Geometric mechanics on product Heisenberg groups

Abstract

In this article, we use Hamiltonian and Lagrangian formalisms to give a detailed discussion of sub-Riemannian geometry, which arose from the sub-Laplacian on the product of Heisenberg groups. In particular, we calculate the sub-Riemannian distances along the geodesics. We also find the complex action function and the volume element on the product group. Using this action function and the volume element, we obtain the fundamental solution and the heat kernel for the operator .

Keywords:

• Heisenberg groups
• sub-Laplacian
• Hamilton–Jacobi equation
• geodesics
• complex action function
• heat kernel

AMS Subject Classifications::

• 53C17
• 53C22
• 35H20

Acknowledgements

D.-C. Chang and J. Hu are partially supported by a Hong Kong RGC competitive earmarked research grant #600607. D.-C. Chang is also partially supported by a competitive research grant from Georgetown University.