群作用下子流形的微分不变量和Monge-Taylor形式

Differential invariants and the Monge-Taylor forms for submanifolds under group actions

通过对等价活动标架方法及群作用下无穷小生成子的高阶延拓和微分不变量全局递推公式的研究,本文给出可自动获取无穷小生成子高阶延拓的定理,并基于符号计算及确定的规范化微分不变量基本集,给出可自动构造无穷多高阶规范化微分不变量的精确递推公式及描述高阶微分不变量之间关系的无穷多syzygies的系统化方法.最后,基于所获高阶微分不变量,构造了群作用下一般子流形的显式Monge-Taylor形式.

Motivated by the equivariant moving frame method and starting from the study about the prolonga- tion of the infinitesimal generators of transformation groups and the universal recurrence formulae for differential invariants, we present some theories which make it possible for the systematic determination of the higher order prolongation of the infinitesimal generators. Farthermore, based on the symbolic computation and the fundamen- tal sets of the determined normalized differential invariants, we give the exact recurrence formulae to construct the infinitely many higher order normalized differential invariants and syzygies which show the relationships be- tween them. In addition, by the obtained normalized differential invariants, the explicit Monge-Taylor forms are obtained extensively for submanifolds under group actions.