GEOMETRY THEORY OF SYSTEMS OF ORDINARY DEFFERENTIAL EQUATIONS

1.本文研究了K（2）阶伪射影系统的几何，并由此导出了一类K阶偏微分方程对合系统的几何。注意这类包括K阶常微分方程组。(1)首先给出了K阶伪射影系统的模符号代数L，并计算了它的延拓G。(2)然后，他证明了每个K阶伪射影系统R都自然地对应于模型空间G/G^<（0）>的一个正规Cartan联络，其中G/G^<（0）>自然地由G构成。(3)进一步研究了K阶标准伪射影系统R_O，并用射影几何刻画了它.头算器也为完全可积系统的Lie-Cartan积分理论提供了严格的证明和理论依据。其中，设一个Cartan系统，它表示流形P上的横截绝对平行性，流形P上有一个完全可积系统E。设g（）表示的无穷小自同构的约化李代数。然后，他严格证明了以下定理由于埃利Cartan：积分E可以进行的平方和积分李的微分方程与简单的组成部分g（）。

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( ).