An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball

Abstract

Spheres are known to be rigid geometric objects: they cannot be deformed isometrically, i.e., while preserving the length of curves, in a twice differentiable way. An unexpected result by Nash (Ann Math 60:383–396, 1954) and Kuiper (Indag Math 17:545–555, 1955) shows that this is no longer the case if one requires the deformations to be only continuously differentiable. A remarkable consequence of their result makes possible the isometric reduction of a unit sphere inside an arbitrarily small ball. In particular, if one views the Earth as a round sphere, the theory allows to reduce its diameter to that of a terrestrial globe while preserving geodesic distances. Here, we describe the first explicit construction and visualization of such a reduced sphere. The construction amounts to solve a nonlinear PDE with boundary conditions. The resulting surface consists of two unit spherical caps joined by a \(C^1\) fractal equatorial belt. An intriguing question then arises about the transition between the smooth and the \(C^1\) fractal geometries. We show that this transition is similar to the one observed when connecting a Koch curve to a line segment.

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Notes

  1. 1.

    Very recently, a formal construction of a deformed isometric sphere was obtained by considering isometric extensions [13, Cor. 1.3]. However, one equator is left unchanged in this approach, which prevents the sphere to be globally reduced.

References

  1. 1.

    François Apéry, Models of the real projective plane: computer graphics of Steiner and Boy surfaces, Springer, 1987.

  2. 2.

    Yurii F. Borisov, \({C}^{1,\alpha }\) isometric immersions of Riemannian spaces, Doklady Akademii Nauk SSSR 163 (1965), no. 1, 11.

    MathSciNet  Google Scholar 

  3. 3.

    Yurii F. Borisov, Irregular \({C}^{1,\beta }\) -surfaces with an analytic metric, Siberian Mathematical Journal 45 (2004), no. 1, 19–52.

    MathSciNet  Article  Google Scholar 

  4. 4.

    Vincent Borrelli, Saïd Jabrane, Francis Lazarus, and Boris Thibert, Flat tori in three dimensional space and convex integration, Proceedings of the National Academy of Sciences of the United States of America (PNAS) 109 (2012), no. 19, 7218–7223.

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Vincent Borrelli, Saïd Jabrane, Francis Lazarus, and Boris Thibert, Isometric embeddings of the square flat torus in ambient space, Ensaios Matemáticos 24 (2013), 1–91.

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Werner Boy, Über die curvatura integra und die topologie geschlossener flächen, Mathematische Annalen 57 (1903), no. 2, 151–184.

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Stephan E. Cohn-Vossen, Die verbiegung von flachen im grossen, Fortschr. Math. Wiss 1 (1936), 33–76.

    Google Scholar 

  8. 8.

    Sergio Conti, Camillo De Lellis, and László Székelyhidi Jr, h-principle and rigidity for \({C}^{1,\alpha }\) isometric embeddings, Nonlinear partial differential equations, Abel Symposia, vol. 7, Springer, 2012, pp. 83–116.

  9. 9.

    Yakov Eliashberg and Nikolai Mishachev, Introduction to the \(h\)-principle, Graduate Studies in Mathematics, vol. 48, A.M.S., Providence, 2002.

    Google Scholar 

  10. 10.

    George Francis and John M. Sullivan, Visualizing a Sphere Eversion, IEEE Trans. Vis. and Comp. Graphics 10 (2004), no. 5, 509–515.

    Article  Google Scholar 

  11. 11.

    Mikhail Gromov, Partial differential relations, Springer-Verlag, 1986.

  12. 12.

    Noel J. Hicks, Notes on differential geometry, Math. Studies, Van Nostrand Reinhold, Princeton, NJ, 1965.

    Google Scholar 

  13. 13.

    Norbert Hungerbühler and Micha Wasem, The one-sided isometric extension problem, arXiv preprint arXiv:1410.0232 (2015).

  14. 14.

    Nicolaas H. Kuiper, On \({C}^1\) -isometric imbeddings, Indag. Math. 17 (1955), 545–555.

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Silvio Levy, Making waves. A guide to the ideas behind outside in, AK Peters, Wellesley, MA, 1995.

    Google Scholar 

  16. 16.

    Benoit B Mandelbrot, The fractal geometry of nature, vol. 173, Macmillan, 1983.

  17. 17.

    David Mumford, Caroline Series, and David Wright, Indra’s pearls: the vision of felix klein, Cambridge University Press, 2002.

  18. 18.

    John F. Nash, \({C}^1\) -isometric imbeddings, Ann. of Math. 60 (1954), no. 3, 383–396.

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    David Spring, Convex integration theory, Monographs in Mathematics, vol. 92, Birkhäuser Verlag, 1998.

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Acknowledgements

The authors thank the anonymous referees for their scrutinous and insightful comments.

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Correspondence to Francis Lazarus.

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Dedicated to the memory of David Spring.

This work is part of the Hevea project and was partly supported by the LabEx Persyval-Lab ANR-11-LABX-0025-01. The first author was in internship at the Institut Camille Jordan. The third author was a postdoc financed by the Matstic grant First from University Joseph Fourier and by Laboratoire Jean Kuntzmann. We are also thankful to the Grenoble University High Performance Computing Centre project (Ciment) for providing access to its computing platform.

Communicated by Philippe G. Ciarlet.

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Bartzos, E., Borrelli, V., Denis, R. et al. An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball. Found Comput Math 18, 1015–1042 (2018). https://doi.org/10.1007/s10208-017-9360-1

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Keywords

  • Isometric embedding
  • Convex integration
  • Sphere reduction
  • Boundary conditions

Mathematics Subject Classification

  • Primary 35-04
  • Secondary 53C21
  • 53C23
  • 53C42
  • 57R40