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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Cylindricity of isometric immersions between hyperbolic spaces
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by S. Alexander and E. Portnoy PDF
Trans. Amer. Math. Soc. 237 (1978), 311-329 Request permission

Abstract:

The motivation for this paper was to prove the following analogue of the Euclidean cylinder theorem: any umbilic-free isometric immersion $\eta :{H^{n - 1}} \to {H^n}$ between hyperbolic spaces takes the form of a hyperbolic $(n - 2)$-cylinder over a uniquely determined parallelizing curve in ${\bar H^n}$. Our approach is through the more general study of isometric immersions generated by one-parameter families of hyperbolic k-planes without focal points. A by-product of this study is a natural extension to curves in ${\bar H^n}$ of the notion of a parallel family of k-planes along a curve in ${H^n}$; the extension is based on spherical symmetry of variation fields. Existence and uniqueness properties of this extended notion of parallelism are considered.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 237 (1978), 311-329
  • MSC: Primary 53C40
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0461379-0
  • MathSciNet review: 0461379