Stratified Convexity &Concavity of Gradient Flows on Manifolds with Boundary

• 作者: Gabriel Katz
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: MORSE Theory Gradient FLOWS CONVEXITY CONCAVITY MANIFOLDS with Boundary
• 年，卷（期）: 2014,（17）
• 页码范围: 2823-2848
• 页数: 26页
• 分类号: O1

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.