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

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 $\mathcal{D}$.