Electronic Resource
College Park, Md.
:
American Institute of Physics (AIP)
Journal of Mathematical Physics
41 (2000), S. 5107-5128
ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
We prove that an m-dimensional unit ball Dm in the Euclidean space Rm cannot be isometrically embedded into a higher-dimensional Euclidean ball Brd⊂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. © 2000 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.533394
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