A classical result in geometry, Frobenius theorem, states that there exist no k-dimensional surface which is tangent to a non-involutive distribution of k-planes.

In this talk I consider some extensions of this statement to weaker notions of surfaces, such as rectifiable sets (and possibly currents).

Following the work of Z. Balogh, I first consider a contact set E, namely a set of points where a k-dimensional surface S is tangent to a distribution of k-planes V, and ask how large E can be, and in particular if it can have positive k-dimensional measure. It turns out that the answer depends on a combination of the regularity of S and of the boundary of E. On the level of proofs, this problem reduces to a question about the (strong) locality of the curl operator in dimension two.

Passing to currents, Frobenius theorem holds for integral currents (a notion which entails a certain regularity of the boundary), but interestingly enough, the answer turns out to be more involved for normal currents.

These results are part of an ongoing research project with Annalisa Massaccesi (University of Padova), Andrea Merlo (University of the Basque Country) and Evgeny Stepanov (Steklov Institute, Saint Petersburg, and Scuola Normale Superiore, Pisa).