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

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 ${\mathbb{S}}^3$.

35 pages, 2 figures

References in corpus (3)

• Invariant Carnot-Caratheodory metrics on $S^3$, $SO(3)$, $SL(2)$ and lens spaces
• The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds
• Sub-Riemannian geodesics on the 3-D sphere