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Prolongations, invariants, and fundamental identities of geometric structures

Jaehyun Hong, Tohru Morimoto

Abstract

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.

Prolongations, invariants, and fundamental identities of geometric structures

Abstract

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 of the complete step prolongation of a normal geometric structure by expanding it into components and establish the fundamental identities for , , . 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 and make clear where the Cartan connections are placed in the perspective of the step prolongations.
Paper Structure (31 sections, 57 theorems, 250 equations, 2 figures)

This paper contains 31 sections, 57 theorems, 250 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Geometric preliminaries
  3. Algebraic preliminaries
  4. Geometric structures of higher order and universal frame bundles
  5. Geometric structures of order $k$
  6. Completed universal frame bundles
  7. Canonical Pfaff class and canonical Pfaff form
  8. Semi-canonical embeddings
  9. Structure equations and structure functions
  10. $W$-normal Step prolongations
  11. Proper geometric structures
  12. $W$-normal reductions
  13. Proper geometric structure of finite type
  14. Fundamental identities
  15. Invariants
  16. ...and 16 more sections

Key Result

Theorem I

The equivalence problem of the proper geometric structures $Q^{(k)}$ of type $(\mathfrak g_-, G_0. \cdots, G_k)$, reduces to the equivalence problem of the absolute parallelisms $(\mathscr S_W Q^{(k)}, \theta)$ of the step prolongations. Moreover if $\mathfrak g[k]$ is of finite type, that is, the p

Figures (2)

  • Figure 1: $\frak g_3(\frak g_-, \frak g_0,\frak g_1, \frak g_2)$ and ${\bf G}_3(\frak g_-, G_0, G_1, G_2)$
  • Figure 2: Completed universal frame bundles

Theorems & Definitions (106)

  • Theorem I: Theorem \ref{['thm:W normal prolongation']}
  • Theorem II: Theorem \ref{['thm:fundamental identities']}
  • Theorem III: Theorem \ref{['thm:complete system of invariants']}
  • Theorem IV: Theorem \ref{['thm:involutive local equivalence']}
  • Theorem V: Theorem \ref{['thm:vanishing of tau implies']}
  • Theorem VI: Theorem \ref{['thm:Cartan connection']}
  • Proposition 1.1: SS65, T70
  • Theorem 1.1: Theorem of finitude, M88
  • Definition 2.1
  • Proposition 2.1
  • ...and 96 more