Involutivity of distributions at points of superdense tangency with respect to normal currents

TL;DR

This paper proves that a $C^1$ distribution of planes must be involutive at almost every superdensity point where it is tangent to a normal current, revealing a geometric regularity condition.

Contribution

It establishes a new involutivity property of distributions at superdensity tangent points with respect to normal currents.

Findings

Involutivity holds at almost every superdensity tangent point.

The result links geometric measure theory with classical involutivity conditions.

Provides conditions under which distributions exhibit involutive behavior at tangent points.

Abstract

Let ${\mathcal D}$ and $T$ be, respectively, a $C^1$ distribution of $k$-planes and a normal $k$-current on ${\mathbf R}^n$. Then ${\mathcal D}$ has to be involutive at almost every superdensity point of the tangency set of $T$ with respect to ${\mathcal D}$.