The left-invariant sub-Riemannian problem on the Heisenberg group is regarded as a cornerstone of sub-Riemannian geometry. It can be formulated as the fastest-time problem with a circle being the set of control parameters. The talk will be devoted to its natural variation, the problem of the slowest time with a hyperbola as a set of control parameters. This variation is the left-invariant sub-Lorentzian problem on the Heisenberg group.
We plan to present the following results in the talk:
1) reachable sets on the group,
2) Pontryagin's maximum principle, parametrization of extremal trajectories by hyperbolic functions, exponential mapping,
3) the diffeomorphism of the exponential mapping, its inversion,
4) optimality of extremal trajectories, optimal synthesis,
5) sub-Lorentz distance: explicit formula, symmetries,
6) sub-Lorentzian spheres of positive and zero radii.
Results 1) and 2) were obtained by M.Grochowski (2006), while other results are new.
We will also discuss open problems.
This is a joint work with E.F. Sachkova.
The talk will be via Zoom at:
https://utoronto.zoom.us/j/99576627828
Passcode: 448487
Note the unusual start time: 12:30pm.