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

2008/09/16 by Chang, Der-Chen, Markina, Irina, Vasil'ev, Alexander
#53C17 #70H05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0809.2735

Abstract

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 S³. 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 S³.

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