Symmetry results for the area formula in homogeneous groups

TL;DR

This paper establishes a link between symmetry properties of the metric unit ball in homogeneous groups and the validity of the standard area formula, also relating two measures under specific geometric conditions.

Contribution

It proves that symmetry of the unit ball implies the standard area formula and shows equality of spherical and centered Hausdorff measures under certain conditions.

Findings

Symmetry of the metric unit ball ensures the standard area formula.

Equality between spherical measure and centered Hausdorff measure is established.

Results apply to both smooth and nonsmooth submanifolds.

Abstract

We prove that if the shape of the metric unit ball in a homogeneous group enjoys a precise symmetry property, then the associated distance yields the standard form of the area formula. The result applies to some classes of smooth and nonsmooth submanifolds. We finally prove the equality between spherical measure and centered Hausdorff measure, under two different geometric conditions on the shape of the metric unit ball.