On the approximability and parametrization of preimages of elements of Carnot groups on sub-Lorentzian structures (Q2113410)

In this paper the level sets of mappings from one Carnot group to another are studied in the case where the preimages are equipped with a multidimensional sub-Lorentzian structure. Also, a new problem is solved, concerning the accuracy of the order of tangency between a surface and a plane, which is sufficient for both classical and sub-Riemannian approximations. The basic result of the study is the equivalence of such approximations: it is established that, in the case of contact mappings of class \(C^1\), the tangent plane adequately approximates the preimage of an element with respect to the sub-Riemannian metric. A consequence of this result is the derivation of explicit analytical expressions for sub-Riemannian and Riemannian measures of the intersection of a sub-Lorentzian ball and the preimage of an element of a Carnot group.