{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:37:57Z","timestamp":1759970277619,"version":"build-2065373602"},"reference-count":79,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T00:00:00Z","timestamp":1736726400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A hypersurface Mn in a real space form Rn+1, Sn+1, or Hn+1 is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and M\u00fcnzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938\u20131940 and two papers of M\u00fcnzner that were published in preprint form in the early 1970s and as journal articles in 1980\u20131981. These papers of Cartan and M\u00fcnzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres.<\/jats:p>","DOI":"10.3390\/axioms14010056","type":"journal-article","created":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T09:49:17Z","timestamp":1736761757000},"page":"56","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Work of Cartan and M\u00fcnzner on Isoparametric Hypersurfaces"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0994-9314","authenticated-orcid":false,"given":"Thomas E.","family":"Cecil","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA"}]},{"given":"Patrick J.","family":"Ryan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S4K1, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF02410700","article-title":"Familles de surfaces isoparam\u00e9triques dans les espaces \u00e0 courbure constante","volume":"17","author":"Cartan","year":"1938","journal-title":"Ann. Mat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/BF01580289","article-title":"Sur des familles remarquables d\u2019hypersurfaces isoparam\u00e9triques dans les espaces sph\u00e9riques","volume":"45","author":"Cartan","year":"1939","journal-title":"Math. Z."},{"key":"ref_3","first-page":"30","article-title":"Sur quelque familles remarquables d\u2019hypersurfaces","volume":"1","author":"Cartan","year":"1939","journal-title":"C.R. Congr\u00e8s Math. Li\u00e8ge"},{"key":"ref_4","first-page":"5","article-title":"Sur des familles d\u2019hypersurfaces isoparam\u00e9triques des espaces sph\u00e9riques \u00e0 5 et \u00e0 9 dimensions","volume":"1","author":"Cartan","year":"1940","journal-title":"Rev. Univ. Tucuman. Ser. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/BF01420281","article-title":"Isoparametrische Hyperfl\u00e4chen in Sph\u00e4ren I","volume":"251","year":"1980","journal-title":"Math. Ann."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/BF01450799","article-title":"Isoparametrische Hyperfl\u00e4chen in Sph\u00e4ren II","volume":"256","year":"1981","journal-title":"Math. Ann."},{"key":"ref_7","unstructured":"Chi, Q.-S. (2018, January 28\u201330). The isoparametric story, a heritage of \u00c9lie Cartan. Proceedings of the International Consortium of Chinese Mathematicians, Taipei, Taiwan."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Cecil, T., and Ryan, P. (2015). Geometry of Hypersurfaces, Springer.","DOI":"10.1007\/978-1-4939-3246-7"},{"key":"ref_9","unstructured":"Kobayashi, S., and Nomizu, K. Foundations of Differential Geometry, Wiley\u2013Interscience. Volume 1, 1963, Volume II, 1969."},{"key":"ref_10","first-page":"974","article-title":"Sulle relazione fra il principio di Huygens e l\u2019ottica geometrica","volume":"54","author":"Somigliana","year":"1918","journal-title":"Atti Acc. Sci. Torino"},{"key":"ref_11","first-page":"666","article-title":"Una propriet\u00e1 caratteristica de tre sistemi \u221e1 di superficie","volume":"59","author":"Segre","year":"1924","journal-title":"Atti Acc. Sc. Torino"},{"key":"ref_12","first-page":"355","article-title":"Famiglie di superficie isoparametrische nell\u2019ordinario spacio euclideo","volume":"26","year":"1937","journal-title":"Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur."},{"key":"ref_13","first-page":"203","article-title":"Famiglie di ipersuperficie isoparametrische negli spazi euclidei ad un qualunque numero di demesioni","volume":"27","author":"Segre","year":"1938","journal-title":"Atti. Accad. Naz Lincei Rend. Cl. Sci. Fis. Mat. Natur."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"341","DOI":"10.2748\/tmj\/1178241128","article-title":"A characterization of metric spheres in hyperbolic space by Morse theory","volume":"26","author":"Cecil","year":"1974","journal-title":"T\u00f4hoku Math. J."},{"key":"ref_15","first-page":"1509","article-title":"Isoparametric hypersurfaces in complex projective spaces","volume":"Volume 3","author":"Wang","year":"1982","journal-title":"Symposium on Differential Geometry and Differential Equations (Beijing, 1980)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s40863-022-00291-2","article-title":"From isoparametric submanifolds to polar foliations","volume":"16","author":"Thorbergsson","year":"2022","journal-title":"S\u00e3o Paulo J. Math. