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Proceedings of the American Mathematical Society

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

 

 

Cylindricity of isometric immersions into Euclidean space


Author: Robert Maltz
Journal: Proc. Amer. Math. Soc. 53 (1975), 428-432
MSC: Primary 53C40
MathSciNet review: 0643658
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Abstract: A simple geometric proof is given for the Hartman-Nirenberg cylindricity theorem and some generalizations. Then the following cylindricity theorem (unpublished) of S. Alexander is proved using the same idea.

Theorem. Let $ f:M \to {E^n}$ be an isometric Euclidean immersion of the Riemannian product $ M = {M_1} \times \ldots \times {M_k} \times {E^m}$ where the $ {M_i}$ are not everywhere flat Riemannian manifolds, and $ {E^m}$ denotes Euclidean $ m$-space. Then $ C \geqslant k$, where $ C$ denotes the codimension of the immersion; and if $ C = k$, then the immersion is cylindrical on the Euclidean factor.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0643658-X
Keywords: Isometric immersions, Riemannian products, cylindricity
Article copyright: © Copyright 1975 American Mathematical Society