Isoparametric and Dupin Hypersurfaces

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 d...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2008
Main Author:	Cecil, T.E.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2008
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/149015
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Cite this:	Isoparametric and Dupin Hypersurfaces / T.E. Cecil // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 171 назв. — англ.

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author	Cecil, T.E.
author_facet	Cecil, T.E.
citation_txt	Isoparametric and Dupin Hypersurfaces / T.E. Cecil // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 171 назв. — англ.
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	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.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 062, 28 pages Isoparametric and Dupin Hypersurfaces? Thomas E. CECIL Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA E-mail: cecil@mathcs.holycross.edu URL: http://mathcs.holycross.edu/faculty/cecil.html Received June 24, 2008, in final form August 28, 2008; Published online September 08, 2008 Original article is available at http://www.emis.de/journals/SIGMA/2008/062/ 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 genera- lization 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 2000 Mathematics Subject Classification: 53C40; 53C42; 53B25 1 Introduction A hypersurface Mn−1 immersed in Euclidean space Rn, the sphere Sn or hyperbolic space Hn is called isoparametric if it has constant principal curvatures. An isoparametric hypersurface in Rn can have at most two distinct principal curvatures, and it must be an open subset of a hyperplane, hypersphere or a spherical cylinder Sk ×Rn−k−1. This was first proven for n = 3 by Somigliana [134] in 1919 (see also B. Segre [128] and Levi-Civita [75]) and for arbitrary n by B. Segre [129] in 1938. A similar result holds in Hn. However, as Élie Cartan [8, 9, 10, 11] showed in a series of four papers published in the period 1938–1940, the theory of isoparametric hypersurfaces in the sphere Sn is much more beautiful and complicated. Cartan produced examples of isoparametric hypersurfaces in spheres with g = 1, 2, 3 or 4 distinct principal curvatures, and he classified those with g ≤ 3. Approximately thirty years later, Münzner [93, 94] wrote two papers which greatly extended Cartan’s work, proving that all isoparametric hypersurfaces are algebraic and that the number g of distinct principal curvatures must be 1, 2, 3, 4 or 6. Since Cartan had classified isoparametric hypersurfaces with g ≤ 3, the classification of those with g = 4 or 6 quickly became the goal of researchers in the field after Münzner’s work. This classification problem has proven to be interesting and difficult, and it was listed as Problem 34 on Yau’s [170] list of important open problems in geometry in 1990. The problem remains open in both cases g = 4 and 6 at this time, although much progress has been made. ?This paper is a contribution to the Special Issue “Élie Cartan and Differential Geometry”. The full collection is available at http://www.emis.de/journals/SIGMA/Cartan.html mailto:cecil@mathcs.holycross.edu http://mathcs.holycross.edu/faculty/cecil.html http://www.emis.de/journals/SIGMA/2008/062/ http://www.emis.de/journals/SIGMA/Cartan.html 2 T.E. Cecil A hypersurface Mn−1 in one of the real space-forms Rn, Sn or Hn 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 clearly a gen- eralization of the isoparametric condition, and it has its roots in nineteenth century differential geometry. Aside from the isoparametric surfaces in R3, the first examples of Dupin hyper- surfaces are the cyclides of Dupin in R3. These are all surfaces that can be obtained from a torus of revolution, a circular cylinder or a circular cone by inversion in a 2-sphere in R3. The cyclides have several other characterizations that will be discussed in this paper (see also [38, pp. 151–166], [26, pp. 148–159], [115]). The proper Dupin property is preserved under Möbius (conformal) transformations, and an important class of compact proper Dupin hypersurfaces in Rn consists of those hypersurfaces that