  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.
