Seminar - Nikolay Shcherbina (G.15.25)
Title. On integrability of non-smooth distributions.
Abstract. A classical theorem of Frobenius gives a necessary and sufficient condition for the existence (and uniqueness) of integral surfaces for smooth distributions. If the given distribution is not smooth, but just continuous, then the Lie brakets of the vector fields generating the destribution are not defined and, moreover, even if the integral surface exists, it will be not unique in general (like in the classical theorem of Peano for the case of continuous vector fields).
In our talk we present a necessary and sufficient condition for the existence (but not uniqueness) of the integral surface in the case of continuous distributions of hyperplanes. In the case of distributions of higher codimension such kind of results are unknown and the question of finding of proper necessary and sufficient conditions for the existence of integral surfaces is (up to my knowledge) completely open.
This result is not recent, but (on my opinion) nice and elementary. Besides of that there are interesting open questions related to it. That is why I have decided to discuss it on our seminar.
