The Shortest Polygonal Chains in the Heisenberg Group

We describe the shortest polygonal chains that connect two points on the first Heisenberg group with the sub-Riemannian structure. The shortest polygonal chain connecting two points with a fixed number of links either is a straight line or consists of segments of the same length such that the projections of their endpoints are inscribed in a circle. The analytical description is obtained for the spheres of the quasimetric generated by the shortest polygonal chains with three links.