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The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows


This paper describes a mechanism by which a traversally generic flow v on a smooth connected \((n+1)\)-dimensional manifold X with boundary produces a compact n-dimensional CW-complex \({\mathcal {T}}(v)\), which is homotopy equivalent to X and such that X embeds in \({\mathcal {T}}(v)\times \mathbb R\). The CW-complex \(\mathcal T(v)\) captures some residual information about the smooth structure on X (such as the stable tangent bundle of X). Moreover, \({\mathcal {T}}(v)\) is obtained from a simplicial origami map\(O: D^n \rightarrow {\mathcal {T}}(v)\), whose source space is a ball \(D^n \subset \partial X\). The fibers of O have the cardinality \((n+1)\) at most. The knowledge of the map O, together with the restriction to \(D^n\) of a Lyapunov function \(f: X \rightarrow \mathbb R\) for v, make it possible to reconstruct the topological type of the pair \((X, {\mathcal {F}}(v))\), were \({\mathcal {F}}(v)\) is the 1-foliation, generated by v. This fact motivates the use of the word “holography” in the title. In a qualitative formulation of the holography principle, for a massive class of ODE’s on a given compact manifold X, the solutions of the appropriately staged boundary value problems are topologically rigid.

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Fig. 1
Fig. 2


  1. i.e., compact 1-dimensional CW-complexes.

  2. In fact, we may assume that \(D^n \subset \partial X\).

  3. \(C^\infty ({\mathcal {T}}(v))\) is just a subalgebra of \(C^\infty (X)\), not an ideal.

  4. that is induced by the origami map \(\Gamma : D^n \rightarrow {\mathcal {T}}(v)\)

  5. see Definition 3.1.

  6. By Theorem 4.1, such vector field exists.


  1. Besson, G., Courtois, G., Gallaot, G.: Minimal entropy and Mostow’s rigidity theorems. Ergod. Theory Dyn. Syst. 16, 623–649 (1996)

    MathSciNet  Article  Google Scholar 

  2. Croke, C.: Scattering rigidity with trapped geogesics, arXiv:1103.5511v2 [mathDG] 21 Nov 2012

  3. Croke, C.: Rigidity theorems in riemannian geometry, chapter in geometric methods in inverse problems and PDE control. The IMA Volumes in Mathematics and its Applications, p. 137. Springer, Berlin (2004)

    Google Scholar 

  4. Croke, C., Eberlein, P., Kleiner, B.: Conjugacy and rigidity for nonpositively curved manifolds of higher rank. Topology 35, 273–286 (1996)

    MathSciNet  Article  Google Scholar 

  5. Fenn, R., Rourke, C.: Nice spines of 3-manifolds, Topology of low-dimensional manifolds, Lecture Notes in Mathematics, no. 722, 31–36

  6. Gilman, D., Rolfsen, D.: Manifolds and their special spines. Contemp. Math. 20, 145–151 (1983)

    MathSciNet  Article  Google Scholar 

  7. Gromov, M.: Singularities, expanders and topology of maps. part I: homology versus volume in the spaces of cycles. Geom. Funct. Anal. 19, 743–841 (2009)

    MathSciNet  Article  Google Scholar 

  8. Katz, G.: Convexity of Morse stratifications and gradient spines of 3-manifolds. JP J. Geom.Topol. 9(1), 1–119 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Katz, G.: Stratified convexity & concavity of gradient flows on manifolds with boundary. Appl. Math. 5, 2823–2848 (2014)

  10. Katz, G.: Traversally generic & versal flows: semi-algebraic models of tangency to the boundary. Asian J. Math., vol. 21, No. 1, 127–168 (2017) (arXiv:1407.1345v1 [mathGT] 4 July, 2014))

  11. Katz, G.: The stratified spaces of real polynomials & trajectory spaces of traversing flows, JP J. Geom. Topol. vol. 19 No. 2 , 95–160 (2016) (arXiv:1407.2984v3 [mathGT] 6 Aug 2014)

  12. Katz, G.: Causal holography of traversing flows, arXiv:1409.0588v1 [mathGT] (2 Sep 2014)

  13. Katz G.: Causal holography in application to the inverse scattering problem. Inverse Problems Imaging J. 13(3), 597–633 June 2019 (arXiv:1703.08874v1 [Math.GT], 27 Mar 2017)

  14. Katz, G.: Holography and Homology of Traversing Flows. This monograph will be published by World Scientific Co. (Singapore)

  15. Katz, G.: Complexity of shadows and traversing flows in terms of the simplicial volume. J. Topol. Anal. 8(3), 501–543 (2016)

    MathSciNet  Article  Google Scholar 

  16. Noether, E., Der, : Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen 28–35 (1926)

  17. Stefanov, P.G., Uhlmann, G.: Boundary rigidity and stability for generic simple metrics. J. Am. Math. Soc 18(4), 975–1003 (2005)

    MathSciNet  Article  Google Scholar 

  18. Stefanov, P., G. Uhlmann, G.: Boundary and lens rigidity, tensor holography, and analytic microlocal analysis, In: Algebraic Analysis of Differential Equations. Springer, Cham (2008)

  19. Stefanov, P.G., Uhlmann, G.: Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds. J. Differ. Geom. 82(2), 383409 (2009)

    MathSciNet  Article  Google Scholar 

  20. Stefanov, P.G., Uhlmann, G., Vasy, A.: Rigidity with partial data. J. Am. Math. Soc. 29, 299–332 (2016).

    MathSciNet  Article  Google Scholar 

  21. Stefanov, P., G. Uhlmann, G., Vasy, A.: Inverting the local geodesic \(X\)-ray transform on tensors, Journal d’Analyse Mathematique, to appear, arXiv:1410.5145

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The author is grateful to the referee for advising to make the original text more self-contained and for encouraging the author to explain better the motivation behind the results.

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Correspondence to Gabriel Katz.

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Katz, G. The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows. Qual. Theory Dyn. Syst. 19, 41 (2020).

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  • Manifolds with boundary
  • Traversing flows
  • Trajectory spaces
  • Boundary value problems
  • Holography