Bridges Between Subriemannian Geometry and Algebraic Geometry

TL;DR

This paper explores the deep connections between nonholonomic geometry and algebraic geometry, introducing new regularization theorems, topological invariants, and experimental links between discrete invariants and algebraic curve properties.

Contribution

It establishes novel bridges between nonholonomic systems and algebraic geometry, including a regularization theorem and topological invariant computations.

Findings

A regularization theorem for curves in nonholonomic geometry.

Computation of topological invariants for classifying spaces.

Experimental links between discrete invariants and algebraic curve properties.

Abstract

We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.