Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Classification of isoparametric hypersurfaces in spheres with $ (g,m)=(6,1)$


Author: Anna Siffert
Journal: Proc. Amer. Math. Soc. 144 (2016), 2217-2230
MSC (2010): Primary 53C40; Secondary 53C55, 53C30
DOI: https://doi.org/10.1090/proc/12924
Published electronically: October 8, 2015
MathSciNet review: 3460180
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We classify the isospectral families $ L(t)=\cos (t)L_0+\sin (t)L_1\in$$ \mbox {Sym}(5,\mathbb{R})$, $ t\in \mathbb{R}$, with $ L_0=$$ \mbox {diag}(\sqrt {3},\tfrac {1}{\sqrt {3}},0, -\tfrac {1}{\sqrt {3}},-\sqrt {3})$. Using this result we provide a classification of isoparametric hypersurfaces in spheres with $ (g,m)=(6,1)$ and thereby give a simplified proof of the fact that any isoparametric hypersurface with $ (g,m)=(6,1)$ is homogeneous. This result was first proven by Dorfmeister and Neher in 1985.


References [Enhancements On Off] (What's this?)

  • [1] U. Abresch, Isoparametric hypersurfaces with four or six distinct principal curvatures. Necessary conditions on the multiplicities, Math. Ann. 264 (1983), no. 3, 283-302. MR 714104 (85g:53052b), https://doi.org/10.1007/BF01459125
  • [2] Uwe Abresch, Notwendige Bedingungen für isoparametrische Hyperflächen in Sphären mit mehr als drei verschiedenen Hauptkrümmungen, Bonner Mathematische Schriften [Bonn Mathematical Publications], 146, Universität Bonn, Mathematisches Institut, Bonn, 1982 (German). MR 701112 (85g:53052a)
  • [3] Josef Dorfmeister and Erhard Neher, Isoparametric hypersurfaces, case $ g=6,\;m=1$, Comm. Algebra 13 (1985), no. 11, 2299-2368. MR 807479 (87d:53096), https://doi.org/10.1080/00927878508823278
  • [4] Reiko Miyaoka, The Dorfmeister-Neher theorem on isoparametric hypersurfaces, Osaka J. Math. 46 (2009), no. 3, 695-715. MR 2583325 (2011d:53126)
  • [5] Reiko Miyaoka, Isoparametric hypersurfaces with $ (g,m)=(6,2)$, Ann. of Math. (2) 177 (2013), no. 1, 53-110. MR 2999038, https://doi.org/10.4007/annals.2013.177.1.2
  • [6] R.Miyaoka, Remarks on the Dorfmeister-Neher theorem on isoparametric hypersurfaces, to appear in Osaka J. Math.
  • [7] Reiko Miyaoka, The linear isotropy group of $ G_2/{\rm SO}(4)$, the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. 30 (1993), no. 2, 179-202. MR 1233508 (95a:53100)
  • [8] Hans Friedrich Münzner, Isoparametrische Hyperflächen in Sphären, Math. Ann. 251 (1980), no. 1, 57-71 (German). MR 583825 (82a:53058), https://doi.org/10.1007/BF01420281
  • [9] Hans Friedrich Münzner, Isoparametrische Hyperflächen in Sphären. II. Über die Zerlegung der Sphäre in Ballbündel, Math. Ann. 256 (1981), no. 2, 215-232 (German). MR 620709 (82m:53053), https://doi.org/10.1007/BF01450799
  • [10] A.Siffert, A new structural approach to isoparametric hypersurfaces in spheres, submitted, arXiv:1410.6206.
  • [11] Stephan Stolz, Multiplicities of Dupin hypersurfaces, Invent. Math. 138 (1999), no. 2, 253-279. MR 1720184 (2001d:53065), https://doi.org/10.1007/s002220050378
  • [12] Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, 2000, pp. 963-995. MR 1736861 (2001a:53097), https://doi.org/10.1016/S1874-5741(00)80013-8
  • [13] Ryoichi Takagi and Tsunero Takahashi, On the principal curvatures of homogeneous hypersurfaces in a sphere, Differential geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 469-481. MR 0334094 (48 #12413)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C40, 53C55, 53C30

Retrieve articles in all journals with MSC (2010): 53C40, 53C55, 53C30


Additional Information

Anna Siffert
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email: asiffert@math.upenn.edu

DOI: https://doi.org/10.1090/proc/12924
Received by editor(s): April 9, 2015
Received by editor(s) in revised form: June 16, 2015
Published electronically: October 8, 2015
Additional Notes: The author would like to thank DFG for supporting this work with the grant SI 2077/1-1.
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society