# On some geometric properties of currents and Frobenius theorem

## TL;DR

This paper explores the geometric structure of currents and their relation to Frobenius theorem, showing that certain currents cannot be tangent to non-involutive distributions and can be foliated when tangent to involutive ones.

## Contribution

It establishes new connections between the theory of currents and Frobenius theorem, providing partial answers to longstanding questions in geometric measure theory.

## Key findings

- Integral currents cannot be tangent to nowhere involutive distributions.
- Normal currents tangent to involutive distributions can be locally foliated.
- Partial resolution of a question by Frank Morgan.

## Abstract

In this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current which is tangent to an involutive distribution of planes can be locally foliated in terms of integral currents (Theorem 4.3). This statement gives a partial answer to a question raised by Frank Morgan in [1].