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

for each v ∈ P (X) and x = ν(v), where ν : P (X) → X is the projection. For a point
v ∈ P (X), we define the symbol subspace ˆ f(v) of D(x) by
ˆ f(v) = {X ∈ t−1(x) | [X, t−2(x)] ⊂ ˆv},
where x = ν(v) ∈ X, ˆv = p−3(v) ⊂ t−3(x) and p−3 : Tx(X) → t−3 = Tx(X)/∂D(x) is the
projection. We put
R(X) = {v ∈ P (X) | codim ˆ f(v) = s}
(R(X) is an open subset of P (X) under the condition (X.3). See below). We denote the
restrictions of differential systems D1 X and D2 X of P (X) to R(X) by the same symbols.
Then we have
Proposition 4.2. (R(X), D1 X , D2 X ) is a P D manifold of second order.
Proof. By (X.1) and the construction of P (X), it follows that D1 X and D2 X are differential
holds on
systems on P (X) of codimension 1 and n + 1 respectively. Moreover ∂D2 X ⊂ D1 X
P (X) by construction. Since P (X) is a submanifold of J(X, m − 1), Realization Lemma
for (P (X); D1 X , ν, X) implies
Ker ¯ι∗ = Ker ν∗ ∩ Ch (D 1 X) = Ch (D 1 X) ∩ Ch (D 2 X) = {0},
where ¯ι : P (X) → J(X, m − 1) is the inclusion. Thus it remains to show that R(X) is
an open subset of P (X) under the condition (X.3) and Ch (D 1 X ) is a subbundle of D2 X of
codimension n on R(X).
For this purpose, let us take a point vo ∈ P (X). We can find locally independent
1-forms ϖ0, . . . , ϖr, π1, . . . , πs on a neighborhood U of xo = ν(vo) ∈ X such that
∂D = {ϖ0 = · · · = ϖr = 0}, D = {ϖ0 = · · · = ϖr = π1 = · · · = πs = 0}.
Here we may assume that vo = {ϖ0 = 0} ⊂ Txo(X). Then we have
dϖi ≡ 0 (mod ϖ0, . . . , ϖr, π1, . . . , πs) for i = 0, . . . , r.
Hence there exist 1-forms β α i such that
dϖi ≡ β 1 i ∧ π1 + · · · + β s i ∧ πs (mod ϖ0, . . . , ϖr) for i = 0, . . . , r.
Thus the Cartan rank of ∂D is less than or equal to s.
Now let us consider
ϖ = ϖ0 + λ1ϖ1 + · · · + λrϖr
on U. Namely we consider a point v ∈ P (X) such that v = {ϖ = 0} ⊂ Tx(X), where
x = ν(v) ∈ U. Here (λ1, . . . , λr) constitutes an inhomogeneous coordinate of the fibres of
ν : P (X) → X. Denoting the pullback on P (X) of 1-forms on X by the same symbol, we
have
D 1 X = {ϖ = 0},
and
dϖ = dϖ0 +
r∑
λidϖi +
i=1
r∑
dλi ∧ ϖi.
on ν −1 (U). From dϖi ≡ ∑ s
α=1 βα i ∧ πα + ∑ r
j=1 γj
i ∧ ϖj (mod ϖ) for i = 0, . . . , r, we
calculate
dϖ ≡
s∑
α=1
(β α 0 +
r∑
i=1
λiβ α i ) ∧ πα +
r∑
i=1
i=1
(dλi + γ i 0 +
17
r∑
λjγ i j) ∧ ϖi (mod ϖ).
j=1