Prolongation of $(8,15)$-Distribution of Type $F_4$ by Singular Curves

TL;DR

This paper investigates the singular curves of Cartan's $(8,15)$-distribution model associated with the exceptional Lie algebra $F_4$, and constructs its prolongation revealing a nilpotent graded Lie algebra related to $F_4$.

Contribution

It introduces a prolongation method for Cartan's $(8,15)$-distribution model based on singular curves, linking it to the graded Lie algebra of $F_4$.

Findings

Constructed the prolongation of Cartan's model.

Identified the nilpotent graded Lie algebra isomorphic to the negative part of $F_4$.

Analyzed abnormal extremals within sub-Riemannian geometry.

Abstract

Cartan gives the model of $(8, 15)$-distribution with the exceptional simple Lie algebra $F_4$ as its symmetry algebra in his paper (1893), which is published one year before his thesis. In the present paper, we study abnormal extremals (singular curves) of Cartan's model from viewpoints of sub-Riemannian geometry and geometric control theory.Then we construct the prolongation of Cartan's model based on the data related to its singular curves, and obtain the nilpotent graded Lie algebra which is isomorphic to the negative part of the graded Lie algebra $F_4$.