Limitations on the smooth confinement of an unstretchable manifold

S. C. Venkataramani, T. A. Witten, E. M. Kramer, R. P. Geroch

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17 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)5107-5128
Number of pages22
JournalJournal of Mathematical Physics
Volume41
Issue number7
DOIs
StatePublished - Jul 2000
Externally publishedYes

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ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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