On geometry of curves of flags of constant type

TL;DR

This paper develops an algebraic framework for the geometry of curves of flags of constant type, generalizing Cartan's method and Tanaka prolongation, applicable to classical groups and their geometries.

Contribution

It introduces an algebraic approach to the geometry of flag curves, extending Cartan and Tanaka methods to a broader class of group actions and geometries.

Findings

Constructs canonical bundles of moving frames for flag curves.

Describes the algebraic structure of the geometry of curves of flags.

Applies to classical groups, including projective and affine geometries.

Abstract

We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$. Under some natural assumptions on the subgroup $G$ and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure Linear Algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.