Asian Journal of Mathematics

Volume 14 (2010)

Number 4

Modified Action and Differential Operators on the 3-D Sub-Riemannian Sphere

Pages: 439 – 474

DOI: http://dx.doi.org/10.4310/AJM.2010.v14.n4.a1

Authors

Der-Chen Chang

Irina Markina

Alexander Vasil'ev

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 $S^3$. 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 $S^3$.

Keywords

Sub-Riemannian geometry; action; sub-Laplacian; heat kernel; geodesic; Hamiltonian system; optimal control

2010 Mathematics Subject Classification

Primary 53C17. Secondary 70H05.

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