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Transactions of the American Mathematical Society

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Some families of isoparametric hypersurfaces and rigidity in a complex hyperbolic space


Author: Micheal H. Vernon
Journal: Trans. Amer. Math. Soc. 312 (1989), 237-256
MSC: Primary 53C40
DOI: https://doi.org/10.1090/S0002-9947-1989-0983871-X
MathSciNet review: 983871
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Abstract: The geometric notion of equivalence for submanifolds in a chosen ambient space is that of congruence. In this study, a certain type of isoparametric hypersurface of a complex hyperbolic space form is shown to have a rigid immersion by utilizing the congruences of a Lorentzian hyperbolic space form that lies as an $ {S^1}$-fiber bundle over the complex hyperbolic space. Several families of isoparametric hypersurfaces (namely tubes and horospheres) are constructed whose immersions are rigid.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0983871-X
Keywords: Real hypersurfaces, complex hyperbolic space, Lorentzian manifold, Anti-De Sitter spacetime, $ {S^1}$-fibration, isoparametric hypersurfaces, tubes, rigidity
Article copyright: © Copyright 1989 American Mathematical Society