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High codimensional 0-tight maps on spheres


Author: T. F. Banchoff
Journal: Proc. Amer. Math. Soc. 29 (1971), 133-137
MSC: Primary 57.20; Secondary 53.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0279820-4
MathSciNet review: 0279820
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Abstract: For smooth immersions of 2-manifolds into $ {E^M}$, the condition of 0-tightness is equivalent to that of minimal total absolute curvature, but for higher dimensional manifolds these notions are quite different. By a result of Chern and Lashof, a smooth n-sphere embedded in $ {E^M}$ with minimal total absolute curvature must bound a convex $ (n + 1)$-cell in an affine $ (n + 1)$-dimensional subspace, but we show that for any $ n > 2$ and any $ M > n$ there is a 0-tight polyhedral embedding of the n-sphere into $ {E^M}$ with image lying in no hyperplane.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0279820-4
Keywords: Tight mapping, minimal total absolute curvature, polyhedral immersion
Article copyright: © Copyright 1971 American Mathematical Society

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