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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Lie sphere transformations and the focal sets of certain taut immersions


Author: Steven G. Buyske
Journal: Trans. Amer. Math. Soc. 311 (1989), 117-133
MSC: Primary 53C42
MathSciNet review: 965743
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Abstract: We study the images of certain taut or Dupin hypersurfaces, including their focal sets, under Lie sphere transformations (generalizations of conformal transformations of euclidean or spherical space). After the introduction, the method of studying hypersurfaces as Lie sphere objects is developed. In two recent papers, Cecil and Chern use submanifolds of the space of lines on the Lie quadric. Here we use submanifolds of the Lie quadric itself instead. The third section extends the concepts of tightness and tautness to semi-euclidean space. The final section shows that if a hypersurface is the Lie sphere image of certain standard constructions (tubes, cylinders, and rotations) over a taut immersion, the resulting family of curvature spheres is taut in the Lie quadric. The sheet of the focal set will be tight in euclidean space if it is compact. In particular, if a hypersurface in euclidean space is the Lie sphere image of an isoparametric hypersurface each compact sheet of the focal set will be tight.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0965743-X
PII: S 0002-9947(1989)0965743-X
Keywords: Taut, tight, Lie sphere, isoparametric, Dupin, semi-euclidean, semi-riemannian, immersions
Article copyright: © Copyright 1989 American Mathematical Society