are obtained from isoparametric hypersurfaces in a sphere Sn via stereographic projection from Sn−{P} to Rn, where P is a point in Sn. Indeed several of the major classification results for compact proper Dupin hypersurfaces involve such hypersurfaces. An important step in the theory of proper Dupin hypersurfaces was the work of Pinkall [112, 113, 114, 115] which situated the study of Dupin hypersurfaces in the setting of Lie sphere geometry. Among other things, Pinkall proved that the proper Dupin property is invariant under the group of Lie sphere transformations, which contains the group of Möbius transformations as a subgroup. The theory of proper Dupin hypersurfaces has both local and global aspects to it, and many natural problems remain open. In this paper, we survey the known results for isoparametric and proper Dupin hypersurfaces with emphasis on results that have been obtained in more recent years, and discuss impor- tant open problems in the field. The reader is also referred to the excellent survey article by Thorbergsson [157] published in the year 2000. 2 Isoparametric hypersurfaces Let M̃n(c) be a simply connected, complete Riemannian manifold of dimension n with constant sectional curvature c, that is, a real space-form. For c = 0, 1,−1, respectively, M̃n(c) is Euclidean space Rn, the unit sphere Sn ⊂ Rn+1, or hyperbolic space Hn. According to the original definition, a 1-parameter family Mt of hypersurfaces in M̃n(c) is called an isoparametric family if each Mt is equal to a level set V −1(t) for some non-constant smooth real-valued function V defined on a connected open subset of M̃n(c) such that the gradient and Laplacian of V satisfy \|gradV \|2 = T ◦ V, 4V = S ◦ V, (1) for some smooth functions S and T . Thus, the two classical Beltrami differential parameters are both functions of V itself, which leads to the name “isoparametric” for such a family of hypersurfaces. Such a function V is called an isoparametric function. (See Thorbergsson [157, pp. 965–967] and Q.-M. Wang [163, 164] for more discussion of isoparametric functions.) Let f : Mn−1 → M̃n(c) be an oriented hypersurface with field of unit normal vectors ξ. The parallel hypersurface to f(Mn−1) at signed distance t ∈ R is the map ft : Mn−1 → M̃n(c) such that for each x ∈ Mn−1, the point ft(x) is obtained by traveling a signed distance t along the geodesic in M̃n(c) with initial point f(x) and initial tangent vector ξ(x). For M̃n(c) = Rn, the formula for ft is ft(x) = f(x) + tξ(x), (2) and for M̃n(c) = Sn, the formula for ft is ft(x) = cos t f(x) + sin t ξ(x). (3) There is a similar formula in hyperbolic space (see, for example, [20]). Isoparametric and Dupin Hypersurfaces 3 Locally, for sufficiently small values of t, the map ft is also an immersed hypersurface. However, the map ft may develop singularities at the focal points of the original hypersur- face f(Mn−1). Specifically, a point p = ft(x) is called a focal point of multiplicity m > 0 of (f(Mn−1), x) if the differential (ft)∗ has nullity m at x. In the case M̃n(c) = Rn, respec- tively Sn, the point p = ft(x) is a focal point of multiplicity m of (f(Mn−1), x) if and only if 1/t, respectively cot t, is a principal curvature of multiplicity m of f(Mn−1) at x (see, for example, [83, pp. 32–38] or [38, pp. 243–247]). Let f : Mn−1 → M̃n(c) be an oriented hypersurface with constant principal curvatures. One can show from the formulas for the principal curvatures of a parallel hypersurface (see, for example, [38, pp. 131–132]) that if f has constant principal curvatures, then each ft that is an immersed hypersurface also has constant principal curvatures. However, since the principal curvatures of f are constant on Mn−1, the focal points along the normal geodesic to f(Mn−1) at f(x) occur for the same values of t independent of the choice of point x ∈ Mn−1. For example, for M̃n(c) = Rn, if µ is a non-zero constant principal curvature of multiplicity m > 0 of Mn−1, the map f1/µ has constant rank n − 1 − m on Mn−1, and the set f1/µ(Mn−1) is an (n− 1−m)-dimensional submanifold of Rn called a focal submanifold of f(Mn−1). One can show (see, for example, [38, pp. 268–274]) that the level hypersurfaces of an isopara- metric function V form a family of