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Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary

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DOI: 10.4236/am.2014.517270    4,436 Downloads   4,782 Views   Citations
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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 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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Katz, G. (2014) Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary. Applied Mathematics, 5, 2823-2848. doi: 10.4236/am.2014.517270.

References

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