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

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