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4487","DOI":"10.1090\/S0002-9947-00-02578-2","article-title":"Principal curvatures of isoparametric hypersurfaces in CPn","volume":"352","author":"Xiao","year":"2000","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1016\/j.aim.2017.05.012","article-title":"Isoparametric hypersurfaces in complex hyperbolic spaces","volume":"314","year":"2017","journal-title":"Adv. Math."},{"key":"ref_19","first-page":"1211","article-title":"Isoparametric foliations on complex projective spaces","volume":"368","year":"2016","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_20","unstructured":"Dominguez-Vazquez, M., and Kollross, A. (2024). On isoparametric foliations of complex and quaternionic projective spaces. arXiv."},{"key":"ref_21","unstructured":"Spivak, M. (1999). A Comprehensive Introduction to Differential Geometry, Publish or Perish. [3rd ed.]. Volumes 1\u20135."},{"key":"ref_22","first-page":"251","article-title":"Hypersurfaces with parallel Ricci tensor","volume":"8","author":"Ryan","year":"1971","journal-title":"Osaka J. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01448439","article-title":"\u00dcber die Komposition der quadratischen Formen","volume":"88","author":"Hurwitz","year":"1923","journal-title":"Math. Ann."},{"key":"ref_24","first-page":"933","article-title":"On the riemannian Geometries admitting an absolute parallelism","volume":"29","author":"Cartan","year":"1926","journal-title":"Proc. Akad. Amst."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kuiper, N. (1980). Tight embeddings and maps. Submanifolds of Geometrical Class Three in En. The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer.","DOI":"10.1007\/978-1-4613-8109-9_6"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/BF00233101","article-title":"Oktaven, Ausnahmegruppen und Oktavengeometrie","volume":"19","author":"Freudenthal","year":"1985","journal-title":"Geom. Dedicata"},{"key":"ref_27","unstructured":"Ewert, H. (1994). \u00dcber Isoparametrische Hyperfl\u00e4chen in R\u00e4umen Konstanter Kr\u00fcmmung. [Ph.D. Thesis, Universit\u00e4t K\u00f6ln]."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"265","DOI":"10.4310\/jdg\/1214433424","article-title":"Manifolds with planar geodesics","volume":"11","author":"Little","year":"1976","journal-title":"J. Differ. Geom."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01265638","article-title":"Projective planes and isoparametric hypersurfaces","volume":"58","author":"Knarr","year":"1995","journal-title":"Geom. Dedicata"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s002290050106","article-title":"Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces","volume":"97","author":"Console","year":"1998","journal-title":"Manuscripta Math."},{"key":"ref_31","first-page":"73","article-title":"Triality and the normal sections of Cartan\u2019s isoparametric hypersurfaces","volume":"52","author":"Sanchez","year":"2011","journal-title":"Rev. Un. Mat. Argentina"},{"key":"ref_32","first-page":"139","article-title":"Projective spaces in the algebraic sets of planar normal sections of homogeneous isoparametric submanifolds","volume":"55","author":"Giunta","year":"2014","journal-title":"Rev. Un. Mat. Argentina"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1090\/S0002-9904-1973-13371-3","article-title":"Some results in E. Cartan\u2019s theory of isoparametric families of hypersurfaces","volume":"79","author":"Nomizu","year":"1973","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1090\/pspum\/027.1\/0423260","article-title":"\u00c9lie Cartan\u2019s work on isoparametric families of hypersurfaces","volume":"27","author":"Nomizu","year":"1975","journal-title":"Proc. Sympos. Pure Math. (Amer. Math. Soc.)"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01219082","article-title":"Cliffordalgebren und neue isoparametrische Hyperfl\u00e4chen","volume":"177","author":"Ferus","year":"1981","journal-title":"Math. Z."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"429","DOI":"10.4310\/jdg\/1214441486","article-title":"Dupin hypersurfaces, group actions, and the double mapping cylinder","volume":"26","author":"Grove","year":"1987","journal-title":"J. Differ. Geom."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"225","DOI":"10.4310\/jdg\/1214433420","article-title":"A class of hypersurfaces with constant principal curvatures in a sphere","volume":"11","author":"Takagi","year":"1976","journal-title":"J. Differ. Geom."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"515","DOI":"10.2748\/tmj\/1178240941","article-title":"On some types of isoparametric hypersurfaces in spheres I","volume":"27","author":"Ozeki","year":"1975","journal-title":"T\u00f4hoku Math. J."