Modified action and differential operators on the 3-D sub-Riemannian sphere

TL;DR

This paper derives a geometrically meaningful formula for fundamental solutions to sub-elliptic differential equations and heat equations on the 3-D sub-Riemannian sphere, using a Hamiltonian approach and a new invariant distance.

Contribution

It introduces a closed-form, sub-Riemannian invariant modified action that serves as a distance on the sphere, advancing the understanding of sub-elliptic operators in this geometry.

Findings

Derived fundamental solutions for sub-elliptic equations on $ ext{S}^3$

Provided a closed-form expression for the modified action

Established a sub-Riemannian invariant distance on $ ext{S}^3$

Abstract

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approach, where the corresponding Hamitonian system is solved with mixed boundary conditions. A closed form of the modified action is given. It is a sub-Riemannian invariant and plays the role of a distance on $\mathbb S^3$.