parallel hypersurfaces (modulo reparametrization of the normal geodesics to take into account the possibility that \|gradV \| is not identically equal to one), and each of these level hypersurfaces has constant principal curvatures. Conversely, one can begin with a connected hypersurface f : Mn−1 → M̃n(c) having constant principal curva- tures and construct an isoparametric function V such that each parallel hypersurface ft of f is contained in a level set of V . Therefore, one can define an isoparametric hypersurface to be a hypersurface with constant principal curvatures, and an isoparametric family of hypersurfaces can be characterized as a family of parallel hypersurfaces, each of which has constant principal curvatures. It is important that isoparametric hypersurfaces always come as a family of parallel hypersurfaces together with their focal submanifolds. As noted in the introduction, an isoparametric hypersurface in Rn must be an open subset of a hyperplane, hypersphere or a spherical cylinder Sk ×Rn−k−1 (Somigliana [134] for n = 3, see also B. Segre [128] and Levi-Civita [75], and B. Segre [129] for arbitrary n). Shortly after the publication of the papers of Levi-Civita and Segre, Cartan [8, 9, 10, 11] undertook the study of isoparametric hypersurfaces in arbitrary real space-forms M̃n(c), c ∈ R, and we now describe his primary contributions. Let f : Mn−1 → M̃n(c) be an isoparametric hypersurface with g distinct principal curvatures µ1, . . . , µg, having respective multiplicities m1, . . . ,mg. If g > 1, Cartan showed that for each i, 1 ≤ i ≤ g, ∑ j 6=i mj c + µiµj µi − µj = 0. (4) This important equation, known as Cartan’s identity, is crucial in Cartan’s work on isoparamet- ric hypersurfaces. For example, using this identity, Cartan was able to classify isoparametric hypersurfaces in the cases c ≤ 0 as follows. In the case c = 0, if g = 1, then f is totally umbilic, and it is well known that f(Mn−1) must be an open subset of a hyperplane or hypersphere. If g ≥ 2, then by taking an appropriate choice of unit normal field ξ, one can assume that at least one of the principal curvatures is positive. If µi is the smallest positive principal curvature, then each term µiµj/(µi − µj) in the sum in equation (4) is non-positive, and thus must equal zero. Therefore, there are at most two distinct principal curvatures, and if there are two, then one of them must be zero. Hence, g = 2 and one can show f(Mn−1) is an open subset of a spherical cylinder by standard methods in Euclidean hypersurface theory. 4 T.E. Cecil In the case c = −1, if g = 1, then f is totally umbilic, and it is well known that f(Mn−1) must be an open subset of a totally geodesic hyperplane, an equidistant hypersurface, a horosphere or a hypersphere in Hn (see, for example, [135, p. 114]). If g ≥ 2, then again one can arrange that at least one of the principal curvatures is positive. Then there must exist a positive principal curvature µi such that no principal curvature lies between µi and 1/µi. (Here µi is either the largest principal curvature between 0 and 1 or the smallest principal curvature greater than or equal to one.) For this µi, each term (−1 + µiµj)/(µi − µj) in the sum in equation (4) is negative unless µj = 1/µi. Thus, there are at most two distinct principal curvatures, and if there are two, then they are reciprocals of each other. Hence, g = 2 and one can show that f(Mn−1) is an open subset of a standard product Sk × Hn−k−1 in hyperbolic space Hn (see Ryan [126]). In the sphere Sn, however, Cartan’s identity does not lead to such strong restrictions on the number g of distinct principal curvatures, and Cartan himself produced examples with g = 1, 2, 3 or 4 distinct principal curvatures. Moreover, he classified isoparametric hypersurfaces f : Mn−1 → Sn with g ≤ 3 as follows. In the case g = 1, the hypersurface f is totally umbilic, and it is well known that f(Mn−1) is an open subset of a great or small hypersphere in Sn (see, for example, [135, p. 112]). If g = 2, then M must be a standard product of two spheres, Sp(r)× Sq(s) ⊂ Sn(1) ⊂ Rp+1 ×Rq+1 = Rn+1, r2 + s2 = 1, (5) where n = p + q + 1 (see, for example, [38, pp. 295–296]). In the case of three distinct principal curvatures, Cartan [10] showed that all the principal curvatures must have the same multiplicity m = 1, 2, 4 or 8, and f(Mn−1) must be a tube of con- stant radius over a standard embedding of a projective plane FP2 into S3m+1 (see, for example, [38, pp. 296–299]), where F is the division algebra R, C, H (quaternions), O (Cayley numbers), for m = 1, 2, 4, 8, respectively. (In the case F = R, a standard embedding is a Veronese surface in S4.) Thus, up to congruence, there is only one such family of isoparametric hypersurfaces for each value of m. For each of these hypersurfaces, the focal set of f(Mn−1) consists of two antipodal standard embeddings of FP2, and f(Mn−1) is a tube of constant radius over each focal submanifold. In the process of proving this theorem, Cartan showed that any isoparametric family with g distinct principal curvatures of the same multiplicity can be defined by an equation of the form F = cos gt (restricted to Sn), where F is a harmonic homogeneous polynomial of degree g on Rn+1 satisfying \|gradF \|2 = g2r2g−2, (6) where r = \|x\| for x ∈ Rn+1, and gradF is the gradient of F in Rn+1. This was a forerunner of Münzner’s general result that every isoparametric hypersurface is algebraic, and its defining polynomial satisfies certain differential equations which generalize those that Cartan found in this special case. In the case g = 4, Cartan produced isoparametric hypersurfaces with four principal cur- vatures of multiplicity one in S5 and four principal curvatures of multiplicity two in S9. He noted all of his examples are homogeneous, each being an orbit of a point under an appro- priate closed subgroup of SO(n + 1). Based on his results and the properties of his ex- amples, Cartan asked the following three questions [10], all of which were answered in the 1970’s. 1. For each positive integer g, does there exist an isoparametric family with g distinct prin- cipal curvatures of the same multiplicity? Isoparametric and Dupin Hypersurfaces 5 2. Does there exist an isoparametric family of hypersurfaces with more than three dis- tinct principal curvatures such that the principal curvatures do not all have the same multiplicity? 3. Does every isoparametric family of hypersurfaces admit a transitive group of isomet- ries? Despite the depth and beauty of Cartan’s work, the subject of isoparametric hypersurfaces in Sn was virtually ignored for thirty years until a revival of the subject in the early 1970’s by several authors. Nomizu [102, 103] wrote two papers describing the highlights of Cartan’s work. He also generalized Cartan’s example with four principal curvatures of multiplicity one to produce examples with four principal curvatures having multiplicities m1 = m2 = m, and m3 = m4 = 1, for any positive integer m. This answered Cartan’s Question 2 in the affirmative. Nomizu also proved that every focal submanifold of every isoparametric hypersurface must be a minimal submanifold of Sn. Takagi and Takahashi [143] gave a complete classification of all homogeneous isoparametric hypersurfaces in Sn, based on the work of Hsiang and Lawson [67]. Takagi and Takahashi showed that each homogeneous isoparametric hypersurface in Sn is an orbit of the isotropy representation of a Riemannian symmetric space of rank 2, and they gave a complete list of examples [143, p. 480]. This list contains examples with six principal curvatures as well as those with g = 1, 2, 3, 4 principal curvatures, and in some cases with g = 4, the principal curvatures do not all have the same multiplicity, so this also provided an affirmative answer to Cartan’s Question 2. At about the same time as the papers of Nomizu and Takagi–Takahashi, Münzner published two preprints that greatly extended Cartan’s work and have served as the basis for much of the research in the field since that time. The preprints were eventually published as papers [93, 94] in 1980–1981. In the first paper [93] (see also Chapter 3 of [38]), Münzner began with a geometric study of the focal submanifolds of an isoparametric hypersurface f : Mn−1 → Sn with g distinct principal curvatures. Using the fact that each focal submanifold of f is obtained as a parallel map ft, where cot t is a principal curvature of f(Mn−1), Münzner computed a