A smooth distribution of rank 
on a smooth manifold 
assigns an 
-dimensional linear subspace 
of the tangent
space 
to every point 
,
with a local basis of 
smooth vector fields near
each point. Thus it specifies the allowed tangent directions of horizontal
curves (Wang and Zelenko 2026).
The distribution is bracket-generating if its local vector fields and their iterated Lie brackets span the full tangent space at every point. Constancy of the dimensions of the successive bracket spans is the equiregularity condition.
