### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 117-133 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 311 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1989 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Dupin
- Immersions
- Isoparametric
- Lie sphere
- Semi-euclidean
- Semiriemannian
- Taut
- Tight

### Cite this

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**Lie sphere transformations and the focal sets of certain taut immersions.** / Buyske, Steven.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Lie sphere transformations and the focal sets of certain taut immersions

AU - Buyske, Steven

PY - 1989/1/1

Y1 - 1989/1/1

N2 - 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.

AB - 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.

KW - Dupin

KW - Immersions

KW - Isoparametric

KW - Lie sphere

KW - Semi-euclidean

KW - Semiriemannian

KW - Taut

KW - Tight

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UR - http://www.scopus.com/inward/citedby.url?scp=70350320228&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1989-0965743-X

DO - 10.1090/S0002-9947-1989-0965743-X

M3 - Article

VL - 311

SP - 117

EP - 133

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -