Area of images of measurable sets on depth 2 Carnot manifolds with sub-Lorentzian structure

The paper is devoted to analysis of metric properties of images of measurable sets in sub-Lorentzian geometry introduced on Carnot manifolds. The current research continues the results obtained earlier for classes of compact sets on Carnot groups. The main difference is that, firstly, the mapping is defined on a measurable set (not necessarily compact), and, secondly, the preimage and image of the mapping do not have a group structure. Also, the definition of sub-Lorentzian analog of Hausdorff measure (which is not a measure in general) is modified: in contrast to earlier research, it does not require “uniform” sub-Riemannian differentiability. One of results is the property of quasi-additivity of this sub-Lorentzian analog. The latter enables to derive its parameterization by sub-Riemannian Hausdorff measure. In turn, this property means that the sub-Lorentzian analog of Hausdorff measure has classical properties of measure on certain class of sets. The sub-Lorentzian area formula on Carnot manifold is the main result of the paper. We also demonstrate the main ideas of its proof and show their specificity.