Limitations on the smooth confinement of an unstretchable manifold

We prove that an m-dimensional unit ball D-m in the Euclidean space R-m can not be isometrically embedded into a higher-dimensional Euclidean ball B(r) (d)subset of R-d of radius r < 1/2 unless one of two conditions is met: (1) the embedding manifold has dimension d greater than or equal to 2m; (2) th e embedding is not smooth. The proof uses differential geometry to show tha t if d < 2m and the embedding is smooth and isometric, we can construct a l ine 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 diamete r of the embedding ball must exceed 1. (C) 2000 American Institute of Physi cs. [S0022-2488(00)00707-6].