Calculus of Variations and Geometric Measure Theory

G. Alberti - A. Massaccesi - E. Stepanov

On the geometric structure of currents tangent to smooth distributions

created by massaccesi on 17 Jul 2019
modified by alberti on 30 Nov 2022


Published Paper

Inserted: 17 jul 2019
Last Updated: 30 nov 2022

Journal: J. Diff. Geom.
Volume: 122
Number: 1
Pages: 1-33
Year: 2022
Doi: 10.4310/jdg/1668186786

ArXiv: 1907.07456 PDF


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.

Keywords: non-involutive distributions, Frobenius theorem, integral and normal currents, geometric property of the boundary