Sub-Riemannian metrics and isoperimetric problems in contact case (Q2771536)

There are three ways to study exponential mapping: 1) if \(s\) is the arc length parameter, then the exponential mapping is considered as the Lagrange mapping and is denoted by \(\bar\varepsilon\); 2) for a fixed \(s\) the exponential mapping is considered as the Legendre mapping and is denoted by \(\varepsilon_{s}\); 3) the exponential mapping is considered as suspension of ordinary smooth mapping of two-dimensional manifolds and is denoted by \(\varepsilon^{S}\).NEWLINENEWLINENEWLINEThe effective calculation of the exponential mapping in the three-dimensional case for the contact general sub-Riemannian metrics, for the isoperimetric metrics, and for Dido's case are presented. The normal forms and invariants of contact sub-Riemannian metrics and isoperimetric problems are studied. The investigation of singularities of \(\bar\varepsilon\), \(\varepsilon_{s}\) and \(\varepsilon^{S}\) is presented and, as consequences of this study, the descriptions of sphere and wave fronts with their singularities and caustics with their singularities are obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00042].