Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 062, 28 pages      arXiv:0809.1433      http://dx.doi.org/10.3842/SIGMA.2008.062
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Isoparametric and Dupin Hypersurfaces

Thomas E. Cecil
Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA

Received June 24, 2008, in final form August 28, 2008; Published online September 08, 2008

Abstract
A hypersurface Mn−1 in a real space-form Rn, Sn or Hn is isoparametric if it has constant principal curvatures. For Rn and Hn, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Élie Cartan showed in a series of four papers in 1938–1940, the subject is much deeper and more complex for hypersurfaces in the sphere Sn. A hypersurface Mn−1 in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on Mn−1, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field.

Key words: isoparametric hypersurface; Dupin hypersurface.

pdf (396 kb)   ps (219 kb)   tex (38 kb)

References

  1. Abresch U., Isoparametric hypersurfaces with four or six distinct principal curvatures, Math. Ann. 264 (1983), 283-302.
  2. Alexandrino M., Singular Riemannian foliations with sections, Illinois J. Math. 48 (2004), 1163-1182, math.DG/0311454.
  3. Atiyah M.F., Bott R., Shapiro A., Clifford modules, Topology 3 (1964), suppl. 1, 3-38.
  4. Banchoff T., The spherical two-piece property and tight surfaces in spheres, J. Differential Geom. 4 (1970), 193-205.
  5. Berndt J., Real hypersurfaces with constant principal curvatures in complex space forms, in Proceedings of the Tenth International Workshop on Differential Geometry, Kyungpook Nat. Univ., Taegu, 2006, 1-12.
  6. Berndt J., Console S., Olmos C., Submanifolds and holonomy, Chapman and Hall/CRC Research Notes in Mathematics, Vol. 434, Chapman and Hall/CRC, Boca Raton, Florida, 2003.
  7. Bott R., Samelson H., Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964-1029.
  8. Cartan É., Familles de surfaces isoparamétriques dans les espaces à courbure constante, Annali di Mat. 17 (1938), 177-191 (see also in Oeuvres Complètes, Partie III, Vol. 2, 1431-1445).
  9. Cartan É., Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z. 45 (1939), 335-367 (see also in Oeuvres Complètes, Partie III, Vol. 2, 1447-1479).
  10. Cartan É., Sur quelque familles remarquables d'hypersurfaces, in C.R. Congrès Math. Liège, 1939, 30-41 (see also in Oeuvres Complètes, Partie III, Vol. 2, 1481-1492).
  11. Cartan É., Sur des familles d'hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions, Revista Univ. Tucuman, Serie A, 1 (1940), 5-22 (see also in Oeuvres Complètes, Partie III, Vol. 2, 1513-1530).
  12. Carter S., Sentürk Z., The space of immersions parallel to a given immersion, J. London Math. Soc. (2) 50 (1994), 404-416.
  13. Carter S., West A., Tight and taut immersions, Proc. London Math. Soc. (3) 25 (1972), 701-720.
  14. Carter S., West A., Totally focal embeddings, J. Differential Geom. 13 (1978), 251-261.
  15. Carter S., West A., Totally focal embeddings: special cases, J. Differential Geom. 16 (1981), 685-697.
  16. Carter S., West A., A characterisation of isoparametric hypersurfaces in spheres, J. London Math. Soc. (2) 26 (1982), 183-192.
  17. Carter S., West A., Isoparametric systems and transnormality, Proc. London Math. Soc. (3) 51 (1985), 520-542.
  18. Carter S., West A., Isoparametric and totally focal submanifolds, Proc. London Math. Soc. (3) 60 (1990), 609-624.
  19. Cayley A., On the cyclide, Quart. J. of Pure and Appl. Math. 12 (1873), 148-165 (see also in Collected Mathematical Papers, Vol. 9, Cambridge U. Press, 1896, 64-78).
  20. Cecil T., A characterization of metric spheres in hyperbolic space by Morse theory, Tôhoku Math. J. (2) 26 (1974), 341-351.
  21. Cecil T., Taut immersions of non-compact surfaces into a Euclidean 3-space, J. Differential Geom. 11 (1976), 451-459.
