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

在这个项目中，我们研究了有限型微分方程组的几何不变量。（k+1）-阶可积型完整微分方程系统（Χ）定义了k-射流空间j^k上的开子流形R和R上的微分系统E。（R； E，F）在R上的k+1阶双度（m+1,1）伪投影结构与（Χ）相关联的符号代数是一个k+1阶双度（n， R）的伪投影基本渐变李代数。N.Tanaka证明了自然存在与e相关的G/G^<(0)>型的正规Cartan连接（P,ω），并且曲率函数K （P,ω）的调和部分HK生成了Cartan连接（P,ω）的基本不变量，它在第二广义Spencer上同调空间H^2（δ）中取值。因此我们需要研究空间H^2（δ）首先得到了伪射影梯度李代数的广义Spencer上同调空间H^2（δ）的非消失子空间。B.Doubrov证明了Cartan连接的曲率函数HK在H^2（δ）的子空间F^1H^2（δ）中取值。当δ是二阶（m+1,1）的k+1阶伪投影渐变李代数时，我们得到了子空间F^1H^2（δ）的生成子空间。

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).