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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the immersion of an $ n$-dimensional manifold in $ n+1$-dimensional Euclidean space


Author: Benjamin Halpern
Journal: Proc. Amer. Math. Soc. 30 (1971), 181-184
MSC: Primary 57.20
MathSciNet review: 0286116
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Abstract: Consider the subset of $ n + 1$-dimensional Euclidean space swept out by the tangent hyperplanes drawn through the points of an immersed compact closed connected n-dimensional smooth manifold. If this is not all of the Euclidean space, then the manifold is diffeomorphic to a sphere, the immersion is an embedding, the image of the immersion is the boundary of a unique open starshaped set, and the set of points not on any tangent hyperplane is the interior of the kernel of the open starshaped set. A converse statement also holds.


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

  • [1] Benjamin Halpern, On the immersion of an n-dimensional manifold in $ (n + 1)$ dimensional Euclidean space, Notices Amer. Math. Soc. 16 (1969), 774. Abstract #667-60.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0286116-3
PII: S 0002-9939(1971)0286116-3
Keywords: Immersion, embedding, Euclidean space, starshaped, tangent hyperplane
Article copyright: © Copyright 1971 American Mathematical Society