Classes of Noncontact Mappings of Carnot Groups and Metric Properties

Abstract

We study the metric properties of level surfaces for classes of smooth noncontact mappings from arbitrary Carnot groups into two-step ones with some constraints on the dimensions of horizontal subbundles and the subbundles corresponding to degree 2 fields. We calculate the Hausdorff dimension of the level surfaces with respect to the sub-Riemannian quasimetric and derive an analytical relation between the Hausdorff measures for the sub-Riemannian quasimetric and the Riemannian metric. As application, we establish a new form of coarea formula, also proving that the new coarea factor is well defined.