Measure of submanifolds in the Engel group

TL;DR

This paper characterizes intrinsic measures of smooth submanifolds in the Engel group, proving their equivalence to spherical Hausdorff measures, using a novel blow-up approach to negligibility.

Contribution

It introduces a new method based on blow-up techniques to analyze the equivalence of intrinsic measures and Hausdorff measures in the Engel group.

Findings

Intrinsic measures are equivalent to spherical Hausdorff measures on submanifolds.

The degree of the submanifold determines the measure.

A new approach to negligibility via blow-up techniques was developed.

Abstract

We find all intrinsic measures of $C^{1,1}$ smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding $d$-dimensional spherical Hausdorff measure restricted to the submanifold. The integer $d$ is the degree of the submanifold. These results follow from a different approach to negligibility, based on a blow-up technique.