GEOMETRY THEORY OF SYSTEMS OF ORDINARY DEFFERENTIAL EQUATIONS

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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( ).