Limitations on the smooth confinement of an unstretchable manifold

TL;DR

This paper proves that an m-dimensional unit ball cannot be smoothly isometrically embedded into a smaller higher-dimensional Euclidean ball unless the embedding dimension is at least twice m or the embedding is not smooth, using differential geometry.

Contribution

It establishes fundamental limitations on smooth isometric embeddings of unit balls into smaller Euclidean balls, highlighting the necessity of high embedding dimension or non-smoothness.

Findings

Smooth isometric embedding requires dimension d >= 2m or non-smoothness.

Embedding into a smaller radius ball is impossible under smooth isometric conditions for d < 2m.

Constructs a geodesic line argument to prove diameter constraints.

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.