A geometric study of invariants of the system of differential equations of finite type.

在这个项目中，我们研究了有限类型微分方程组的几何不变量。可积型(k，1)阶完整微分方程组(Χ)定义了k-JET空间j^k的开子流形R和R上的微分系统E。与(Χ)相关的(R；E，F)在R上的k 1次(m 1，1)阶伪射影结构的符号代数是(n，r)阶伪射影基本分次李代数.N.Tanaka证明了与E相联系的自然存在G/G^&lt；(0)&gt；型正规Cartan联络(P，ω)。此外，(P，ω)的曲率函数K的调和部分HK生成CartanConnection(P，ω)的基本不变量，它取值于第二广义Spencer上同调空间H^2(δ)。因此，我们需要研究空间H^2(δ)。首先得到了伪射影分次李代数的广义Spencer上同调空间H^2(δ)的非零子空间。B.Doubrov证明了Cartan联络的曲率函数Hk取值于H^2(δ)的子空间F^1H^2(δ)。当δ是k阶伪射影分次李代数时，得到了F^1H^2(δ)子空间的生成元.

In this project, we reserched the geometric invariants of the system of differential equtations of finite type. The system (Χ) of (k+1)-th order holonomic differential equations of integrable type defines an open submanifold R of k-jet space j^k and a differential system E on R. The symbol algebra of the pseudo-projective structure of order k+1 bidegree (m+1,1) of (R ; E,F) on R associated with (Χ) is a pseudo-projective fundamental graded Lie algebra of order k+1 of bidegree (n,r). N.Tanaka showed that there is naturally a normal Cartan connection (P,ω) of type G/G^<(0)> associated with E. Moreover the harmonic part HK of the curvature function K of (P,ω) generates the fundamental invariants of the Cartanconnection (P,ω), which takes values in the second generalized Spencer cohomology space H^2(δ). Therefore we need to investigate the space H^2(δ). We first obtained the non-vanishing subspaces of the generalized Spencer cohomology space H^2(δ) of pseudo-projective graded Lie algebras. B.Doubrov showed that the curvature function HK of the Cartan connection takes values in the subspace F^1H^2(δ) of H^2(δ). We otained the generators of subspaces F^1H^2(δ) in case δ is a pseudo-projectiv graded Lie algebra of order k+1 of bidegree (m+1,1).