As has been observed by Morse [1], any generic vector field
v on a compact smooth manifold
X with boundary gives rise to a stratification of the boundary
by compact submanifolds
, where
. Our main observation is that this stratification re-flects the stratified convexity/concavity of the boundary
with respect to the
v-flow. We study the behavior of this stratification under deformations of the vector field
v. We also investigate the restrictions that the existence of a convex/concave traversing
v-flow imposes on the topology of
X. Let
be the orthogonal projection of
on the tangent bundle of
. We link the dynamics of the
on the boundary with the property of
in
X being convex/concave. This linkage is an instance of more general phenomenon that we call “holography of traversing fields”—a subject of a different paper to follow.