Vector distributions with very large symmetries via rational normal curves

TL;DR

This paper constructs specific rank 3 distributions on high-dimensional manifolds with exponentially large symmetry groups, contrasting with classical geometries, and relates these models to symplectic flag curves and rational normal curves.

Contribution

It introduces new distributions with very large symmetry groups and analyzes their geometric properties, especially their relation to rational normal curves and symplectic flags.

Findings

Symmetry group dimension grows exponentially with dimension n.

Maximal order of weighted jet grows quadratically in n.

Models differ from parabolic geometries in symmetry growth and jet order.

Abstract

We construct a sequence of rank 3 distributions on $n$-dimensional manifolds for any $n\geq 7$ such that the dimension of their symmetry group grows exponentially in $n$ (more precisely it is equal to $\operatorname{Fib}_{n-1}+n+2$, where $\operatorname{Fib}_n$ is the $n$-th Fibonacci number, starting with $\operatorname{Fib}_1=\operatorname{Fib}_2=1$) and such that the maximal order of weighted jet needed to determine these symmetries grows quadratically in $n$. These examples are in sharp contrast with the parabolic geometries where the dimension of a symmetry group grows polynomially with respect to the dimension of the ambient manifold and the corresponding maximal order of weighted jet space is equal to the degree of nonholonomy of the underlying distribution plus $1$. Our models are closely related to the geometry of certain curves of symplectic flags and of the rational normal curves.