TITLE:
Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary
AUTHORS:
Gabriel Katz
KEYWORDS:
Morse Theory, Gradient Flows, Convexity, Concavity, Manifolds with Boundary
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.17,
October
29,
2014
ABSTRACT: 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 theon 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.