Flat tori in three-dimensional space and convex integration.

@article{Borrelli2012FlatTI,
  title={Flat tori in three-dimensional space and convex integration.},
  author={Vincent Borrelli and Sa{\"i}d Jabrane and Francis Lazarus and Boris Thibert},
  journal={Proceedings of the National Academy of Sciences of the United States of America},
  year={2012},
  volume={109 19},
  pages={7218-23}
}
It is well-known that the curvature tensor is an isometric invariant of C(2) 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 C(1). This unexpected flexibility has many paradoxical consequences, one of them is the existence of C(1) isometric embeddings of flat tori into Euclidean three-dimensional space. In the 1970s and… CONTINUE READING

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