# Canonical bundles of moving frames for parametrized curves in Lagrangian Grassmannians: algebraic approach

## TL;DR

This paper develops a detailed algebraic method for constructing canonical moving frames and differential invariants for parametrized curves in Lagrangian Grassmannians, connecting classical and modern geometric theories.

## Contribution

It provides a detailed algebraic approach to canonical bundles of moving frames in Lagrangian Grassmannians, clarifying normalization conditions and linking to generalized flag variety theory.

## Key findings

- Constructs canonical bundles of moving frames for parametrized curves in Lagrangian Grassmannians.
- Clarifies the origin of normalization conditions in the construction.
- Connects classical results with modern geometric theory of flag varieties.

## Abstract

The aim of these notes is to describe how to construct canonical bundles of moving frames and differential invariants for parametrized curves in Lagrangian Grassmannians, at least in the monotonic case. Such curves appear as Jacobi curves of sub-Riemannian extremals [1,2] . Originally this construction was done in [6,7], where it uses the specifics of Lagrangian Grassmannian. In later works [3,4] a much more general theory for construction of canonical bundles of moving frames for parametrized or unparametrized curves in the so-called generalized flag varieties was developed, so that the problem which is discussed here can be considered as a particular case of this general theory. Although this was briefly discussed at the very end of [3], the application of the theory of [3,4] to obtain the results of [6,7] were never written in detail and this is our goal here. We believe that this exposition gives a more conceptual point of view on the original results of [6,7] and especially clarifies the origin of the normalization conditions of the canonical bundles of moving frames there, which in fact boil down to a choice of a complement to a certain subspace of the symplectic Lie algebra. The notes in almost the same form are included under my authorship as the Appendix in the book "A Comprehensive Introduction to sub-Riemannian Geometry from Hamiltonian view point" by A. Agrachev D. Barilari, U. Boscain. Following the style of this book, some not difficult statements are given as exercises for pedagogical purposes.