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1971 (Publication #10)

High codimensional 0-tight mappings on spheres

Proc. Amer. Math. Soc., 29 (1971), 133-137.

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ABSTRACT

The condition of 0-tightness for smooth immersions of 2-manifolds in E^M is equivalent to minimal total absolute curvature, but these notions do not coincide in higher dimensions. By a result of Chern and Lashof, if a smooth n-sphere embedded in E^M has minimal total absolute curvature, then it must bound a convex (n+1)-cell in an affine (n+1)-dimensional subspace. We will show that for any n > 2 and M > n, there exists a 0-tight polyhedral embedding of the n-sphere into E^M whose image lies in no hyperplane.