Geometric analysis on quaternion \(\mathbb H\)-type groups
The authors consider seven examples of \(\mathbb{H}\)-type groups, which are related to the quaternion numbers. Three of them have a one-dimensional center, and actually, are isomorphic to the Heisenberg group \(\mathbb{H}^2\). Three others have a two-dimensional center and the last one has a three dimensional center which is isomorphic to imaginary quaternions. In each case the authors construct the Hamiltonian function associated with the sub-Laplacian. They solve the Hamiltonian system of differential equations and give exact solutions describing the geodesics on the groups. Considering different positions of points, the number of geodesics connecting these points is studied. Kepler's law is proved, that gives a nice geometric interpretation of the length of a part of a horizontal curve belonging to a center of the group and the area of a certain surface. The authors also consider complex geodesics and find a relation between the complex action function and the Carnot-Carathéodory length of geodesics.
scientific article; zbMATH DE number 5015571
scientific article