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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Parallel tangent hyperplanes

Author: Sol Schwartzman
Journal: Proc. Amer. Math. Soc. 130 (2002), 1457-1458
MSC (2000): Primary 53A07, 55M20
Published electronically: December 27, 2001
MathSciNet review: 1879969
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Abstract: Let $\Sigma^{2n}$ be a smooth strictly convex closed hypersurface in $R^{2n+1}$and let $M^{2n}$ be any oriented smooth connected manifold immersed in $R^{2n+1}.$ Suppose that $f$ is a continuous function from $\Sigma^{2n}$ to $M^{2n}.$ Then there is at least one point $p \in \Sigma^{2n}$ such that the hyperplane tangent to $\Sigma^{2n}$ at $p$ is parallel to the hyperplane tangent to the immersed manifold $M^{2n}$ at the point corresponding to $f(p).$If there did not exist at least two such points, $M^{2n}$ would have to be compact and the Hurewicz homomorphism of $\pi_{2n}(M^{2n})$ into $\mbox{H}_{2n}(M^{2n})$ would have to be surjective. If in addition our immersion was an embedding, the Euler characteristic of $M^{2n}$ would have to be equal to $\pm 2.$ For any $\Sigma^{2n}$ and any immersed $M^{2n}$ we could always get maps $f$ for which the number of points $p$ satisfying the conditions of our theorem exactly equaled two. An example can be given in which both $\Sigma^{2n}$ and $M^{2n}$ are the unit sphere about the origin in $R^{2n+1}$and there is only one such point $p$.

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Additional Information

Sol Schwartzman
Affiliation: Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881

Received by editor(s): August 1, 2000
Published electronically: December 27, 2001
Communicated by: Ralph Cohen
Article copyright: © Copyright 2001 American Mathematical Society