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

Chang, Der-Chen; Markina, Irina; Vasil'ev, Alexander

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

Authors

Chang, Der-Chen

Markina, Irina

Vasil'ev, Alexander

Description

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