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

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

2008-09-16

arxiv.org/abs/0809.2735v1

Mathematics

Differential Geometry

35 pages, 2 figures

Scientific paper

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 $\mathbb S^3$.

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