GEOMETRY THEORY OF SYSTEMS OF ORDINARY DEFFERENTIAL EQUATIONS

1. 首先研究了k阶伪射影系统的几何，它给出了某一类k阶偏微分方程对合系统的几何，注意这一类包括k阶常微分方程系统。(1)首先确定了k阶伪射影系统的模型符号代数l；(2)证明了每一个k阶伪射影系统r都有一个模型空间g / g ^<(0)>的正构连接，其中g / g ^<(0)>是由g构造而成的。(3)进一步详细研究了k阶标准伪射影系统r_0，并用射影几何对其进行了刻画。作者还给出了完全可积系统的lie-cartan积分理论的严密公式和证明。其中设一个卡坦系统，它表示流形P上具有完全可积系统e的横向绝对平行，设g（）表示的无穷小自同构的简化李代数。然后，他严格地证明了下列定理：E的积分可以通过正交和与g（）的简单分量相关的李氏微分方程的积分来实现。

1. THE HEAD INVESTIGATOR STUDIED THE GEOMETRY OFPSEUDO-PROJECTIVE SYSTEMS OF ORDER K( 2), WHICH FORMULATES THE GEOMETRY OF A CERTAIN CLASS OF INVOLUTIVE SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS OF ORDER K. NOTE THAT THIS CLASS INCLUDES THE CLASS OF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS OF ORDER K.(1) FIRST OF ALL HE DETERMINDE THE MODEL SYMBOL ALGEBRA L OF PSEUDO-PROJECTIVE SYSTEM OF ORDER K, AND CALCULATED ITS PROLONGATION G.(2) THEN HE SHOWED THTA THERE IS NATURALLY ASSOCIATED TO EVERY PSEUDO-PROJECTIVE SYSTEM R OF ORDER K A NORMAL CARTAN CONNECTION OF MODEL SPACE G/G^<(0)>, WHERE G/G^<(0)> IS NATURALLY CONSTRUCTED FROM G.(3) FURTHERMORE HE STUDIED THE STANDARD PSEUDO-PROJECTIVE SYSTEM R_O OF ORDER K IN DETAIL, AND CHARACTERIZED IT IN TERMS OF THE PROJECTIVE GEOMETRY.2. THE HEAD INVESTIGATOR ALSO GAVE A RIGOROUS FORMULATION AS WELL AS PROOF OF LIE-CARTAN'S INTEGRATION THEORY FOR COMPLETELY INTEGRABLE SYSTEMS. AMONG OTHERS LET BE A CARTAN SYSTEM, WHICH MEANS A TRANSVERSE ABSOLUTE PARALLELISM ON A MANIFOLD P WITH A COMPLETELY INTEGRABLE SYSTEM E. LET g() DENOTE THE REDUCED LIE ALGEBRA OF INFINITESIMAL AUTOMORPHISMS OF . THEN HE RIGOROUSLY PROVED THE FOLLOWING THEOREM DUE TO ELIE CARTAN: THE INTEGRATION OF E CAN BE CARRIED OUT BY QUADRACTURES AND THE INTEGRATIONS OF LIE'S DIFFERENTIAL EQUATINS ASSOCIATED WITH THE SIMPLE COMPONENTS OF g( ).