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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The two-piece property and tight $ n$-manifolds-with-boundary in $ E\sp{n}$

Author: Thomas F. Banchoff
Journal: Trans. Amer. Math. Soc. 161 (1971), 259-267
MSC: Primary 57.20; Secondary 53.00
MathSciNet review: 0287556
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Abstract: The two-piece property for a set A is a generalization of convexity which reduces to the condition of minimal total absolute curvature if A is a compact 2-manifold. We show that a connected compact 2-manifold-with-boundary in $ {E^2}$ has the TPP if and only if each component of the boundary has the TPP. The analogue of this result is not true in higher dimensions without additional conditions, and we introduce a stronger notion called k-tightness and show that an $ (n + 1)$-manifold-with-boundary $ {M^{n + 1}}$ embedded in $ {E^{n + 1}}$ is 0- and $ (n - 1)$-tight if and only if its boundary is 0- and $ (n - 1)$-tight.

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Keywords: Tight manifolds, two-piece property, minimal total absolute curvature
Article copyright: © Copyright 1971 American Mathematical Society

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