  22. Cecil T., Reducible Dupin submanifolds, Geom. Dedicata 32 (1989), 281-300.
  23. Cecil T., On the Lie curvatures of Dupin hypersurfaces, Kodai Math. J. 13 (1990), 143-153.
  24. Cecil T., Lie sphere geometry and Dupin submanifolds, in Geometry and Topology of Submanifolds III (Leeds, 1990), Editors L. Verstraelen and A. West, World Scientific, River Edge, NJ, 1991, 90-107.
  25. Cecil T., Taut and Dupin submanifolds, in Tight and Taut Submanifolds (Berkeley, CA, 1994), Editors T. Cecil and S.-S. Chern, Math. Sci. Res. Inst. Publ., Vol. 32, Cambridge Univ. Press, Cambridge, 1997, 135-180.
  26. Cecil T., Lie sphere geometry, with applications to submanifolds, 2nd ed., Universitext, Springer, New York, 2008.
  27. Cecil T., Chern S.-S., Tautness and Lie sphere geometry, Math. Ann. 278 (1987), 381-399.
  28. Cecil T., Chern S.-S., Dupin submanifolds in Lie sphere geometry, in Differential Geometry and Topology (Tianjin, 1986-87), Editors B. Jiang et al., Lecture Notes in Math., Vol. 1369, Springer, Berlin - New York, 1989, 1-48.
  29. Cecil T., Chi Q.-S., Jensen G., Isoparametric hypersurfaces with four principal curvatures, Ann. of Math. (2) 166 (2007), 1-76.
  30. Cecil T., Chi Q.-S., Jensen G., Dupin hypersurfaces with four principal curvatures. II, Geom. Dedicata 128 (2007), 55-95.
  31. Cecil T., Chi Q.-S., Jensen G., Classifications of Dupin hypersurfaces, in Pure and Applied Differential Geometry, PADGE 2007, Editors F. Dillen and I. van de Woestyne, Shaker Verlag, Aachen, 2007, 48-56.
  32. Cecil T., Chi Q.-S., Jensen G., On Kuiper's question whether taut submanifolds are algebraic, Pacific J. Math. 234 (2008), 229-247.
  33. Cecil T., Jensen G., Dupin hypersurfaces with three principal curvatures, Invent. Math. 132 (1998), 121-178.
  34. Cecil T., Jensen G., Dupin hypersurfaces with four principal curvatures, Geom. Dedicata 79 (2000), 1-49.
  35. Cecil T., Ryan P., Focal sets, taut embeddings and the cyclides of Dupin, Math. Ann. 236 (1978), 177-190.
  36. Cecil T., Ryan P., Conformal geometry and the cyclides of Dupin, Canad. J. Math. 32 (1980), 767-782.
  37. Cecil T., Ryan P., Tight spherical embeddings, in Global Differential Geometry and Analysis (Berlin 1979), Editors D. Ferus, W. Kühnel, U. Simon and B. Wegner, Lecture Notes in Math., Vol. 838, Springer, Berlin - New York, 1981, 94-104.
  38. Cecil T., Ryan P., Tight and taut immersions of manifolds, Research Notes in Math., Vol. 107, Pitman, London, 1985.
  39. Chern S.-S., An introduction to Dupin submanifolds, in Differential Geometry: A Symposium in Honour of Manfredo do Carmo (Rio de Janeiro, 1988), Editors H.B. Lawson and K. Tenenblat, Pitman Monographs Surveys Pure Appl. Math., Vol. 52, Longman Sci. Tech., Harlow, 1991, 95-102.
  40. Chi Q.-S., Isoparametric hypersurfaces with four principal curvatures revisited, in Nagoya Math. J., to appear, arXiv:0803.1284.
  41. Christ U., Homogeneity of equifocal submanifolds, J. Differential Geom. 62 (2002), 1-15.
  42. Dadok J., Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), 125-137.
  43. Dajczer M., Florit L., Tojeiro R., Reducibility of Dupin submanifolds, Illinois J. Math. 49 (2005), 759-791.
  44. Degen W., Generalized cyclides for use in CAGD, in Computer-Aided Surface Geometry and Design: the Mathematics of Surfaces IV (Bath, 1990), Editor A. Bowyer, Inst. Math. Appl. Conf. New Ser., Vol. 48, Oxford Univ. Press, New York, 1994, 349-363.
