  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
$\mathbb S^3$.

Differential Geometry;Analysis of PDEs