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"7","DOI":"10.2748\/tmj\/1178240877","article-title":"On some types of isoparametric hypersurfaces in spheres II","volume":"28","author":"Ozeki","year":"1976","journal-title":"T\u00f4hoku Math. J."},{"key":"ref_40","unstructured":"Takagi, R., and Takahashi, T. (1972). On the principal curvatures of homogeneous hypersurfaces in a sphere. Differ. Geom. in honor of K. Yano, 469\u2013481. Kinokuniya, Tokyo, Japan."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4310\/jdg\/1214429775","article-title":"Minimal submanifolds of low cohomogeneity","volume":"5","author":"Hsiang","year":"1971","journal-title":"J. Differ. Geom."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1112\/plms\/s3-51.3.520","article-title":"Isoparametric systems and transnormality","volume":"51","author":"Carter","year":"1985","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1112\/plms\/s3-60.3.609","article-title":"Isoparametric and totally focal submanifolds","volume":"60","author":"Carter","year":"1990","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_44","unstructured":"West, A. (1989). Isoparametric systems. Geometry and Topology of Submanifolds I, World Scientific."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"79","DOI":"10.4310\/jdg\/1214439466","article-title":"Isoparametric submanifolds and their Coxeter groups","volume":"21","author":"Terng","year":"1985","journal-title":"J. Differ. Geom."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/BF01390326","article-title":"Convexity theorem for isoparametric submanifolds","volume":"85","author":"Terng","year":"1986","journal-title":"Invent. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF01457280","article-title":"Submanifolds with flat normal bundle","volume":"277","author":"Terng","year":"1987","journal-title":"Math. Ann."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"9","DOI":"10.4310\/jdg\/1214442631","article-title":"Proper Fredholm submanifolds of Hilbert space","volume":"29","author":"Terng","year":"1989","journal-title":"J. Differ. Geom."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1090\/pspum\/054.1\/1216600","article-title":"Recent progress in submanifold geometry","volume":"54","author":"Terng","year":"1993","journal-title":"Proc. Sympos. Pure Math., Amer. Math. Soc."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"423","DOI":"10.4310\/jdg\/1214442003","article-title":"The topology of isoparametric submanifolds","volume":"27","author":"Hsiang","year":"1988","journal-title":"J. Differ. Geom."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/BF02584674","article-title":"Isoparametric families of submanifolds","volume":"13","author":"Harle","year":"1982","journal-title":"Bol. Soc. Bras. Mat."},{"key":"ref_52","first-page":"367","article-title":"Isoparametric submanifolds","volume":"20","year":"1986","journal-title":"Geom. Dedicata"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"429","DOI":"10.2307\/2944343","article-title":"Isoparametric foliations and their buildings","volume":"133","author":"Thorbergsson","year":"1991","journal-title":"Ann. Math."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"964","DOI":"10.2307\/2372843","article-title":"Applications of the theory of Morse to symmetric spaces","volume":"80","author":"Bott","year":"1958","journal-title":"Amer. J. Math."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"203","DOI":"10.4310\/jdg\/1214428257","article-title":"Minimal imbeddings of R-spaces","volume":"2","author":"Takeuchi","year":"1968","journal-title":"J. Differ. Geom."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1016\/S1874-5741(00)80013-8","article-title":"A survey on isoparametric hypersurfaces and their generalizations","volume":"Volume I","author":"Thorbergsson","year":"2000","journal-title":"Handbook of Differential Geometry"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1112\/blms\/15.5.493","article-title":"Dupin hypersurfaces","volume":"15","author":"Thorbergsson","year":"1983","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Grove, L.C., and Benson, C.T. (1985). Finite Reflection Groups, Springer. [2nd ed.]. Graduate Texts in Mathematics 99.","DOI":"10.1007\/978-1-4757-1869-0"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/BF01459125","article-title":"Isoparametric hypersurfaces with four or six distinct principal curvatures","volume":"264","author":"Abresch","year":"1983","journal-title":"Math. Ann."},{"key":"ref_60","first-page":"1237","article-title":"Isoparametric hypersurfaces with four distinct principal curvatures","volume":"36","author":"Tang","year":"1991","journal-title":"Chin. Sci. Bull."