  45. Dorfmeister J., Neher E., An algebraic approach to isoparametric hypersurfaces. I, Tôhoku Math. J. (2) 35 (1983), 187-224.
    Dorfmeister J., Neher E., An algebraic approach to isoparametric hypersurfaces. II, Tôhoku Math. J. (2) 35 (1983), 225-247.
  46. Dorfmeister J., Neher E., Isoparametric triple systems of algebra type, Osaka J. Math. 20 (1983), 145-175.
  47. Dorfmeister J., Neher E., Isoparametric triple systems of FKM-type. I, Abh. Math. Sem. Hamburg 53 (1983), 191-216.
  48. Dorfmeister J., Neher E., Isoparametric triple systems of FKM-type. II, Manuscripta Math. 43 (1983), 13-44.
  49. Dorfmeister J., Neher E., Isoparametric hypersurfaces, case g = 6, m = 1, Comm. Algebra 13 (1985), 2299-2368.
  50. Dorfmeister J., Neher E., Isoparametric triple systems with special Z-structure, Algebras Groups Geom. 7 (1990), 21-94.
  51. Dupin C., Applications de géométrie et de méchanique, Bachelier, Paris, 1822.
  52. Eschenburg J.-H., Schroeder V., Tits distance of Hadamard manifolds and isoparametric hypersurfaces, Geom. Dedicata 40 (1991), 97-101.
  53. Fang F., Multiplicities of principal curvatures of isoparametric hypersurfaces, Preprint, Max Planck Institut für Mathematik, Bonn, 1996.
  54. Fang F., On the topology of isoparametric hypersurfaces with four distinct principal curvatures, Proc. Amer. Math. Soc. 127 (1999), 259-264.
  55. Fang F., Topology of Dupin hypersurfaces with six principal curvatures, Math. Z. 231 (1999), 533-555.
  56. Ferapontov E.V., Dupin hypersurfaces and integrable hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants, Differential Geom. Appl. 5 (1995), 121-152.
  57. Ferapontov E.V., Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and N-wave systems, Differential Geom. Appl. 5 (1995), 335-369.
  58. Ferus D., Karcher H., Münzner H.-F., Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z. 177 (1981), 479-502.
  59. Fladt K., Baur A., Analytische Geometrie spezieller Flächen und Raumkurven, Friedr. Vieweg and Sohn, Braunschweig, 1975.
  60. Geatti L., Gorodski C., Polar orthogonal representations of real reductive algebraic groups, J. Algebra, to appear, arXiv:0801.0574.
  61. Grove K., Halperin S., Dupin hypersurfaces, group actions, and the double mapping cylinder, J. Differential Geom. 26 (1987), 429-459.
  62. Hahn J., Isoparametric hypersurfaces in the pseudo-Riemannian space forms, Math. Z. 187 (1984), 195-208.
  63. Hahn J., Isotropy representations of semisimple symmetric spaces and homogeneous hypersurfaces, J. Math. Soc. Japan 40 (1988), 271-288.
  64. Harle C., Isoparametric families of submanifolds, Bol. Soc. Brasil Mat. 13 (1982), no. 2, 35-48.
  65. Heintze E., Liu X., Homogeneity of infinite dimensional isoparametric submanifolds, Ann. of Math. (2) 149 (1999), 149-181, math.DG/9901150.
  66. Heintze E., Olmos C., Thorbergsson G., Submanifolds with constant principal curvatures and normal holonomy groups, Internat. J. Math. 2 (1991), 167-175.
  67. Hsiang W.-Y., Lawson H. B., Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1-38.
  68. Hsiang W.-Y., Palais R., Terng C.-L., The topology of isoparametric submanifolds, J. Differential Geom. 27 (1988), 423-460.
  69. Hu Z., Li D., Möbius isoparametric hypersurfaces with three distinct principal curvatures, Pacific J. Math. 232 (2007), 289-311.
  70. Hu Z., Li H.-Z., Classification of Möbius isoparametric hypersurfaces in S4, Nagoya Math. J. 179 (2005), 147-162.
  71. Hu Z., Li H.-Z., Wang C.-P., Classification of Möbius isoparametric hypersurfaces in S5, Monatsh. Math. 151 (2007), 201-222.
