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.
Geometrical structures on the prolongation of a pre-Lie algebroid on fibered manifolds and application to Partial Finsler geometry on foliated anchored bundle
Overdetermined systems, conformal geometry, and the BGG complex
这是IMA暑期课程“对称性和偏微分方程的超定组“的两个系列讲座的扩展版本。本文的主要部分描述了最近与T. P. Branson，M.G.伊斯特伍德和A.R.哥佛首先讨论一个简单的特殊情况，然后一般地描述（黎曼）过程。延拓过程是从简化构造某些几何结构的不变微分算子的Bernstein-Gelfand-Gelfand（BGG）序列导出的。接下来描述用于共形结构的这种构造的版本。最后，我们讨论了推广的延长过程和建设不变的运营商到其他几何结构。
