Prolongation of Tanaka structures: an alternative approach

TL;DR

This paper introduces an alternative constructive method for Tanaka's prolongation of geometric structures, leveraging quasi-gradations, G-structures, and torsion functions to enhance understanding of invariants and automorphisms.

Contribution

It develops a new approach to Tanaka's prolongation procedure using quasi-gradations, providing a potentially more explicit and constructive framework.

Findings

Provides a new constructive method for Tanaka prolongation

Enhances the analysis of local invariants and automorphisms

Applicable to a broad class of geometric structures

Abstract

The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated to second order differential equations and structures defined by gradings of Lie algebras (in the setting of parabolic geometry). Tanaka's prolongation procedure associates to a Tanaka structure of finite order a manifold with an absolute parallelism. It is a very fruitful method for the description of local invariants, investigation of the automorphism group and the equivalence problem. In this paper we develop an alternative constructive approach for Tanaka's prolongation procedure, based on the theory of quasi-gradations in filtered vector spaces, G-structures and their torsion functions.