  72. Immervoll S., On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres, Ann. Math., to appear.
  73. Ivey T., Surfaces with orthogonal families of circles, Proc. Amer. Math. Soc. 123 (1995), 865-872.
  74. Kamran N., Tenenblat K., Laplace transformation in higher dimensions, Duke Math. J. 84 (1996), 237-266.
  75. Levi-Civita T., Famiglie di superficie isoparametrische nell'ordinario spacio euclideo, Atti. Accad. naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 26 (1937), 355-362.
  76. Li H.-Z., Liu H.-L., Wang C.-P., Zhao G.-S., Möbius isoparametric hypersurfaces in Sn+1 with two distinct principal curvatures, Acta Math. Sin. (Engl. Ser.) 18 (2002), 437-446.
  77. Li Z.-Q., Xie X.-H., Space-like isoparametric hypersurfaces in Lorentzian space forms, J. Nanchang Univ. Natur. Sci. Ed. 28 (2004), 113-117 (English transl: Front. Math. China 1 (2006), 130-137).
  78. Lilienthal R., Besondere Flächen, in Encyklopädie der Math. Wissenschaften, Vol. III, Teubner, Leipzig, 1902-1927, 269-354.
  79. Liouville J., Note au sujet de l'article précedént, J. de Math. Pure et Appl. (1) 12 (1847), 265-290.
  80. Lytchak A., Thorbergsson G., Variationally complete actions on nonnegatively curved manifolds, Illinois J. Math. 51 (2007), 605-615.
  81. Magid M., Lorentzian isoparametric hypersurfaces, Pacific J. Math. 118 (1985), 165-197.
  82. Maxwell J.C., On the cyclide, Quart. J. of Pure and Appl. Math. 34 (1867), 111-126 (see also in Scientific papers, Vol. 2, Cambridge U. Press, 1890, 144-159).
  83. Milnor J., Morse theory, Ann. Math. Stud., Vol. 51, Princeton U. Press, Princeton, NJ, 1963.
  84. Miyaoka R., Compact Dupin hypersurfaces with three principal curvatures, Math. Z. 187 (1984), 433-452.
  85. Miyaoka R., Taut embeddings and Dupin hypersurfaces, in Differential Geometry of Submanifolds (Kyoto, 1984), Editor K. Kenmotsu, Lecture Notes in Math., Vol. 1090, Springer, Berlin - New York, 1984, 15-23.
  86. Miyaoka R., Dupin hypersurfaces and a Lie invariant, Kodai Math. J. 12 (1989), 228-256.
  87. Miyaoka R., Dupin hypersurfaces with six principal curvatures, Kodai Math. J. 12 (1989), 308-315.
  88. Miyaoka R., The linear isotropy group of G2/SO(4), the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. 30 (1993), 179-202.
  89. Miyaoka R., The Dorfmeister-Neher's theorem on isoparametric hypersurfaces, math.DG/0602519.
  90. Miyaoka R., Ozawa T., Construction of taut embeddings and Cecil-Ryan conjecture, in Geometry of Manifolds, Editor K. Shiohama, Perspect. Math., Vol. 8, Academic Press, Boston, 1989, 181-189.
  91. Morse M., Cairns S., Critical point theory in global analysis and differential topology, Academic Press, New York, 1969.
  92. Mullen S., Isoparametric systems on symmetric spaces, in Geometry and Topology of Submanifolds VI, Editors F. Dillen et al., World Scientific, River Edge, NJ, 1994, 152-154.
  93. Münzner H.-F., Isoparametrische Hyperflächen in Sphären, Math. Ann. 251 (1980), 57-71.
  94. Münzner H.-F., Isoparametrische Hyperflächen in Sphären II: Über die Zerlegung der Sphäre in Ballbündel, Math. Ann. 256 (1981), 215-232.
  95. Niebergall R., Dupin hypersurfaces in R5. I, Geom. Dedicata 40 (1991), 1-22.
  96. Niebergall R., Dupin hypersurfaces in R5. II, Geom. Dedicata 41 (1992), 5-38.
  97. Niebergall R., Ryan P., Isoparametric hypersurfaces - the affine case, in Geometry and Topology of Submanifolds V, Editors F. Dillen et al., World Scientific, River Edge, NJ, 1993, 201-214.