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"181","DOI":"10.4310\/AJM.1998.v2.n1.a7","article-title":"Multiplicities of equifocal hypersurfaces in symmetric spaces","volume":"2","author":"Tang","year":"1998","journal-title":"Asian J. Math."},{"key":"ref_62","unstructured":"Fang, F. (1996). Multiplicities of Principal Curvatures of Isoparametric Hypersurfaces, Max Planck Institut f\u00fcr Mathematik. preprint."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1090\/S0002-9939-99-04490-1","article-title":"On the topology of isoparametric hypersurfaces with four distinct principal curvatures","volume":"127","author":"Fang","year":"1999","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s002220050378","article-title":"Multiplicities of Dupin hypersurfaces","volume":"138","author":"Stolz","year":"1999","journal-title":"Invent. Math."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/PL00004742","article-title":"Topology of Dupin hypersurfaces with six principal curvatures","volume":"231","author":"Fang","year":"1999","journal-title":"Math. Z."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4007\/annals.2007.166.1","article-title":"Isoparametric hypersurfaces with four principal curvatures","volume":"166","author":"Cecil","year":"2007","journal-title":"Ann. Math."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.4007\/annals.2008.168.1011","article-title":"On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres","volume":"168","author":"Immervoll","year":"2008","journal-title":"Ann. Math."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1017\/S0027763000026064","article-title":"Isoparametric hypersurfaces with four principal curvatures revisited","volume":"193","author":"Chi","year":"2009","journal-title":"Nagoya Math. J."},{"key":"ref_69","first-page":"127","article-title":"A note on the paper \u201cIsoparametric hypersurfaces with four principal curvatures\u201d","volume":"3","author":"Chi","year":"2011","journal-title":"Hongyou Meml. Vol. Pac. J. Appl. Math."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1215\/00277630-1431813","article-title":"Isoparametric hypersurfaces with four principal curvatures II","volume":"204","author":"Chi","year":"2011","journal-title":"Nagoya Math. J."},{"key":"ref_71","first-page":"133","article-title":"A new look at Condition A","volume":"49","author":"Chi","year":"2012","journal-title":"Osaka J. Math."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"469","DOI":"10.4310\/jdg\/1370979335","article-title":"Isoparametric hypersurfaces with four principal curvatures III","volume":"94","author":"Chi","year":"2013","journal-title":"J. Differ. Geom."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"225","DOI":"10.4310\/jdg\/1589853626","article-title":"Isoparametric hypersurfaces with four principal curvatures IV","volume":"115","author":"Chi","year":"2020","journal-title":"J. Differ. Geom."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"2299","DOI":"10.1080\/00927878508823278","article-title":"Isoparametric hypersurfaces, case g = 6, m = 1","volume":"13","author":"Dorfmeister","year":"1985","journal-title":"Commun. Algebra"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"2217","DOI":"10.1090\/proc\/12924","article-title":"Classification of isoparametric hypersurfaces in spheres with (g, m) = (6, 1)","volume":"144","author":"Siffert","year":"2016","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"53","DOI":"10.4007\/annals.2013.177.1.2","article-title":"Isoparametric hypersurfaces with (g, m) = (6, 2)","volume":"177","author":"Miyaoka","year":"2013","journal-title":"Ann. Math."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.4007\/annals.2016.183.3.7","article-title":"Errata on isoparametric hypersurfaces with (g, m) = (6, 2)","volume":"183","author":"Miyaoka","year":"2016","journal-title":"Ann. Math."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s10455-017-9563-3","article-title":"A new structural approach to isoparametric hypersurfaces in spheres","volume":"52","author":"Siffert","year":"2017","journal-title":"Ann. Global Anal. Geom."},{"key":"ref_79","first-page":"062","article-title":"Isoparametric and Dupin hypersurfaces","volume":"4","author":"Cecil","year":"2008","journal-title":"SIGMA"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/56\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:27:54Z","timestamp":1759919274000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,13]]},"references-count":79,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010056"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010056","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,13]]}}}