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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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Abstract

We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \subset {\mathbb 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 {\mathbb R}^d. Since such a line has length 1, the diameter of the embedding ball must exceed 1.

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S. C. Venkataramani, T. A. Witten, E. M. Kramer, R. P. Geroch. 2000-07-03. Limitations on the smooth confinement of an unstretchable manifold. https://doi.org/10.1063/1.533394

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