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
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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 Then the following cylindricity theorem (unpublished) of S. Alexander is proved using the same idea.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.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