Isoparametric hypersurfaces with $(g,m)=(6,2)$

Abstract

We prove that isoparametric hypersurfaces with $(g,m)=(6,2)$ are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and solves Yau’s problem in the case $g=6$.

  • [A] Go to document U. Abresch, "Isoparametric hypersurfaces with four or six distinct principal curvatures. Necessary conditions on the multiplicities," Math. Ann., vol. 264, pp. 283-302, 1983.
    @article{A,
      author={Abresch, U.},
      TITLE={Isoparametric hypersurfaces with four or six distinct principal curvatures. {N}ecessary conditions on the multiplicities},
      JOURNAL={Math. Ann.},
      VOLUME={264},
      YEAR={1983},
      PAGES={283--302},
      MRNUMBER = {0714104},
      ZBLNUMBER = {0505.53027},
      DOI = {10.1007/BF01459125},
     }
  • [C1] Go to document &. Cartan, "Familles de surfaces isoparamétriques dans les espaces à courbure constante," Ann. Mat. Pura Appl., vol. 17, iss. 1, pp. 177-191, 1938.
    @article {C1, MRKEY = {1553310},
      AUTHOR = {Cartan, {É}lie},
      TITLE = {Familles de surfaces isoparamétriques dans les espaces à courbure constante},
      JOURNAL = {Ann. Mat. Pura Appl.},
      FJOURNAL = {Annali di Matematica Pura ed Applicata},
      VOLUME = {17},
      YEAR = {1938},
      NUMBER = {1},
      PAGES = {177--191},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1553310},
      DOI = {10.1007/BF02410700},
      ZBLNUMBER = {0020.06505},
      }
  • [C2] Go to document &. Cartan, "Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques," Math. Z., vol. 45, pp. 335-367, 1939.
    @article {C2, MRKEY = {0000169},
      AUTHOR = {Cartan, {É}lie},
      TITLE = {Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {45},
      YEAR = {1939},
      PAGES = {335--367},
      ISSN = {0025-5874},
      MRCLASS = {14.0X},
      MRNUMBER = {0000169},
      MRREVIEWER = {T. Y. Thomas},
      ZBLNUMBER = {0021.15603},
      DOI = {10.1007/BF01580289},
     }
  • [C3] &. Cartan, "Sur quelques familles remarquables d’hypersurfaces," C. R. Congrès Math. Liège, pp. 30-41, 1939.
    @article{C3,
      author={Cartan, {É}lie},
      TITLE = {Sur quelques familles remarquables d'hypersurfaces},
      JOURNAL={C. R. Congrès Math. Liège},
      YEAR={1939},
      PAGES={30--41},
     ZBLNUMBER = {0025.36601},
     }
  • [C4] &. Cartan, "Sur des familles d’hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions," Univ. Nac. Tucumán. Revista A., vol. 1, pp. 5-22, 1940.
    @article {C4, MRKEY = {0004519},
      AUTHOR = {Cartan, {É}lie},
      TITLE = {Sur des familles d'hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions},
      JOURNAL = {Univ. Nac. Tucumán. Revista A.},
      VOLUME = {1},
      YEAR = {1940},
      PAGES = {5--22},
      MRCLASS = {53.0X},
      MRNUMBER = {0004519},
      MRREVIEWER = {T. Y. Thomas},
      ZBLNUMBER = {0025.22603},
      }
  • [CCJ] Go to document T. E. Cecil, Q. Chi, and G. R. Jensen, "Isoparametric hypersurfaces with four principal curvatures," Ann. of Math., vol. 166, iss. 1, pp. 1-76, 2007.
    @article {CCJ, MRKEY = {2342690},
      AUTHOR = {Cecil, Thomas E. and Chi, Quo-Shin and Jensen, Gary R.},
      TITLE = {Isoparametric \hbox{hypersurfaces} with four principal curvatures},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {166},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {1--76},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {53C42},
      MRNUMBER = {2342690},
      MRREVIEWER = {Luiz A. M. Sousa, Jr.},
      DOI = {10.4007/annals.2007.166.1},
      ZBLNUMBER = {1143.53058},
      }
  • [Ch1] Go to document Q. Chi, "Isoparametric hypersurfaces with four principal curvatures revisited," Nagoya Math. J., vol. 193, pp. 129-154, 2009.
