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Flows in Flatland: A Romance of Few Dimensions

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Abstract

This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on n-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensional worlds to popularize our approach to the Morse theory on smooth manifolds with boundary. In this approach, the boundary effects take the central stage.

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Notes

  1. In Abbott (1992), the drama unfolds in the Euclidean plane, the Flatland, while here we will tell a story that takes place in the 2D-land, a compact Riemannian surface.

  2. The flows and the vector fields here are considered up to the natural action of the diffeomorphism group \(\mathsf {Diff}(X)\) on them.

  3. In the case of annulus, this set is empty.

  4. That is, for an open and dense set in the space of all traversing vector fields on X.

  5. It is possible to have a vector field v for which some trajectories will be cubically tangent to the boundary, but the majority of 2-dimensional vector fields v avoid such cubic tangencies Katz (2014b).

  6. see the paragraph that follows Theorem 1.1.

  7. This excludes the disk and the annulus.

  8. In particular, a traversally generic v is boundary generic.

  9. Even traversally generic

  10. Note that Eq. (6.3) resembles Eq. (6.1).

  11. Considered up to diffeomorphisms

  12. This is equivalent to saying that the \((k-1)\)-st jet at x of \(z|_\gamma \) vanishes, but the k-th jet does not.

  13. In the forthcoming paper we will see that, for the geodesic flows \(v^g\) on the tangent spherical bundle \(SM \rightarrow M\) over a smooth connected Riemannian n-manifold (M, g) with boundary, \(C_{v^g}\) is the scattering map, a version of the billiard map.

  14. Such metrics form an open set in the space of all Riemannian metrics.

  15. This allows for local maxima/minima only and excludes the \(\theta \)-vertical inflections.

  16. Note that the second bullet excludes the triple intersections of \(C_t\).

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Acknowledgements

We are grateful to the referees; their thoughtful and detailed recommendations helped to improve the quality of our presentation.

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    Gabriel Katz

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Katz, G. Flows in Flatland: A Romance of Few Dimensions. Arnold Math J. 3, 281–317 (2017). https://doi.org/10.1007/s40598-016-0059-1

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