### Abstract

We prove that an m-dimensional unit ball D_{m} in the Euclidean space R^{m} cannot be isometrically embedded into a higher-dimensional Euclidean ball B^{d}_{r}⊂.ℝ^{d} of radius r<1/2 unless one of two conditions is met: (1) the embedding manifold has dimension d≥2m; (2) the embedding is not smooth. The proof uses differential geometry to show that if d<2m and the embedding is smooth and isometric, we can construct a line from the center of D^{m} to the boundary that is geodesic in both D^{m} and in the embedding manifold ℝ^{d}. Since such a line has length 1, the diameter of the embedding ball must exceed 1.

Original language | English (US) |
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Pages (from-to) | 5107-5128 |

Number of pages | 22 |

Journal | Journal of Mathematical Physics |

Volume | 41 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2000 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Venkataramani, S. C., Witten, T. A., Kramer, E. M., & Geroch, R. P. (2000). Limitations on the smooth confinement of an unstretchable manifold.

*Journal of Mathematical Physics*,*41*(7), 5107-5128. https://doi.org/10.1063/1.533394