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首页> 外文期刊>Proceedings of the National Academy of Sciences of the United States of America >Flat tori in three-dimensional space and convex integration
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Flat tori in three-dimensional space and convex integration

机译：三维空间中的扁平花托和凸积分

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摘要

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 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 C~1 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 C~1 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.
机译：众所周知，曲率张量是C〜2黎曼流形的等距不变量。该不变性是在黎曼几何中观察到的刚度的起源。 1950年代中期，纳什证明了这种刚性按正则性C〜1分解，令世界数学界感到惊讶。这种出乎意料的灵活性带来了许多矛盾的后果，其中之一就是平托环的C〜1等距嵌入到欧几里得三维空间中。在1970年代和1980年代，M。Gromov重新审视了Nash的结果，提出了凸积分理论，为解决这类几何问题提供了一个通用框架。在这项研究中，我们将凸积分理论转换为可生成平面花托等距图的算法。我们提供了凸积分过程的实现，该过程导致嵌入扁平圆环的图像。所得的表面显示出C〜1的分形结构：尽管切线平面定义在各处，但法向矢量表现出分形行为。因此，扁平圆托的等距嵌入可以显示为同时出现C〜1和分形的结构的几何形状。除了这些结果之外，我们的实现还证明了凸积分（一种仍然仅限于专家的理论）可以产生偏微分关系在计算上易于处理的解。

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    Vincent Borrelli; Said Jabrane; Francis Lazarus; Boris Thibert;

  • 作者单位

    Institut Camille Jordan, Universite Lyon I, 69622 Villeurbanne, France;

    Institut Camille Jordan, Universite Lyon I, 69622 Villeurbanne, France;

    Centre National de la Recherche Scientifique, Laboratoire Grenoble Image Parole Signal Automatique, 38402 Grenoble, France;

    Laboratoire Jean Kuntzmann, Universite de Grenoble, 38041 Grenoble, France;

  • 收录信息 美国《科学引文索引》(SCI);美国《生物学医学文摘》(MEDLINE);美国《化学文摘》(CA);
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  • 正文语种 eng
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