formula for the shape operator of the focal submanifold ft(Mn−1) in terms of the shape operator of f(Mn−1) itself. In particular, his calculation shows that the assumption that f(Mn−1) has constant principal curvatures implies that the eigenvalues of the shape operator Aη are the same for every unit normal η at every point of the focal submanifold ft(Mn−1). Then by using a symmetry argument, he proved that if the principal curvatures of f(Mn−1) are written as cot θk, 0 < θ1 < · · · < θg < π, with multiplicities mk, then θk = θ1 + (k − 1) g π, 1 ≤ k ≤ g, (7) and the multiplicities satisfy mk = mk+2 (subscripts mod g). Thus, if g is odd, all of the multiplicities must be equal, and if g is even, there are at most two distinct multiplicities. Münzner’s calculation shows further that the focal submanifolds must be minimal submanifolds of Sn, as Nomizu [102] had shown by a different proof, and that Cartan’s identity is equivalent to the minimality of the focal submanifolds (see also [38, p. 251]). If cot t is not a principal curvature of f , then the map ft in equation (3) is also an isopara- metric hypersurface with g distinct principal curvatures cot(θ1 − t), . . . , cot(θg − t). If t = θk (mod π), then the map ft is constant along each leaf of the mk-dimensional principal folia- tion Tk, and the image of ft is a smooth focal submanifold of f of codimension mk +1 in Sn. All of the hypersurfaces ft in a family of parallel isoparametric hypersurfaces have the same focal submanifolds. 6 T.E. Cecil In a crucial step in the theory, Münzner then showed that f(Mn−1) and its parallel hypersur- faces and focal submanifolds are each contained in a level set of a homogeneous polynomial F of degree g satisfying the following Cartan–Münzner differential equations on the Euclidean differential operators gradF and Laplacian 4F on Rn+1, \|gradF \|2 = g2r2g−2, r = \|x\|, (8) 4F = crg−2, c = g2(m2 −m1)/2, where m1, m2 are the two (possibly equal) multiplicities of the principal curvatures on f(Mn−1). This generalized Cartan’s polynomial in equation (6) for the case of g principal curvatures with the same multiplicity. Conversely, the level sets of the restriction F \|Sn of a function F satisfying equation (8) constitute an isoparametric family of hypersurfaces and their focal submanifolds, and F is called the Cartan–Münzner polynomial associated to this family. Furthermore, Münzner showed that the level sets of F are connected, and thus any connected isoparametric hypersurface in Sn lies in a unique compact, connected isoparametric hypersurface obtained by taking the whole level set. The values of the restriction F \|Sn range between −1 and +1. For −1 < t < 1, the level set Mt = (F \|Sn)−1(t) is an isoparametric hypersurface, while M+ = (F \|Sn)−1(1) and M− = (F \|Sn)−1(−1) are focal submanifolds. Thus, there are exactly two focal submanifolds for the isoparametric family, regardless of the number g of distinct principal curvatures. Each principal curvature cot θk, 1 ≤ k ≤ g, gives rise to two antipodal focal points corresponding to the values t = θk and t = θk + π in equation (7). The 2g focal points are evenly spaced at intervals of length π/g along a normal geodesic to the isoparametric family, and they lie alternately on the two focal submanifolds M+ and M−, which have respective codimensions m1 + 1 and m2 + 1 in Sn. Using this information, Münzner showed that each isoparametric hypersurface Mt in the family separates the sphere Sn into two connected components D1 and D2, such that D1 is a disk bundle with fibers of dimension m1 + 1 over M+, and D2 is a disk bundle with fibers of dimension m2 + 1 over M−, where m1 and m2 are the multiplicities of the principal curvatures that give rise to the focal submanifolds M+ and M−, respectively. This topological situation has been the basis for many important results in this field concern- ing the number g of distinct principal curvatures and the multiplicities m1 and m2. In particular, in his second paper, Münzner [94] assumed that M is a compact, connected embedded hyper- surface that separates Sn into two disk bundles D1 and D2 over compact manifolds with fibers of dimensions m1 + 1 and m2 + 1, respectively. From this hypothesis, Münzner proved that