    @article {Ch1, MRKEY = {2502911},
      AUTHOR = {Chi, Quo-Shin},
      TITLE = {Isoparametric hypersurfaces with four principal curvatures revisited},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {193},
      YEAR = {2009},
      PAGES = {129--154},
      ISSN = {0027-7630},
      CODEN = {NGMJA2},
      MRCLASS = {53C42 (53C40)},
      MRNUMBER = {2502911},
      MRREVIEWER = {Thomas E. Cecil},
      URL = {http://projecteuclid.org/euclid.nmj/1236089984},
      ZBLNUMBER = {1165.53032},
      }
  • [Ch2] Q. Chi, "A note on the paper “Isoparametric hypersurfaces with four principal curvatures” [MR2342690]," Pac. J. Appl. Math., vol. 3, iss. 1-2, pp. 127-134, 2011.
    @article {Ch2, MRKEY = {2918561},
      AUTHOR = {Chi, Quo-Shin},
      TITLE = {A note on the paper ``{I}soparametric hypersurfaces with four principal curvatures'' [{MR}2342690]},
      JOURNAL = {Pac. J. Appl. Math.},
      FJOURNAL = {Pacific Journal of Applied Mathematics},
      VOLUME = {3},
      YEAR = {2011},
      NUMBER = {1-2},
      PAGES = {127--134},
      ISSN = {1941-3963},
      MRCLASS = {53Cxx},
      MRNUMBER = {2918561},
      }
  • [Ch3] Go to document Q. Chi, "Isoparametric hypersurfaces with four principal curvatures, II," Nagoya Math. J., vol. 204, pp. 1-18, 2011.
    @article {Ch3, MRKEY = {2863363},
      AUTHOR = {Chi, Quo-Shin},
      TITLE = {Isoparametric hypersurfaces with four principal curvatures, {II}},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {204},
      YEAR = {2011},
      PAGES = {1--18},
      ISSN = {0027-7630},
      MRCLASS = {53Cxx},
      MRNUMBER = {2863363},
      URL = {http://projecteuclid.org/euclid.nmj/1323107835},
      ZBLNUMBER = {1243.53094},
      }
  • [Ch4] Go to document Q. Chi, "A new look at Condition A," Osaka J. of Math., pp. 133-166, 2012.
    @article{Ch4,
      author={Chi, Quo-Shin},
      TITLE = {A new look at Condition {A}},
      JOURNAL={Osaka J. of Math.},
      PAGES = {133--166},
      MRNUMBER = {2903258},
      ZBLNUMBER = {06029031},
      URL = {http://projecteuclid.org/euclid.ojm/1332337242},
      YEAR={2012},
      SORTYEAR={2013},
     }
  • [Ch5] Q. Chi, Isoparametric hypersurfaces with four principal curvatures, III, 2011.
    @misc{Ch5, MRKEY = {2863363},
      AUTHOR = {Chi, Quo-Shin},
      TITLE = {Isoparametric hypersurfaces with four principal curvatures, {III}},
      NOTE={preprint},
      YEAR={2011},
      ARXIV= {1104.3249},
      SORTYEAR={2012},
     }
  • [DN1] Go to document J. Dorfmeister and E. Neher, "An algebraic approach to isoparametric hypersurfaces in spheres. I," Tôhoku Math. J., vol. 35, iss. 2, pp. 187-224, 1983.
    @article {DN1, MRKEY = {0699927},
      AUTHOR = {Dorfmeister, Josef and Neher, Erhard},
      TITLE = {An algebraic approach to isoparametric hypersurfaces in spheres. {I}},
      JOURNAL = {Tôhoku Math. J.},
      FJOURNAL = {The Tôhoku Mathematical Journal. Second Series},
      VOLUME = {35},
      YEAR = {1983},
      NUMBER = {2},
      PAGES = {187--224},
      ISSN = {0040-8735},
      CODEN = {TOMJAM},
      MRCLASS = {53C40},
      MRNUMBER = {0699927},
      MRREVIEWER = {Th. Friedrich},
      DOI = {10.2748/tmj/1178229051},
      ZBLNUMBER = {0507.53038},
     }
  • [DN2] Go to document J. Dorfmeister and E. Neher, "Isoparametric hypersurfaces, case $g=6,\;m=1$," Comm. Algebra, vol. 13, iss. 11, pp. 2299-2368, 1985.
