### 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) |
---|---|

Pages (from-to) | 5107-5128 |

Number of pages | 22 |

Journal | Journal of Mathematical Physics |

Volume | 41 |

Issue number | 7 |

State | Published - Jul 2000 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Mathematical Physics*,

*41*(7), 5107-5128.

**Limitations on the smooth confinement of an unstretchable manifold.** / Venkataramani, Shankar C; Witten, T. A.; Kramer, E. M.; Geroch, R. P.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 41, no. 7, pp. 5107-5128.

}

TY - JOUR

T1 - Limitations on the smooth confinement of an unstretchable manifold

AU - Venkataramani, Shankar C

AU - Witten, T. A.

AU - Kramer, E. M.

AU - Geroch, R. P.

PY - 2000/7

Y1 - 2000/7

N2 - We prove that an m-dimensional unit ball Dm in the Euclidean space Rm cannot be isometrically embedded into a higher-dimensional Euclidean ball Bdr⊂.ℝd 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 Dm to the boundary that is geodesic in both Dm and in the embedding manifold ℝd. Since such a line has length 1, the diameter of the embedding ball must exceed 1.

AB - We prove that an m-dimensional unit ball Dm in the Euclidean space Rm cannot be isometrically embedded into a higher-dimensional Euclidean ball Bdr⊂.ℝd 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 Dm to the boundary that is geodesic in both Dm and in the embedding manifold ℝd. Since such a line has length 1, the diameter of the embedding ball must exceed 1.

UR - http://www.scopus.com/inward/record.url?scp=0034397684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034397684&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034397684

VL - 41

SP - 5107

EP - 5128

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 7

ER -