CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY*

doi:10.1007/s10473-024-0101-7

Abstract

Abstract: This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of $S^n$ (or ${\bf R}^n$), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

Key words: Dupin hypersurfaces, isoparametric hypersurfaces, Lie sphere geometry, Lie sphere transformations, Lie curvatures

CLC Number:

53A07

Thomas E. Cecil. CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY^*[J].Acta mathematica scientia,Series B, 2024, 44(1): 1-36.

Trendmd

References

[1] Abresch U. Isoparametric hypersurfaces with four or six distinct principal curvatures. Math Ann, 1983, 264: 283-302
[2] Blaschke W.Vorlesungenüber Differentialgeometrie und Geometrische Grundlagen von Einsteins Relativitätstheorie. Vol 3. Berlin: Springer, 1929
[3] Cartan E. Familles de surfaces isoparamétriques dans les espaces à courbure constante. Annali di Mat, 1938, 17: 177-191
[4] Cartan E. Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques. Math Z, 1939, 45: 335-367
[5] Cartan E. Sur quelque familles remarquables d'hypersurfaces. C R Congrès Math Liège, 1939: 30-41
[6] Cartan E. Sur des familles d'hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions. Revista Univ Tucuman, Serie A, 1940, 1: 5-22
[7] Cecil T. On the Lie curvatures of Dupin hypersurfaces. Kodai Math J, 1990, 13: 143-153
[8] Cecil T. Lie sphere geometry and Dupin submanifolds//Verstraelen L, West A. Geometry and Topology of Submanifolds III. River Edge, NJ: World Scientific, 1991: 90-107
[9] Cecil T.Lie Sphere Geometry, with Applications to Submanifolds. 2nd ed. New York: Springer, 2008
[10] Cecil T. Isoparametric and Dupin hypersurfaces. Symmetry Inteyrablity and Geometry-Method and Application, 2008, 4: Art 062
[11] Cecil T. Compact Dupin hypersurfaces. ICCM Notices, 2021, 9(1): 57-68
[12] Cecil T, Chern S S. Tautness and Lie sphere geometry. Math Ann, 1987, 278: 381-399
[13] Cecil T, Chern S S. Dupin submanifolds in Lie sphere geometry//Jiang B, et al. Differential Geometry and Topology: Proceedings of the Special Year at Nankai Institute of Mathematics, Tianjin1986-87. Lecture Notes in Math 1369. Berlin: Springer, 1989: 1-48
[14] Cecil T, Chern S S.Dupin submanifolds in Lie sphere geometry (updated version). arXiv:2010.06429v1 [math.DG] 13 Oct 2020
[15] Cecil T, Chi Q S, Jensen G. Isoparametric hypersurfaces with four principal curvatures. Ann Math, 2007, 166: 1-76
[16] Cecil T, Chi Q S, Jensen G. Dupin hypersurfaces with four principal curvatures II. Geom Dedicata, 2007, 128: 55-95
[17] Cecil T, Chi Q S, Jensen G. Classifications of Dupin hypersurfaces//Dillen F, Van de Woestyne I. Pure and Applied Differential Geometry, PADGE2007. Aachen: Shaker Verlag, 2007: 48-56
[18] Cecil T, Chi Q S, Jensen G. On Kuiper's question whether taut submanifolds are algebraic. Pacific J Math, 2008, 234: 229-247
[19] Cecil T, Jensen G. Dupin hypersurfaces with three principal curvatures. Invent Math, 1998, 132: 121-178
[20] Cecil T, Jensen G. Dupin hypersurfaces with four principal curvatures. Geom Dedicata, 2000, 79: 1-49
[21] Cecil T, Ryan P. Focal sets, taut embeddings and the cyclides of Dupin. Math Ann, 1978, 236: 177-190
[22] Cecil T, Ryan P.Tight and Taut Immersions of Manifolds. Research Notes in Math 107. London: Pitman, 1985
[23] Cecil T, Ryan P.Geometry of Hypersurfaces. New York: Springer, 2015
[24] Chi Q S. Isoparametric hypersurfaces with four principal curvatures revisited. Nagoya Math J, 2009, 193: 129-154
[25] Chi Q S. A note on the paper "Isoparametric hypersurfaces with four principal curvatures''. Hongyou Wu Memorial Volume. Pacific Journal of Applied Mathematics, 2011, 3: 127-134
[26] Chi Q S. Isoparametric hypersurfaces with four principal curvatures II. Nagoya Math J, 2011, 204: 1-18
[27] Chi Q S. A new look at Condition A. Osaka J Math, 2012, 49: 133-166
[28] Chi Q S. Isoparametric hypersurfaces with four principal curvatures III. J Differential Geom, 2013, 94: 469-504
[29] Chi Q S. Isoparametric hypersurfaces with four principal curvatures IV. J Differential Geom, 2020, 115: 225-301
[30] Chi Q S. The isoparametric story, a heritage of élie Cartan. arXiv:2007.02137 [math.DG]
[31] Dorfmeister J, Neher E. Isoparametric hypersurfaces, case $g = 6, m = 1$. Communications in Algebra, 1985, 13: 2299-2368
[32] Dupin C.Applications de Géométrie et de Méchanique. Paris: Bachelier, 1822
[33] Ferus D, Karcher H, Münzner H F.Cliffordalgebren und neue isoparametrische Hyperflächen. Math Z, 1981, 177: 479-502 (see also an English translation by T.E. Cecil, arXiv:1112.2780v1 [mathDG] 13 Dec 2011)
[34] Grove K, Halperin S. Dupin hypersurfaces, group actions,the double mapping cylinder. J Diff Geom, 1987, 26: 429-459
