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Flat tori in three-dimensional space and convex integration

Vincent Borrelli, Saïd Jabrane, Francis Lazarus, and Boris Thibert
PNAS May 8, 2012 109 (19) 7218-7223; https://doi.org/10.1073/pnas.1118478109
Vincent Borrelli
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Saïd Jabrane
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Francis Lazarus
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Boris Thibert
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  1. Edited by Yakov Eliashberg, Stanford University, Stanford, CA, and approved March 9, 2012 (received for review November 9, 2011)

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Abstract

It is well-known that the curvature tensor is an isometric invariant of C2 Riemannian manifolds. This invariant is at the origin of the rigidity observed in Riemannian geometry. In the mid 1950s, Nash amazed the world mathematical community by showing that this rigidity breaks down in regularity C1. This unexpected flexibility has many paradoxical consequences, one of them is the existence of C1 isometric embeddings of flat tori into Euclidean three-dimensional space. In the 1970s and 1980s, M. Gromov, revisiting Nash’s results introduced convex integration theory offering a general framework to solve this type of geometric problems. In this research, we convert convex integration theory into an algorithm that produces isometric maps of flat tori. We provide an implementation of a convex integration process leading to images of an embedding of a flat torus. The resulting surface reveals a C1 fractal structure: Although the tangent plane is defined everywhere, the normal vector exhibits a fractal behavior. Isometric embeddings of flat tori may thus appear as a geometric occurrence of a structure that is simultaneously C1 and fractal. Beyond these results, our implementation demonstrates that convex integration, a theory still confined to specialists, can produce computationally tractable solutions of partial differential relations.

Footnotes

  • ↵1To whom correspondence should be addressed. E-mail: vincent.borrelli{at}math.univ-lyon1.fr.
  • Author contributions: V.B., S.J., F.L., and B.T. performed research and wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1118478109/-/DCSupplemental.

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Flat tori in three-dimensional space
Vincent Borrelli, Saïd Jabrane, Francis Lazarus, Boris Thibert
Proceedings of the National Academy of Sciences May 2012, 109 (19) 7218-7223; DOI: 10.1073/pnas.1118478109

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Flat tori in three-dimensional space
Vincent Borrelli, Saïd Jabrane, Francis Lazarus, Boris Thibert
Proceedings of the National Academy of Sciences May 2012, 109 (19) 7218-7223; DOI: 10.1073/pnas.1118478109
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