Abstract:
After a brief overview of the Heisenberg group and its sub-Lorentzian structure, I will give a description of how to compute the geodesics of this space by means of an isoperimetric problem in the Minkowski plane. Such problem is completely analogous to Dido's problem in the Riemannian setting and allows us to find geodesics without relying on the Pontryagin's Maximum Principle.
I will also discuss (time permitting) how one can use the expression of the geodesics given by the exponential map to disprove the validity of the condition for any choice of dimension . Such phenomenon does not occur in the Riemannian Heisenberg group and thus provides a relatively well-behaved example of a geometric property which holds in the Riemannian setting but does not in its Lorentzian counterpart.