Explicit formulae for geodesics in left invariant sub-Finsler problems on Heisenberg groups via convex trigonometry

Abstract

In the present paper, we obtain explicit formulae for geodesics in some left-invariant sub-Finsler problems on Heisenberg groups ${\mathbb{H}}_{2n+1}$. Our main assumption is the following: the compact convex set of unit velocities at identity admits a generalization of spherical coordinates. This includes convex hulls and sums of coordinate 2-dimensional sets, all left-invariant sub-Riemannian structures on ${\mathbb{H}}_{2n+1}$, and unit balls in $L_p$-metric for $1\le p\le \infty$. In the last case, extremals are obtained in terms of incomplete Euler integral of the first kind.

arXiv admin note: text overlap with arXiv:2004.10194