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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cylindricity of isometric immersions into Euclidean space
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by Robert Maltz
Proc. Amer. Math. Soc. 53 (1975), 428-432
DOI: https://doi.org/10.1090/S0002-9939-1975-0643658-X

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 Then the following cylindricity theorem (unpublished) of S. Alexander is proved using the same idea.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.
References
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Bibliographic Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 53 (1975), 428-432
  • MSC: Primary 53C40
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0643658-X
  • MathSciNet review: 0643658