# On the geometric structure of currents tangent to smooth distributions

## TL;DR

This paper explores how integral and normal currents relate to smooth distributions, revealing that integral currents behave like smooth surfaces while normal currents exhibit more complex behaviors, especially regarding their boundaries.

## Contribution

It extends classical geometric results to currents, showing integral currents mirror smooth surfaces and analyzing the boundary properties of normal currents.

## Key findings

- Integral currents behave like smooth surfaces in relation to distributions.
- Normal currents have multifaceted behaviors, differing from smooth surfaces.
- Boundary properties of currents are crucial to understanding their geometric behavior.

## Abstract

It is well known that a k-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of k-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces, namely integral and normal currents. We find out that integral currents behave to this regard exactly as smooth surfaces, while the behaviour of normal currents is rather multifaceted. This issue is strictly related to a geometric property of the boundary of currents, which is also discussed in details.