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Dupin hypersurfaces with four principal Curvatures, II

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Abstract

If M is an isoparametric hypersurface in a sphere S n with four distinct principal curvatures, then the principal curvatures κ1, . . . , κ4 can be ordered so that their multiplicities satisfy m 1 = m 2 and m 3 = m 4, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R n (or S n) with four distinct principal curvatures with multiplicities m 1 = m 2 ≥  1 and m 3 = m 4 = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.

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References

  1. Cartan E. (1939). Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math. Z. 45: 335–367

    Article  MATH  MathSciNet  Google Scholar 

  2. Cecil T.E. (1989). Reducible Dupin submanifolds. Geom. Dedicata 32(3): 281–300

    Article  MATH  MathSciNet  Google Scholar 

  3. Cecil T.E. (1992). Lie Sphere Geometry. Universitext. Springer-Verlag, New York

    Google Scholar 

  4. Cecil, T.E., Chi, Q.S., Jensen, G.R.: On Kuiper’s conjecture (2005). Preprint

  5. Cecil T.E. and Jensen G.R. (1998). Dupin hypersurfaces with three principal curvatures. Invent. Math. 132(1): 121–178

    Article  MATH  MathSciNet  Google Scholar 

  6. Cecil T.E. and Jensen G.R. (2000). Dupin hypersurfaces with four principal curvatures. Geom. Dedicata 79(1): 1–49

    Article  MATH  MathSciNet  Google Scholar 

  7. Cecil T.E. and Ryan P.J. (1978). Focal sets, taut embeddings and the cyclides of Dupin. Math. Ann. 236(2): 177–190

    Article  MATH  MathSciNet  Google Scholar 

  8. Cecil, T.E., Ryan, P.J.: Tight and taut immersions of manifolds. In: Research Notes in Mathematics, vol. 107. Pitman (Advanced Publishing Program): Boston, MA (1985)

  9. Ferus D., Karcher H., Münzner H.F., (1981). Cliffordalgebren und neue isoparametrische Hyperflächen. Math. Z. 177(4): 479–502

    Article  MATH  MathSciNet  Google Scholar 

  10. Grove K. and Halperin S. (1987). Dupin hypersurfaces, group actions and the double mapping cylinder. J. Differential Geom. 26(3): 429–459

    MATH  MathSciNet  Google Scholar 

  11. Miyaoka R. (1984). Compact Dupin hypersurfaces with three principal curvatures. Math. Z. 187(4): 433–452

    Article  MATH  MathSciNet  Google Scholar 

  12. Miyaoka R. (1989). Dupin hypersurfaces and a Lie invariant. Kodai Math. J. 12(2): 228–256

    Article  MATH  MathSciNet  Google Scholar 

  13. Miyaoka R. (1989). Dupin hypersurfaces with six principal curvatures. Kodai Math. J. 12(3): 308–315

    Article  MATH  MathSciNet  Google Scholar 

  14. Miyaoka R., Ozawa, T.: Construction of taut embeddings and Cecil-Ryan conjecture. In: Geometry of Manifolds (Matsumoto, 1988), Perspect. Math., vol. 8, pp. 181–189. Academic Press, Boston, MA (1989)

  15. Münzner H.F. (1980). Isoparametrische Hyperflächen in Sphären. Math. Ann. 251(1): 57–71

    Article  MATH  MathSciNet  Google Scholar 

  16. Münzner H.F. (1981). Isoparametrische Hyperflächen in Sphären. II. Über die Zerlegung der Sphäre in Ballbündel. Math. Ann. 256(2): 215–232

    Article  MATH  MathSciNet  Google Scholar 

  17. Pinkall U. (1985). Dupin hypersurfaces. Math. Ann. 270: 427–440

    Article  MATH  MathSciNet  Google Scholar 

  18. Pinkall U. (1985). Dupinsche Hyperflächen in E 4. Manuscripta Math. 51(1–3): 89–119

    Article  MATH  MathSciNet  Google Scholar 

  19. Pinkall U. and Thorbergsson G. (1989). Deformations of Dupin hypersurfaces. Proc. Amer. Math. Soc. 107(4): 1037–1043

    Article  MATH  MathSciNet  Google Scholar 

  20. Singley D.H. (1975). Smoothness theorems for the principal curvatures and principal vectors of a hypersurface. Rocky Mountain J. Math. 5: 135–144

    Article  MATH  MathSciNet  Google Scholar 

  21. Stolz and S. (1999). Multiplicities of Dupin hypersurfaces. Invent. Math. 138(2): 253–279

    Article  MATH  MathSciNet  Google Scholar 

  22. Takagi and R. (1976). A class of hypersurfaces with constant principal curvatures in a sphere. J. Diff. Geom. 11: 225–233

    MATH  Google Scholar 

  23. Thorbergsson G. (1983). Dupin hypersurfaces. Bull. London Math. Soc. 15(5): 493–498

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA, 01610-2395, USA

    Thomas E. Cecil

  2. Department of Mathematics, Washington University, Campus Box 1146, St. Louis, MO, 63130, USA

    Quo-Shin Chi & Gary R. Jensen

Authors
  1. Thomas E. Cecil
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  2. Quo-Shin Chi
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  3. Gary R. Jensen
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Correspondence to Gary R. Jensen.

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Cecil, T.E., Chi, QS. & Jensen, G.R. Dupin hypersurfaces with four principal Curvatures, II. Geom Dedicata 128, 55–95 (2007). https://doi.org/10.1007/s10711-007-9183-3

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  • DOI: https://doi.org/10.1007/s10711-007-9183-3

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