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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics