order equations with these symbols
have the property that their symbols are the only invariants under contact transformations.
We will explain this fact for the type f3 (r) in §6.1 by utilizing the First Reduction
Theorem in §4. The other cases will be explained by utilizing the two step reduction
procedure in Part II.
4. P D manifolds of Second Order.
We will here formulate the submanifold theory for (L(J), E) as the geometry of P D
manifolds of second order ([Y1]) and discuss the First Reduction Theorem.
4.1. Realization Theorem. Let R be a submanifold of L(J) satisfying the following
condition:
(R.0) p : R → J ; submersion,
where p = π |R and π : L(J) → J is the projection. Let D 1 and D 2 be differential systems
on R obtained by restricting C 1 = ∂E and C 2 = E to R. Moreover we denote by the same
symbols those 1-forms obtained by restricting the defining 1-forms {ϖ, ϖ1, · · · , ϖn} of the
canonical system E to R. Then it follows from (R.0) that these 1-forms are independent
at each point on R and that
D 1 = {ϖ = 0}, D 2 = {ϖ = ϖ1 = · · · = ϖn = 0}.
In fact (R; D 1 , D 2 ) further satisfies the following conditions:
(R.1) D 1 and D 2 are differential systems of codimension 1 and n + 1 respectively.
(R.2) ∂D 2 ⊂ D 1 .
(R.3) Ch (D 1 ) is a subbundle of D 2 of codimension n.
(R.4) Ch (D 1 )(v) ∩ Ch (D 2 )(v) = {0} at each v ∈ R.
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