Stratified Convexity &Concavity of Gradient Flows on Manifolds with Boundary

作     者：Gabriel Katz 

作者机构：5 Bridle Path Circle Framingham MA USA 

出 版 物：《Applied Mathematics》 (应用数学（英文）)

年 卷 期：2014年第5卷第17期

页      面：2823-2848页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Morse Theory Gradient Flows Convexity Concavity Manifolds with Boundary 

摘      要：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.