    @article {DN2, MRKEY = {0807479},
      AUTHOR = {Dorfmeister, Josef and Neher, Erhard},
      TITLE = {Isoparametric hypersurfaces, case {$g=6,\;m=1$}},
      JOURNAL = {Comm. Algebra},
      FJOURNAL = {Communications in Algebra},
      VOLUME = {13},
      YEAR = {1985},
      NUMBER = {11},
      PAGES = {2299--2368},
      ISSN = {0092-7872},
      CODEN = {COALDM},
      MRCLASS = {53C40},
      MRNUMBER = {0807479},
      MRREVIEWER = {K. Nomizu},
      DOI = {10.1080/00927878508823278},
      ZBLNUMBER = {0578.53041},
      }
  • [FKM] Go to document D. Ferus, H. Karcher, and H. F. Münzner, "Cliffordalgebren und neue isoparametrische Hyperflächen," Math. Z., vol. 177, iss. 4, pp. 479-502, 1981.
    @article {FKM, MRKEY = {0624227},
      AUTHOR = {Ferus, Dirk and Karcher, Hermann and M{ü}nzner, Hans Friedrich},
      TITLE = {Cliffordalgebren und neue isoparametrische {H}yperflächen},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {177},
      YEAR = {1981},
      NUMBER = {4},
      PAGES = {479--502},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {53C40},
      MRNUMBER = {0624227},
      MRREVIEWER = {Authors' review},
      DOI = {10.1007/BF01219082},
      ZBLNUMBER = {0443.53037},
     }
  • [HL] Go to document W. Hsiang and B. H. Lawson Jr., "Minimal submanifolds of low cohomogeneity," J. Differential Geometry, vol. 5, pp. 1-38, 1971.
    @article {HL, MRKEY = {0298593},
      AUTHOR = {Hsiang, Wu-yi and Lawson, Jr., H. Blaine},
      TITLE = {Minimal submanifolds of low cohomogeneity},
      JOURNAL = {J. Differential Geometry},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {5},
      YEAR = {1971},
      PAGES = {1--38},
      ISSN = {0022-040X},
      MRCLASS = {53C40},
      MRNUMBER = {0298593},
      MRREVIEWER = {Bang-yen Chen},
      ZBLNUMBER = {0219.53045},
      URL = {http://projecteuclid.org/euclid.jdg/1214429775},
     }
  • [I] Go to document S. Immervoll, "On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres," Ann. of Math., vol. 168, iss. 3, pp. 1011-1024, 2008.
    @article {I, MRKEY = {2456889},
      AUTHOR = {Immervoll, Stefan},
      TITLE = {On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {168},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {1011--1024},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {53C42},
      MRNUMBER = {2456889},
      MRREVIEWER = {Luiz A. M. Sousa, Jr.},
      DOI = {10.4007/annals.2008.168.1011},
      ZBLNUMBER = {1176.53057},
     }
  • [KN] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. II, New York: Interscience Publishers John Wiley & Sons, Inc., 1969, vol. 15.
    @book{KN,
      author={Kobayashi, S. and Nomizu, K.},
      TITLE={Foundations of Differential Geometry. Vol. II},
      SERIES = {Interscience Tracts in Pure and Applied Math.},
      VOLUME = {15},
      PUBLISHER = {Interscience Publishers John Wiley \& Sons, Inc.},
      ADDRESS = {New York},
      MRNUMBER = {0238225},
      ZBLNUMBER = {0175.48504},
     YEAR={1969},
     }
  • [M1] Go to document R. Miyaoka, "Dupin hypersurfaces with six principal curvatures," Kodai Math. J., vol. 12, iss. 3, pp. 308-315, 1989.
    @article {M1, MRKEY = {1023536},
      AUTHOR = {Miyaoka, Reiko},
      TITLE = {Dupin hypersurfaces with six principal curvatures},
      JOURNAL = {Kodai Math. J.},
      FJOURNAL = {Kodai Mathematical Journal},
      VOLUME = {12},
      YEAR = {1989},
      NUMBER = {3},
      PAGES = {308--315},
      ISSN = {0386-5991},
      MRCLASS = {53C40},
      MRNUMBER = {1023536},
      MRREVIEWER = {Gudlaugur Thorbergsson},
      DOI = {10.2996/kmj/1138039098},
      ZBLNUMBER = {0711.53049},
      }
  • [M2] Go to document R. Miyaoka, "The linear isotropy group of $G_2/{ SO}(4)$, the Hopf fibering and isoparametric hypersurfaces," Osaka J. Math., vol. 30, iss. 2, pp. 179-202, 1993.