  98. Niebergall R., Ryan P., Affine isoparametric hypersurfaces, Math. Z. 217 (1994), 479-485.
  99. Niebergall R., Ryan P., Focal sets in affine geometry, in Geometry and Topology of Submanifolds VI, Editors F. Dillen et al., World Scientific, River Edge, NJ, 1994, 155-164.
  100. Niebergall R., Ryan P., Affine Dupin surfaces, Trans. Amer. Math. Soc. 348 (1996), 1093-1117.
  101. Niebergall R., Ryan P., Real hypersurfaces in complex space forms, in Tight and Taut Submanifolds (Berkeley, CA, 1994), Editors T. Cecil and S.-S. Chern, Math. Sci. Res. Inst. Publ., Vol. 32, Cambridge Univ. Press, Cambridge, 1997, 233-305.
  102. Nomizu K., Some results in E. Cartan's theory of isoparametric families of hypersurfaces, Bull. Amer. Math. Soc. 79 (1973), 1184-1188.
  103. Nomizu K., Élie Cartan's work on isoparametric families of hypersurfaces, in Differential geometry (Stanford, 1973), Editors S.-S. Chern and R. Osserman, Proc. Sympos. Pure Math., Vol. 27, Amer. Math. Soc., Providence, RI, 1975, Part 1, 191-200.
  104. Nomizu K., On isoparametric hypersurfaces in Lorentzian space forms, Japan. J. Math. 7 (1981), 217-226.
  105. Olmos C., Isoparametric submanifolds and their homogeneous structure, J. Differential Geom. 38 (1993), 225-234.
  106. Ozawa T., On the critical sets of distance functions to a taut submanifold, Math. Ann. 276 (1986), 91-96.
  107. Ozeki H., Takeuchi M., On some types of isoparametric hypersurfaces in spheres. I, Tôhoku Math. J. 27 (1975), 515-559.
  108. Ozeki H., Takeuchi M., On some types of isoparametric hypersurfaces in spheres. II, Tôhoku Math. J. 28 (1976), 7-55.
  109. Palais R., Terng C.-L., A general theory of canonical forms, Trans. Amer. Math. Soc. 300 (1987), 771-789.
  110. Palais R., Terng C.-L., Critical point theory and submanifold geometry, Lecture Notes in Math., Vol. 1353, Springer, Berlin - New York, 1988.
  111. Peng C.-K., Hou Z., A remark on the isoparametric polynomials of degree 6, in Differential Geometry and Topology (Tianjin, 1986-87), Editors B. Jiang et al., Lecture Notes in Math., Vol 1369, Springer, Berlin - New York, 1989, 222-224.
  112. Pinkall U., Dupin'sche Hyperflächen, Dissertation, Univ. Freiburg, 1981.
  113. Pinkall U., Letter to T. Cecil, December 5, 1984.
  114. Pinkall U., Dupin'sche Hyperflächen in E4, Manuscripta Math. 51 (1985), 89-119.
  115. Pinkall U., Dupin hypersurfaces, Math. Ann. 270 (1985), 427-440.
  116. Pinkall U., Curvature properties of taut submanifolds, Geom. Dedicata 20 (1986), 79-83.
  117. Pinkall U., Thorbergsson G., Deformations of Dupin hypersurfaces, Proc. Amer. Math. Soc. 10 (1989), 1037-1043.
  118. Pinkall U., Thorbergsson G., Examples of infinite dimensional isoparametric submanifolds, Math. Z. 205 (1990), 279-286.
  119. Pratt M.J., Cyclides in computer aided geometric design, Comput. Aided Geom. Design 7 (1990), 221-242.
  120. Pratt M.J., Cyclides in computer aided geometric design. II, Comput. Aided Geom. Design 12 (1995), 131-152.
  121. Reckziegel H., On the eigenvalues of the shape operator of an isometric immersion into a space of constant curvature, Math. Ann. 243 (1979), 71-82.