[35] Immervoll S. On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres. Ann Math, 2008, 168: 1011-1024
[36] Levi-Civita T. Famiglie di superficie isoparametrische nell'ordinario spacio euclideo. Atti Accad naz Lincei Rend Cl Sci Fis Mat Natur, 1937, 26: 355-362
[37] Lie S. über Komplexe, inbesondere Linien- und Kugelkomplexe, mit Anwendung auf der Theorie der partieller Differentialgleichungen. Math Ann, 1872, 5: 145-208, 209-256
[38] Miyaoka R. Compact Dupin hypersurfaces with three principal curvatures. Math Z, 1984, 187: 433-452
[39] Miyaoka R. Dupin hypersurfaces and a Lie invariant. Kodai Math J, 1989, 12: 228-256
[40] Miyaoka R. Dupin hypersurfaces with six principal curvatures. Kodai Math J, 1989, 12: 308-315
[41] Miyaoka R. Isoparametric hypersurfaces with $(g,m) = (6,2)$. Ann Math, 2013, 177: 53-110
[42] Miyaoka R. Errata on isoparametric hypersurfaces with $(g,m) = (6,2)$. Ann Math, 2016, 183: 1057-1071
[43] Miyaoka R, Ozawa T.Construction of taut embeddings and Cecil-Ryan conjecture//Shiohama K. Geometry of Manifolds. Perspect Math 8. New York: Academic Press, 1989: 181-189
[44] Münzner H F. Isoparametrische Hyperflächen in Sphären. Math Ann, 1980, 251: 57-71 (see also an English translation by T.E. Cecil, Mathematics Department Faculty Scholarship 14. College of the Holy Cross.\\ https://crossworks.holycross.edu/math_fac_scholarship/14)
[45] Münzner H F. Isoparametrische Hyperflächen in Sphären II: über die Zerlegung der Sphäre in Ballbündel. Math Ann, 1981, 256: 215-232 (see also an English translation by T.E. Cecil, Mathematics Department Faculty Scholarship 13. College of the Holy Cross.\\ https://crossworks.holycross.edu/math_fac_scholarship/13)
[46] Niebergall R. Dupin hypersurfaces in ${\bf R}^5$ I. Geom Dedicata, 1991, 40: 1-22
[47] Niebergall R. Dupin hypersurfaces in ${\bf R}^5$ II. Geom Dedicata, 1992, 41: 5-38
[48] Nomizu K. Characteristic roots and vectors of a differentiable family of symmetric matrices. Lin and Multilin Alg, 1973, 2: 159-162
[49] Nomizu K. Some results in E. Cartan's theory of isoparametric families of hypersurfaces. Bull Amer Math Soc, 1973, 79: 1184-1188
[50] Nomizu K. élie Cartan's work on isoparametric families of hypersurfaces//Proc Sympos Pure Math 27, Part 1. Providence, RI: Amer Math Soc, 1975: 191-200
[51] Ozeki H, Takeuchi M. On some types of isoparametric hypersurfaces in spheres I, II. Tôhoku Math J, 1975, 27: 515-559; 1976, 28: 7-55
[52] Pinkall U.Dupin'sche Hyperflächen [D]. Freiburg: Univ Freiburg, 1981 (see also an English translation by T.E. Cecil, Mathematics Department Faculty Scholarship 15. College of the Holy Cross.\\ https://crossworks.holycross.edu/math_fac_scholarship/15)
[53] Pinkall U.Letter to T. Cecil. December, 5, 1984
[54] Pinkall U. Dupin'sche Hyperflächen in $E^4$. Manuscr Math, 1985, 51: 89-119
[55] Pinkall U. Dupin hypersurfaces. Math Ann, 1985, 270: 427-440
[56] Pinkall U, Thorbergsson G. Deformations of Dupin hypersurfaces. Proc Amer Math Soc, 1989, 107: 1037-1043
[57] Reckziegel H. On the eigenvalues of the shape operator of an isometric immersion into a space of constant curvature. Math Ann, 1979, 243: 71-82
[58] Ryan P. Homogeneity and some curvature conditions for hypersurfaces. Tôhoku Math J, 1969, 21: 363-388
[59] Samuel P. Projective Geometry. Berlin: Springer, 1988
[60] 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, 1938, 27: 203-207
[61] Siffert A. Classification of isoparametric hypersurfaces in spheres with $(g,m) = (6,1)$. Proc Amer Math Soc, 2016, 144: 2217-2230
[62] Siffert A. A new structural approach to isoparametric hypersurfaces in spheres. Ann Global Anal Geom, 2017, 52: 425-456
[63] Singley D. Smoothness theorems for the principal curvatures and principal vectors of a hypersurface. Rocky Mountain J Math, 1975, 5: 135-144
[64] Somigliana C. Sulle relazione fra il principio di Huygens e l'ottica geometrica. Atti Acc Sc Torino, 1918-1919, 54: 974-979
[65] Stolz S. Multiplicities of Dupin hypersurfaces. Invent Math, 1999, 138: 253-279
[66] Takagi R. A class of hypersurfaces with constant principal curvatures in a sphere. J Diff Geom, 1976, 11: 225-233
[67] Takagi R, Takahashi T. On the principal curvatures of homogeneous hypersurfaces in a sphere//Differential Geometry in honor of K. Yano. Tokyo: Kinokuniya, 1972: 469-481
[68] Thorbergsson G. Dupin hypersurfaces. Bull London Math Soc, 1983, 15: 493-498
[69] Thorbergsson G. A survey on isoparametric hypersurfaces and their generalizations//Handbook of Differential Geometry, Vol I. Amsterdam: North-Holland, 2000: 963-995

CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY^*

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 1

Metrics

Comments

Recommended 0