论文标题
几何结构的延长，不变和基本身份
Prolongations, invariants, and fundamental identities of geometric structures
论文作者
论文摘要
在Nilpotent几何形状的框架中，我们为几何结构的等效问题提供了一个统一的方案，该方案扩展并整合了Cartan，Singer-Sternberg，Tanaka和Morimoto的早期作品。通过给出高阶几何结构和通用框架束的新表述，我们重建了歌手 - - - - - - - - - - - - 塞伯格）和田中的步骤延长。然后，我们通过将其扩展到组件中$γ=κ+τ+σ$并建立了$κ$，$τ$，$σ$的基本身份，从而研究了正常几何结构的完整步骤延长的结构函数$γ$。然后，这使我们能够研究一般性的几何结构的等效问题，并将应用程序扩展到不一定是cartan连接的几何结构。 在所有中，我们给出了一种算法，以使用与几何结构符号相关的广义Spencer共同体学组来为任何高阶的常规几何结构构建完整的不变系统系统。然后，我们在无限类型和有限类型的情况下彻底讨论几何结构的等效问题。我们还通过结构函数$τ$对cartan连接进行表征，并清楚在步骤延长的角度将cartan连接放置在何处。
Working in the framework of nilpotent geometry, we give a unified scheme for the equivalence problem of geometric structures which extends and integrates the earlier works by Cartan, Singer-Sternberg, Tanaka, and Morimoto. By giving a new formulation of the higher order geometric structures and the universal frame bundles, we reconstruct the step prolongation of Singer-Sternberg and Tanaka. We then investigate the structure function $γ$ of the complete step prolongation of a normal geometric structure by expanding it into components $γ= κ+ τ+ σ$ and establish the fundamental identities for $κ$, $τ$, $σ$. This then enables us to study the equivalence problem of geometric structures in full generality and to extend applications largely to the geometric structures which have not necessarily Cartan connections. Among all we give an algorithm to construct a complete system of invariants for any higher order normal geometric structure of constant symbol by making use of generalized Spencer cohomology group associated to the symbol of the geometric structure. We then discuss thoroughly the equivalence problem for geometric structure in both cases of infinite and finite type. We also give a characterization of the Cartan connections by means of the structure function $τ$ and make clear where the Cartan connections are placed in the perspective of the step prolongations.
