Stratified convexity & concavity of gradient flows on manifolds with boundary

As has been observed by Morse \cite{Mo}, any generic vector field $v$ on a compact smooth manifold $X$ with boundary gives rise to a stratification of the boundary $\d X$ by compact submanifolds $\{\d_j^\pm X(v)\}_{1 \leq j \leq \dim(X)}$, where $\textup{codim}(\d_j^\pm X(v))= j$. Our main observation is that this stratification reflects the stratified convexity/concavity of the boundary $\d X$ 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 $v_1$ be the orthogonal projection of $v$ on the tangent bundle of $\d X$. We link the dynamics of the $v_1$-flow on the boundary with the property of $v$ 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.