    @article {M2, MRKEY = {1233508},
      AUTHOR = {Miyaoka, Reiko},
      TITLE = {The linear isotropy group of {$G\sb 2/{\rm SO}(4)$},
      the {H}opf fibering and isoparametric hypersurfaces},
      JOURNAL = {Osaka J. Math.},
      FJOURNAL = {Osaka Journal of Mathematics},
      VOLUME = {30},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {179--202},
      ISSN = {0030-6126},
      CODEN = {OJMAA7},
      MRCLASS = {53C42 (53C40)},
      MRNUMBER = {1233508},
      MRREVIEWER = {Thomas E. Cecil},
      URL = {http://projecteuclid.org/euclid.ojm/1200784357},
      ZBLNUMBER = {0815.53070},
      }
  • [M3] Go to document R. Miyaoka, "The Dorfmeister-Neher theorem on isoparametric hypersurfaces," Osaka J. Math., vol. 46, iss. 3, pp. 695-715, 2009.
    @article {M3, MRKEY = {2583325},
      AUTHOR = {Miyaoka, Reiko},
      TITLE = {The {D}orfmeister-{N}eher theorem on isoparametric hypersurfaces},
      JOURNAL = {Osaka J. Math.},
      FJOURNAL = {Osaka Journal of Mathematics},
      VOLUME = {46},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {695--715},
      ISSN = {0030-6126},
      CODEN = {OJMAA7},
      MRCLASS = {53C40 (53C42)},
      MRNUMBER = {2583325},
      MRREVIEWER = {Thomas E. Cecil},
      URL = {http://projecteuclid.org/euclid.ojm/1256564202},
      ZBLNUMBER = {1185.53059},
      }
  • [M4] R. Miyaoka, "Isoparametric geometry and related fields," in Surveys on Geometry and Integrable Systems, Tokyo: Math. Soc. Japan, 2008, vol. 51, pp. 315-337.
    @incollection {M4, MRKEY = {2509798},
      AUTHOR = {Miyaoka, Reiko},
      TITLE = {Isoparametric geometry and related fields},
      BOOKTITLE = {Surveys on Geometry and Integrable Systems},
      SERIES = {Adv. Stud. Pure Math.},
      VOLUME = {51},
      PAGES = {315--337},
      PUBLISHER = {Math. Soc. Japan},
      ADDRESS = {Tokyo},
      YEAR = {2008},
      MRCLASS = {53C42 (37K25)},
      MRNUMBER = {2509798},
      MRREVIEWER = {Zhong Hua Hou},
      ZBLNUMBER = {1161.53047},
      }
  • [M5] Go to document R. Miyaoka, "Geometry of $G_2$ orbits and isoparametric hypersurfaces," Nagoya Math. J., vol. 203, pp. 175-189, 2011.
    @article {M5, MRKEY = {2834253},
      AUTHOR = {Miyaoka, Reiko},
      TITLE = {Geometry of {$G\sb 2$} orbits and isoparametric hypersurfaces},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {203},
      YEAR = {2011},
      PAGES = {175--189},
      ISSN = {0027-7630},
      MRCLASS = {53C40 (53C28 53C30 53D12)},
      MRNUMBER = {2834253},
      MRREVIEWER = {Luiz A. M. Sousa, Jr.},
      ZBLNUMBER = {1231.53051},
      URL = {http://projecteuclid.org/euclid.nmj/1313682316},
      }
  • [Mu1] Go to document H. F. Münzner, "Isoparametrische Hyperflächen in Sphären. I," Math. Ann., vol. 251, iss. 1, pp. 57-71, 1980.
