Prolongation of Tanaka structures: an alternative approach

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 with second-order differential equations, and structures defined by gradings of Lie algebras (in the framework of parabolic geometries). Tanaka’s prolongation procedure associates with 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 equivalence problem. In this paper, we develop an alternative constructive approach for Tanaka’s prolongation procedure, based on the theory of quasi-gradations of filtered vector spaces, G-structures, and their torsion functions.