Limitations on the smooth confinement of an unstretchable manifold

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

Research output: Contribution to journalArticle

17 Citations (Scopus)

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
StatePublished - Jul 2000
Externally publishedYes

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embedding
Ball
Line
Differential Geometry
balls
Isometric
Unit ball
Geodesic
Euclidean space
Euclidean
Exceed
High-dimensional
Radius
differential geometry
Euclidean geometry
radii

ASJC Scopus subject areas

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

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.

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

In: Journal of Mathematical Physics, Vol. 41, No. 7, 07.2000, p. 5107-5128.

Research output: Contribution to journalArticle

Venkataramani, SC, Witten, TA, Kramer, EM & Geroch, RP 2000, 'Limitations on the smooth confinement of an unstretchable manifold', Journal of Mathematical Physics, vol. 41, no. 7, pp. 5107-5128.
Venkataramani, Shankar C ; Witten, T. A. ; Kramer, E. M. ; Geroch, R. P. / Limitations on the smooth confinement of an unstretchable manifold. In: Journal of Mathematical Physics. 2000 ; Vol. 41, No. 7. pp. 5107-5128.
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