    @article {Mu1, MRKEY = {0583825},
      AUTHOR = {M{ü}nzner, Hans Friedrich},
      TITLE = {Isoparametrische {H}yperflächen in {S}phären. {I}},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {251},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {57--71},
      ISSN = {0025-5831},
      CODEN = {MAANA3},
      MRCLASS = {53C40},
      MRNUMBER = {0583825},
      MRREVIEWER = {Th. Friedrich},
      DOI = {10.1007/BF01420281},
      ZBLNUMBER = {0417.53030},
     }
  • [Mu2] Go to document H. F. Münzner, "Isoparametrische Hyperflächen in Sphären. II. Über die Zerlegung der Sphäre in Ballbündel," Math. Ann., vol. 256, iss. 2, pp. 215-232, 1981.
    @article {Mu2, MRKEY = {0620709},
      AUTHOR = {M{ü}nzner, Hans Friedrich},
      TITLE = {Isoparametrische {H}yperflächen in {S}phären. {II}. Über die {Z}erlegung der {S}phäre in {B}allbündel},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {256},
      YEAR = {1981},
      NUMBER = {2},
      PAGES = {215--232},
      ISSN = {0025-5831},
      CODEN = {MAANA3},
      MRCLASS = {53C40},
      MRNUMBER = {0620709},
      MRREVIEWER = {Th. Friedrich},
      DOI = {10.1007/BF01450799},
      ZBLNUMBER = {0438.53050},
     }
  • [N1] Go to document K. Nomizu, "Sur les algèbres de Lie de générateurs de Killing et l’homogénéité d’une variété riemannienne," Osaka Math. J., vol. 14, pp. 45-51, 1962.
    @article {N1, MRKEY = {0141054},
      AUTHOR = {Nomizu, Katsumi},
      TITLE = {Sur les algèbres de {L}ie de générateurs de {K}illing et l'homogénéité d'une variété riemannienne},
      JOURNAL = {Osaka Math. J.},
      VOLUME = {14},
      YEAR = {1962},
      PAGES = {45--51},
      MRCLASS = {53.66},
      MRNUMBER = {0141054},
      MRREVIEWER = {A. Nijenhuis},
      ZBLNUMBER = {0103.38603},
      URL = {http://projecteuclid.org/euclid.ojm/1200690496},
     }
  • [N2] K. Nomizu, "Élie Cartan’s work on isoparametric families of hypersurfaces," in Differential Geometry, Providence, R.I.: Amer. Math. Soc., 1975, vol. 27, pp. 191-200.
    @incollection{N2,
      author={Nomizu, Katsumi},
      TITLE = {\'{E}lie {C}artan's work on isoparametric families of hypersurfaces},
      BOOKTITLE={Differential Geometry},
      SERIES={Proc. Symp. Pure Math.},
      VOLUME={27},
      YEAR={1975},
      VENUE={Stanford Univ., Stanford, CA 1973},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      PAGES={191--200},
      MRNUMBER = {0423260},
      ZBLNUMBER = {0318.53052},
     }
  • [O] Go to document C. Olmos, "Isoparametric submanifolds and their homogeneous structures," J. Differential Geom., vol. 38, iss. 2, pp. 225-234, 1993.
    @article {O, MRKEY = {1237484},
      AUTHOR = {Olmos, Carlos},
      TITLE = {Isoparametric submanifolds and their homogeneous structures},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {38},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {225--234},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {53C42 (53C35)},
      MRNUMBER = {1237484},
      MRREVIEWER = {Sheila Carter},
      URL = {http://projecteuclid.org/euclid.jdg/1214454294},
      ZBLNUMBER = {0791.53051},
      }
  • [OT] Go to document H. Ozeki and M. Takeuchi, "On some types of isoparametric hypersurfaces in spheres. II," Tôhoku Math. J., vol. 28, iss. 1, pp. 7-55, 1976.
    @article {OT, MRKEY = {0454889},
      AUTHOR = {Ozeki, Hideki and Takeuchi, Masaru},
      TITLE = {On some types of isoparametric hypersurfaces in spheres. {II}},
      JOURNAL = {Tôhoku Math. J.},
      FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
      VOLUME = {28},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {7--55},
      ISSN = {0040-8735},
      MRCLASS = {53C40},
      MRNUMBER = {0454889},
      MRREVIEWER = {Dirk Ferus},
      ZBLNUMBER = {0359.53012},
      DOI = {10.2748/tmj/1178240877},
      }
  • [R] H. Reckziegel, "Krümmungsflächen von isometrischen Immersionen in Räume konstanter Krümmung," Math. Ann., vol. 223, iss. 2, pp. 169-181, 1976.