  122. Riveros C.M.C., Rodrigues L.A., Tenenblat K., On Dupin hypersurfaces with constant Möbius curvature, Pacific J. Math. 236 (2008), 89-103.
  123. Riveros C.M.C., Tenenblat K., On four dimensional Dupin hypersurfaces in Euclidean space, An. Acad. Brasil Ciênc. 75 (2003), 1-7.
  124. Riveros C.M.C., Tenenblat K., Dupin hypersurfaces in R5, Canad. J. Math. 57 (2005), 1291-1313.
  125. Rodrigues L.A., Tenenblat K., A characterization of Moebius isoparametric hypersurfaces of the sphere, Preprint, 2008.
  126. Ryan P.J., Hypersurfaces with parallel Ricci tensor, Osaka J. Math. 8 (1971), 251-259.
  127. Schrott M., Odehnal B., Ortho-circles of Dupin cyclides, J. Geom. Graph. 10 (2006), 73-98.
  128. Segre B., Una proprietá caratteristica de tre sistemi 1 di superficie, Atti Acc. Sc. Torino 59 (1924), 666-671.
  129. Segre B., Famiglie di ipersuperficie isoparametrische negli spazi euclidei ad un qualunque numero di demesioni, Atti. Accad. naz Lincie Rend. Cl. Sci. Fis. Mat. Natur. 27 (1938), 203-207.
  130. Singley D., Smoothness theorems for the principal curvatures and principal vectors of a hypersurface, Rocky Mountain J. Math. 5 (1975), 135-144.
  131. Solomon B., The harmonic analysis of cubic isoparametric minimal hypersurfaces. I. Dimensions 3 and 6, Amer. J. Math. 112 (1990), 157-203.
  132. Solomon B., The harmonic analysis of cubic isoparametric minimal hypersurfaces. II. Dimensions 12 and 24, Amer. J. Math. 112 (1990), 205-241.
  133. Solomon B., Quartic isoparametric hypersurfaces and quadratic forms, Math. Ann. 293 (1992), 387-398.
  134. Somigliana C., Sulle relazione fra il principio di Huygens e l'ottica geometrica, Atti Acc. Sc. Torino 54 (1918-1919), 974-979 (see also in Memorie Scelte, 434-439).
  135. Spivak M., A comprehensive introduction to differential geometry, Vol. 4, Publish or Perish, Boston, 1975.
  136. Srinivas Y.L., Dutta D., Blending and joining using cyclides, ASME Trans. J. Mechanical Design 116 (1994), 1034-1041.
  137. Srinivas Y.L., Dutta D., An intuitive procedure for constructing complex objects using cyclides, Computer-Aided Design 26 (1994), 327-335.
  138. Srinivas Y.L., Dutta D., Cyclides in geometric modeling: computational tools for an algorithmic infrastructure, ASME Trans. J. Mechanical Design 117 (1995), 363-373.
  139. Srinivas Y.L., Dutta D., Rational parametrization of parabolic cyclides, Comput. Aided Geom. Design 12 (1995), 551-566.
  140. Stolz S., Multiplicities of Dupin hypersurfaces, Invent. Math. 138 (1999), 253-279.
  141. Strübing W., Isoparametric submanifolds, Geom. Dedicata 20 (1986), 367-387.
  142. Takagi R., A class of hypersurfaces with constant principal curvatures in a sphere, J. Differential Geom. 11 (1976), 225-233.
  143. Takagi R., Takahashi T., On the principal curvatures of homogeneous hypersurfaces in a sphere, in Differential Geometry (in Honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, 469-481.
  144. Takeuchi M., Proper Dupin hypersurfaces generated by symmetric submanifolds, Osaka Math. J. 28 (1991), 153-161.
  145. Takeuchi M., Kobayashi S., Minimal imbeddings of R-spaces, J. Differential Geom. 2 (1968), 203-215.
  146. Tang Z.-Z., Isoparametric hypersurfaces with four distinct principal curvatures, Chinese Sci. Bull. 36 (1991), 1237-1240.
