Optimal estimates of the number of links of basis horizontal broken lines for 2-step Carnot groups with horizontal distribution of corank 1

A. V. Greshnov ; R. I. Zhukov

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 147, pp. 244-254

Résumé

For a 2-step Carnot group $\Bbb D_n,$ $\dim\Bbb D_n=n+1,$ with horizontal distribution of corank 1, we proved that the minimal number $N_{\mathcal{X}_{\Bbb D_n}}$ such that any two points $u,v\in\Bbb D_n$ can be joined by some basis horizontal $k$-broken line (i.e. a broken line consisting of $k$ links) $L^{\mathcal{X}_{\Bbb D_n}}_k(u,v),$ $k\leq N_{\mathcal{X}_{\Bbb D_n}},$ does not exeed $n+2.$ The examples of $\Bbb D_n$ such that $N_{\mathcal{X}_{\Bbb D_n}}=n+i,$ $i=1,2,$ were found. Here $\mathcal{X}_{\Bbb D_n}=\{X_1,\ldots,X_n\}$ is the set of left invariant basis horizontal vector fields of the Lie algebra of the group $\Bbb D_n,$ and every link of $L^{\mathcal{X}_{\Bbb D_n}}_k(u,v)$ has the form $\exp(asX_i)(w),$ $s\in[0,s_0],$ $a=const.$

Détail

Keywords: horizontal curves, broken lines, Rashevskii–Chow theorem, $2$-step Carnot groups, basis vector fields