Hamilton–Jacobi theory and the heat kernel on Heisenberg groups

Real and Complex Hamiltonian Mechanics on Some Subriemannian Manifolds

“…Section 7 is devoted to complex Hamiltonian mechanics. In particular, we show that the critical points of the modified complex action function f (τ ) yield the lengths of the geodesics starting from the origin; in the step 2 case this recovers results of [1].…”

“…Consider the subRiemannian geometry induced by the vector fields (1). In this section we shall prove the following global connectivity result: By Chow's theorem [7], arbitrary points P and Q can be joined by a piece-wise horizontal curve.…”

“…We mention some work relevant to this paper. Beals, Gaveau and Greiner [1] study the subRiemannian geodesics on the Heisenberg group H n . In particular, they show that for isotropic H n the t-axis is the set of conjugate points for geodesics starting from the origin.…”

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

“…This contrasts the case of non-compact sub-Riemannian manifolds considered earlier in, e.g., [6,8,9,18], where the authors had to avoid nonintegrable singularities introducing complex modified action.…”

The geometry on a step 3 Grushin model

“…This is no longer true in the sub-elliptic case. A very careful study of the subRiemannian geometry on Heisenberg groups [6] shows that every point in the center of the group is connected to the origin by an infinite number of geodesics of different lengths. A similar situation happens in some other cases, see e.g., [8], [9], and [10].…”