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

For the real version of this theorem, we refer the reader to Theorem 5.3 [Y5].
Now the Parabolic Geometry is a geometry modeled after the homogeneous space
G/G ′ , where G is a (semi-)simple Lie group and G ′ is a parabolic subgroup of G (cf. [Bai]).
Precisely, in this paper, we mean, by a Parabolic Geometry, the Geometry associated
with the Simple Graded Lie Algebra in the sense of N.Tanaka ([T4]).
In fact, let g = ⊕
p∈Z gp be a simple graded Lie algebra over R satisfying the generating
condition. Let M be a manifold with a G ♯
0-structure of type m in the sense of [T4]
(for the precise definition, see §2 of [T4]). In [T4], under the assumption that g is the
prolongation of (m, g0), N. Tanaka constructed the Normal Cartan Connection (P, ω)
of Type g over M, which settles the equivalence problem for the G ♯
0-structure of type
m in the following sense: Let M and ˆ M be two manifolds with G ♯
0-structures of type m.
Let (P, ω) and ( ˆ P , ˆω) be the normal connections of type g over M and ˆ M respectively.
Then a diffeomorphism ϕ of M onto ˆ M preserving the G ♯
0-structures lifts uniquely to an
isomorphism ϕ♯ of (P, ω) onto ( ˆ P , ˆω) and vice versa ([T4], Theorem 2.7).
Here we note that, if g is the prolongation of m, a G ♯
0-structure on M is nothing but
a regular differential system of type m (see [T4, §2.2]). Thus a Parabolic Geometry
modeled after G/G ′ is the geometry of P D manifold of second order with the symbol
algebra s = s−3 ⊕ s−2 ⊕ s−1, if g is the prolongation of m and m is isomorphic to s.
Hence, among simple graded Lie algebras g = ⊕ µ
p=−µ gp ∼ = (Xℓ, ∆1), we will seek those
algebras such that m = ⊕
p