Asian Journal of Mathematics
Volume 14 (2010)
Modified Action and Differential Operators on the 3-D Sub-Riemannian Sphere
Pages: 439 – 474
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 $S^3$. Our method is based on the Hamiltonian-Jacobi 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 $S^3$.
Sub-Riemannian geometry; action; sub-Laplacian; heat kernel; geodesic; Hamiltonian system; optimal control
2010 Mathematics Subject Classification
Primary 53C17. Secondary 70H05.