    @article {R, MRKEY = {0425846},
      AUTHOR = {Reckziegel, Helmut},
      TITLE = {Krümmungsflächen von isometrischen {I}mmersionen in {R}äume konstanter {K}rümmung},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {223},
      YEAR = {1976},
      NUMBER = {2},
      PAGES = {169--181},
      ISSN = {0025-5831},
      MRCLASS = {53C40},
      MRNUMBER = {0425846},
      MRREVIEWER = {David A. Hoffman},
      ZBLNUMBER = {0319.53042},
     }
  • [Si] Go to document I. M. Singer, "Infinitesimally homogeneous spaces," Comm. Pure Appl. Math., vol. 13, pp. 685-697, 1960.
    @article {Si, MRKEY = {0131248},
      AUTHOR = {Singer, I. M.},
      TITLE = {Infinitesimally homogeneous spaces},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {13},
      YEAR = {1960},
      PAGES = {685--697},
      ISSN = {0010-3640},
      MRCLASS = {53.66},
      MRNUMBER = {0131248},
      MRREVIEWER = {G. So{ó}s},
      ZBLNUMBER = {0171.42503},
      DOI = {10.1002/cpa.3160130408},
      }
  • [T] Go to document C. Terng, "Isoparametric submanifolds and their Coxeter groups," J. Differential Geom., vol. 21, iss. 1, pp. 79-107, 1985.
    @article {T, MRKEY = {0806704},
      AUTHOR = {Terng, Chuu-Lian},
      TITLE = {Isoparametric submanifolds and their {C}oxeter groups},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {21},
      YEAR = {1985},
      NUMBER = {1},
      PAGES = {79--107},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {53C40 (53C42)},
      MRNUMBER = {0806704},
      MRREVIEWER = {Sheila Carter},
      URL = {http://projecteuclid.org/euclid.jdg/1214439466},
      ZBLNUMBER = {0615.53047},
      }
  • [Th1] Go to document G. Thorbergsson, "Isoparametric foliations and their buildings," Ann. of Math., vol. 133, iss. 2, pp. 429-446, 1991.
    @article {Th1, MRKEY = {1097244},
      AUTHOR = {Thorbergsson, Gudlaugur},
      TITLE = {Isoparametric foliations and their buildings},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {133},
      YEAR = {1991},
      NUMBER = {2},
      PAGES = {429--446},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {53C40 (51B25 57R30)},
      MRNUMBER = {1097244},
      MRREVIEWER = {Werner Ballmann},
      DOI = {10.2307/2944343},
      ZBLNUMBER = {0727.57028},
      }
  • [Th2] Go to document G. Thorbergsson, "A survey on isoparametric hypersurfaces and their generalizations," in Handbook of Differential Geometry, Vol. I, Amsterdam: North-Holland, 2000, pp. 963-995.
    @incollection {Th2, MRKEY = {1736861},
      AUTHOR = {Thorbergsson, Gudlaugur},
      TITLE = {A survey on isoparametric hypersurfaces and their generalizations},
      BOOKTITLE = {Handbook of Differential Geometry, {V}ol. {I}},
      PAGES = {963--995},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {2000},
      MRCLASS = {53C42 (53C40 58B20)},
      MRNUMBER = {1736861},
      MRREVIEWER = {Thomas E. Cecil},
      DOI = {10.1016/S1874-5741(00)80013-8},
      ZBLNUMBER = {0979.53002},
      }
  • [Y] Go to document S. Yau, "Open problems in geometry," in Chern—A Great Geometer of the Twentieth Century, Int. Press, Hong Kong, 1992, pp. 275-319.
    @incollection {Y, MRKEY = {1201369},
      AUTHOR = {Yau, Shing-Tung},
      TITLE = {Open problems in geometry},
      BOOKTITLE = {Chern---A Great Geometer of the Twentieth Century},
      PAGES = {275--319},
      PUBLISHER = {Int. Press, Hong Kong},
      YEAR = {1992},
      MRCLASS = {53-02 (01A67)},
      MRNUMBER = {1201369},
      ZBLNUMBER = {0826.01010},
      DOI = {10.1007/BF01360880},
      }

Authors

Reiko Miyaoka

Mathematical Institute
Tohoku University
Sendai 980-8578
Japan