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

⎧ ⎛
0
⎜0
⎪⎨
⎜
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⎜
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⎝
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0
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−ξ1
0
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tξ 0 0
⎞
⎟
⎟
⎟
⎟
⎟ ξ ∈ K
⎟
⎟
⎠
0
k ⎫
⎪⎬
, ξ1 ∈ K ,
⎪⎭
⎧ ⎛
0
⎜ 0
⎪⎨
⎜
⎜x1
⎜ 0
⎜ 0
⎝
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0
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t 0 0 0
x
0
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0
0
⎞
⎟
⎟
⎟
⎟
⎟ = ˆx + ˆx1 x ∈ K
⎟
⎟
0
⎠
0
k ⎫
⎪⎬
, x1 ∈ K ,
⎪⎭
⎧ ⎛
0
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⎪⎨
⎜ 0
⎜ 0
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⎝
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⎞
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⎟
⎟
⎟
⎟ = ˆx0 + â1 + â
⎟
0⎟
0
⎠
0
x0, a1 ∈ K,
a ∈ K k
⎫
⎪⎬
,
⎪⎭
ˇg0
⎧ ⎛
b
⎜0
⎪⎨
⎜
⎜0
⎜
= ⎜0
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⎜0
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c
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e
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B
0
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⎞
⎫
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0 ⎟
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0 ⎟
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0 ⎟ b, c, e ∈ K, B ∈ o(k)
⎟
0 ⎟
0
⎠
⎪⎭
−b
ˇg−3 =
ˇg−2 =
ˇg−1 =
ˇgℓ = { t X | X ∈ g−ℓ }, (ℓ = 1, 2, 3, 4, 5),
Then, for X = ˆx + ˆx1 + ˆx0 and A = â + â1 , we calculate
Thus we obtain
[[A, X], X] =
(2x1 t ax − a1 t xx) ∈ ˇg−5
f = 〈{2e ∗ 1 ⊚ e ∗ 2, . . . , 2e ∗ 1 ⊚ e ∗ k+1, e ∗ 2 ⊚ e ∗ 2 + · · · + e ∗ k+1 ⊚ e ∗ k+1〉 ⊂ S 2 (E ⊥ ),
where {e0, e1, . . . , ek+1} is a basis of V and E = 〈{e0}〉 is the Cauchy characteristic
direction. In case k = 1, f = 〈{2e∗ 1 ⊚ e∗ 2, e∗ 2 ⊚ e∗ 2}〉 is an involutive subspace of S2 (E⊥ ).
Hence (Rg; D1 , D2 ) is involutive when g = ⊕
p∈Z gp is of type (B3, {α1, α2, α3}). In case
k > 1, we have
f ⊥ = 〈{e1 ⊚ e1, ei ⊚ ej(2 ≦ i < j ≦ k + 1),
e2 ⊚ e2 − ek+1 ⊚ ek+1, . . . , ek ⊚ ek − ek+1 ⊚ ek+1}〉 ⊂ S 2 (W ),
where W = 〈{e1, . . . , ek+1}〉. Then we see that (f (1) ) ⊥ contains every ei ⊚ ei ⊚ ei for
i = 1, . . . , k + 1, which implies f is of finite type. We can also check that exceptional
cases other than G2 are of finite type by utilizing R-space orbit (Rg; D 1 , D 2 ), whereas
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