  147. Tang Z.-Z., Multiplicities of equifocal hypersurfaces in symmetric spaces, Asian J. Math. 2 (1998), 181-214.
  148. Terng C.-L., Isoparametric submanifolds and their Coxeter groups, J. Differential Geom. 21 (1985), 79-107.
  149. Terng C.-L., Convexity theorem for isoparametric submanifolds, Invent. Math. 85 (1986), 487-492.
  150. Terng C.-L., Submanifolds with flat normal bundle, Math. Ann. 277 (1987), 95-111.
  151. Terng C.-L., Proper Fredholm submanifolds of Hilbert space, J. Differential Geom. 29 (1989), 9-47.
  152. Terng C.-L., Recent progress in submanifold geometry, in Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, 1990), Editors R. Greene and S.T. Yau, Proc. Sympos. Pure Math., Vol. 54, Amer. Math. Soc., Providence, RI, 1993, Part 1, 439-484.
  153. Terng C.-L., Thorbergsson G., Submanifold geometry in symmetric spaces, J. Differential Geom. 42 (1995), 665-718.
  154. Terng C.-L., Thorbergsson G., Taut immersions into complete Riemannian manifolds, in Tight and Taut Submanifolds, Editors T. Cecil and S.-S. Chern, Math. Sci. Res. Inst. Publ., Vol. 32, Cambridge Univ. Press, Cambridge, 1997, 181-228.
  155. Thorbergsson G., Dupin hypersurfaces, Bull. London Math. Soc. 15 (1983), 493-498.
  156. Thorbergsson G., Isoparametric foliations and their buildings, Ann. Math. 133 (1991), 429-446.
  157. Thorbergsson G., A survey on isoparametric hypersurfaces and their generalizations, in Handbook of Differential Geometry, Vol. I, Editors F. Dillen and L. Verstraelen, Elsevier Science, Amsterdam, 2000, 963-995.
  158. Töben D., Parallel focal structure and singular Riemannian foliations, Trans. Amer. Math. Soc. 358 (2006), 1677-1704, math.DG/0403050.
  159. Verhóczki L., Isoparametric submanifolds of general Riemannian manifolds, in Differential Geometry and Its Applications (Eger, 1989), Editors J. Szenthe and L. Tamássy, Colloq. Math. Soc. János Bolyai, Vol. 56, North-Holland, Amsterdam, 1992, 691-705.
  160. Wang C.-P., Surfaces in Möbius geometry, Nagoya Math. J. 125 (1992), 53-72.
  161. Wang C.-P., Möbius geometry for hypersurfaces in S4, Nagoya Math. J. 139 (1995), 1-20.
  162. Wang C.-P., Möbius geometry of submanifolds in Sn, Manuscripta Math. 96 (1998), 517-534.
  163. Wang Q.-M., Isoparametric maps of Riemannian manifolds and their applications, in Advances in Science of China, Mathematics, Vol. 2, Editors C.H. Gu and Y. Wang, Wiley-Interscience, New York, 1986, 79-103.
  164. Wang Q.-M., Isoparametric functions on Riemannian manifolds. I, Math. Ann. 277 (1987), 639-646.
  165. Wang Q.-M., On the topology of Clifford isoparametric hypersurfaces, J. Differential Geom. 27 (1988), 55-66.
  166. West A., Isoparametric systems, in Geometry and Topology of Submanifolds, Editors J.-M. Morvan and L. Verstraelen, World Scientific, River Edge, NJ, 1989, 222-230.
  167. West A., Isoparametric systems on symmetric spaces, in Geometry and Topology of Submanifolds V, Editors F. Dillen et al., World Scientific, River Edge, NJ, 1993, 281-287.
  168. Wu B., Isoparametric submanifolds of hyperbolic spaces, Trans. Amer. Math. Soc. 331 (1992), 609-626.
  169. Wu B., A finiteness theorem for isoparametric hypersurfaces, Geom. Dedicata 50 (1994), 247-250.
  170. Yau S.-T., Open problems in geometry, in Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, 1990), Editors R. Greene and S.T. Yau, Proc. Sympos. Pure Math., Vol. 54, Amer. Math. Soc., Providence, RI, 1993, Part 1, 439-484.
  171. Zhao Q., Isoparametric submanifolds of hyperbolic spaces, Chinese J. Contemp. Math. 14 (1993), 339-346.

Previous article   Next article   Contents of Volume 4 (2008)