Sub-Riemannian coarea formula for classes of noncontact mappings of Carnot groups
A Carnot manifold is a smooth manifold \(M\) with a distribution \(HM \subset TM\) which generates \(TM\) via the Lie bracket:\N\[\NHM = H_1 \subsetneq H_2 \subsetneq H_3 \subsetneq \cdots \subsetneq H_k = TM,\]\Nwith \(H_k =\mathrm{span}\{H_{k-1}, [H_1, H_{k-1}]\}\),\Nwhere \(k\) is called the depth of the Carnot manifold. The author describes the metric properties of a class of level sets of non-contact mappings from a Carnot group of arbitrary depth to a two-step Carnot group. In [Math. Notes 111, No. 1, 152--156 (2022; Zbl 1494.53037); translation from Mat. Zametki 111, No. 1, 140--144 (2022)] she described the metric properties of level sets of contact mappings of sub-Lorentzian structures. However the author writes that the construction of nontrivial examples of contact mappings is an independent difficult problem, due to severe restrictions on the structure of the differential, i.e., the images of horizontal fields must be only either horizontal or degenerate. In turn, there exist quite a few examples of non-contact mappings. Therefore, the following question arises: Is it possible to extract some ``sub-Riemannian'' properties for such mappings? In the present paper, the author solves this problem for the model case of classes of mappings of Carnot groups.
scientific article; zbMATH DE number 7878665