Description:

Quaternion Heisenberg groups can be referred as subRiemannian manifolds, of which, solving heat equations and characterizing vector fields being conservative are main topics to be explored in this work. Using the Hamiltonian formalism, an integral representation for the heat kernel of a subLaplacian is obtained. Lengths and formulas of geodesics are also calculated for some certain circumstances. On the other hand, a necessary and sufficient condition for a vector field being conservative is deduced. In which case, the corresponding potential function can be written out explicitly.