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Transactions of the American Mathematical Society

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



Cylindricity of isometric immersions between hyperbolic spaces

Authors: S. Alexander and E. Portnoy
Journal: Trans. Amer. Math. Soc. 237 (1978), 311-329
MSC: Primary 53C40
MathSciNet review: 0461379
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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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  • [1] D. Ferus, On isometric immersions between hyperbolic spaces, Math. Ann. 205 (1973), 193-200. MR 0336665 (49:1438)
  • [2] P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901-920. MR 0126812 (23:A4106)
  • [3] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 1, Interscience, New York, 1963. MR 0152974 (27:2945)
  • [4] K. Nomizu, Isometric immersions of the hyperbolic plane into the hyperbolic space, Math. Ann. 205 (1973), 181-192. MR 0336664 (49:1437)
  • [5] B. O'Neill and E. Stiel, Isometric immersions of constant curvature manifolds, Michigan Math. J. 10 (1963), 335-339. MR 0158329 (28:1554)
  • [6] E. Portnoy, Developable surfaces in hyperbolic space, Pacific J. Math. 57 (1975), 281-288. MR 0380641 (52:1538)

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