Abstract
We present different types of rotational symmetries for distances in homogeneous groups, showing that the area formula for the associated spherical measure takes a simple form.
Highlights
The problem of computing the spherical measure of submanifolds in homogeneous groups still has a number of questions to be investigated
The anisotropic infinitesimal behavior of the submanifold in many cases leads to new geometric questions. Our motivations have their roots in the wider project to expand results and tools of Geometric Measure Theory in the framework of noncommutative homogeneous groups
A fundamental concept is that of area, that is obtained using the spherical measure with respect to the distance of the group
Summary
The problem of computing the spherical measure of submanifolds in homogeneous groups still has a number of questions to be investigated. Vertical subgroups are the homogeneous tangent spaces at some points of transversal submanifolds, [23] As a result, such rotationally symmetric distances imply a simpler integral form for the spherical measure of transversal submaniofolds, according to Theorem 1.2 below. As a consequence of (1.5) in the case of one codimensional vertical subgroups, combining [9, (4.22)], [9, (4.23)] and [22, Theorem 1.3], we obtain that the spherical measure equals the centered Hausodorff measure of dimension Q − 1 on all G-rectifiable sets This result extends [9, Theorem 4.28] to all homogeneous distances whose metric unit ball is convex. Another application of Theorem 1.4 is for low codimensional H-regular surfaces in Heisenberg groups Hn. When the metric unit ball is convex, formula (1.5) was used in [5] to prove that spherical measure and centered. Similar applications of (1.3) can be obtained for multiradial distances, starting from Theorems 5.3 and 5.5
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