On a step \(2(k+1)\) sub-Riemannian manifold
The article deals with the sub-Riemannian structure on \(\mathbb{R}^3= \mathbb{R}^2_x\times \mathbb{R}_t\) \((x= (x_1,x_2))\) determined by the vector fields \[ X_1= \frac{\partial}{\partial x_1}+ 2x_2| x|^{2k}\frac{\partial}{\partial t},\;X_2= \frac{\partial}{\partial x_2}- 2x_1| x|^{2k} {\partial\over\partial t}\quad (k= 0,1,\dots). \] The vector fields \(X_1\), \(X_2\) and their iterated Lie brackets span the tangent space of \(\mathbb{R}^3\) and the number of needful brackets plus 1, called the step of the space, is equal to \(2(k+1)\). By using polar coordinates, the authors solve the Euler-Lagrange system \[ \dot x= r\dot\varphi(\dot\varphi+ 4(k+1)\vartheta r^{2k}),\quad r^2(\dot\varphi+ 2\vartheta r^{2k})= C,\quad \vartheta= D \] for the geodesics, compute the sub-Riemannian distance explicitly and determine the number of geodesics between the origin (then \(C=0\)) and any other point.
scientific article; zbMATH DE number 2070477