Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary
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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 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.
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