Sub-Riemannian coarea formula for classes of noncontact mappings of Carnot groups
type: Item
instance of: scholarly article
title: Sub-Riemannian coarea formula for classes of noncontact mappings of Carnot groups
author: M. B. Karmanova
published in: Mathematical Notes
publication date: 2024-07-09T00:00:00Z
review text: 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.
reviewed by: M. A. Malakhaltsev
Mathematics Subject Classification ID: 53C17
zbMATH DE Number: 7878665
zbMATH Keywords: Carnot groups
zbMATH Keywords: coarea factor
zbMATH Keywords: non-contact mappings
zbMATH Keywords: Hausdorff measure
MaRDI profile type: Publication
cites work: On the approximability and parametrization of preimages of elements of Carnot groups on sub-Lorentzian structures
cites work: A coarea formula for smooth contact mappings of Carnot-Carathéodory spaces
cites work: Foundations for the theory of quasiconformal mappings on the Heisenberg group
cites work: Contact and quasiconformal mappings on real model filiform groups
cites work: Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
cites work: Q3517273
cites work: Rectifiability and perimeter in the Heisenberg group
cites work: Усреднения компактных отображений группы Энгеля
cites work: Q6062755
cites work: Hardy Spaces on Homogeneous Groups. (MN-28), Volume 28
cites work: Q4812071
DOI: 10.1134/S0001434624030167
description: scientific article; zbMATH DE number 7878665