Limitations on the smooth confinement of an unstretchable manifold (Q2738101)

It is proved 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 \(B^d_r\subset {\mathbb{R}}^d\) of radius \(r<1/2\) unless one of two conditions is met: (1) the embedding manifold has dimension \(d\geq 2m\); (2) the embedding is not smooth. The proof of this result showing that if \(d<2m\) and the embedding is smooth and isometric, one can construct a line from the center of \(D^m\) to the boundary that is a geodesic in both \(D^m\) and in the embedding manifold \({\mathbb{R}}^d\). Since the length of such a line is 1, the diameter of the embedding ball must exceed 1. This is a fine development on the interesting problem of wether a given manifold can be embedded isometrically into another manifold isometrically by methods of differential geometry. The method used in this paper might be generalized to the situation where the embedded manifold and the embedding space are not intrisically flat.