the dimension of the cohomological ring H∗(M,Z2) must be 2α, where α is one of the numbers 1, 2, 3, 4 or 6. He then proved that if M is a compact, connected isoparametric hypersurface with g distinct principal curvatures, then dim H∗(M,Z2) = 2g. Combining these two results, Münzner obtained his major theorem that the number g of distinct principal curvatures of an isoparametric hypersurface in a sphere Sn must be 1, 2, 3, 4 or 6. Münzner also obtained some restrictions on the possible values of the multiplicities m1 and m2. These restrictions have been improved by several authors also using topological ar- guments, as we will describe later in this section. In particular, in the case g = 4, Abresch [1] showed that if m1 = m2 = m, then the only possible values for m are 1 and 2, the values in Cartan’s examples. In fact, up to congruence, Cartan’s examples are the only isoparametric hypersurfaces with four principal curvatures of the same multiplicity m. This was proven by Takagi [142] for m = 1 and by Ozeki and Takeuchi [107] for m = 2. Münzner’s result gave a negative answer to Cartan’s Question 1, but it pointed towards an affirmative answer to Cartan’s Question 3, since the possible values 1, 2, 3, 4 or 6 for Isoparametric and Dupin Hypersurfaces 7 the number g of distinct principal curvatures of an isoparametric hypersurface agreed with the values of g for the homogeneous isoparametric hypersurfaces on the list of Takagi and Takahashi. However, in an important 2-part paper, Ozeki and Takeuchi [107, 108] used representations of Clifford algebras to produce two infinite series of isoparametric families with four principal curvatures, most of which are necessarily inhomogeneous, because their multiplicities do not agree with the examples on the list of Takagi and Takahashi. These papers of Ozeki and Takeuchi were then vastly generalized by Ferus, Karcher and Münzner [58], who also used representations of Clifford algebras to produce a large and important class of isoparametric families with four principal curvatures which contains all known examples with g = 4 with the exception of two homogeneous families. We will now briefly describe the construction of Ferus, Karcher and Münzner (see also, [26, pp. 95–112]). For each integer m ≥ 0, the Clifford algebra Cm is the associative algebra over R that is generated by a unity 1 and the elements e1, . . . , em subject only to the relations e2 i = −1, eiej = −ejei, i 6= j, 1 ≤ i, j ≤ m. (9) One can show that the set {1, ei1 · · · eir \| i1 < · · · < ir, 1 ≤ r ≤ m}, forms a basis for the underlying vector space Cm, and thus dim Cm = 2m. A representation of Cm on Rl (of degree l) corresponds to a set of skew-symmetric matrices E1, . . . , Em in the orthogonal group O(l) such that E2 i = −I, EiEj = −EjEi, i 6= j, 1 ≤ i, j ≤ m. (10) Atiyah, Bott and Shapiro [3] determined all of the Clifford algebras according to the following table, and they showed that the Clifford algebra Cm−1 has an irreducible representation of degree l if and only if l = δ(m) as in the table. m Cm−1 δ(m) 1 R 1 2 C 2 3 H 4 4 H⊕H 4 5 H(2) 8 6 C(4) 8 7 R(8) 8 8 R(8)⊕R(8) 8 k + 8 Ck−1(16) 16δ(k) Reducible representations of Cm−1 on Rl for l = kδ(m), k > 1, can be obtained by taking a direct sum of k irreducible representations of Cm−1 on Rδ(m). Given a representation of Cm−1 on Rl corresponding to skew-symmetric matrices E1, . . ., Em−1 satisfying equation (10), Ferus, Karcher and Münzner [58] construct an isoparametric family of hypersurfaces in the sphere S2l−1 ⊂ R2l = Rl×Rl with four principal curvatures such that one of the focal submanifolds is the manifold V2(Cm−1) = { (u, v) ∈ S2l−1 \| \|u\| = \|v\| = 1√ 2 , u · v = 0, Eiu · v = 0, 1 ≤ i ≤ m− 1 } . (11) 8 T.E. Cecil This is the Clifford–Stiefel manifold of Clifford orthogonal 2-frames of length 1/ √ 2 in Rl (see Pinkall and Thorbergsson [117]), where vectors u and v in Rl are said to be Clifford orthogonal if u · v = E1u · v = · · · = Em−1u · v = 0, where u · v is the usual Euclidean inner product in Rl. Since V2(Cm−1) has codimension m+1 in S2l−1, one of the principal curvatures of an isopara- metric hypersurface M in this family has multiplicity m1 = m. The other multiplicity m2 must satisfy the equation 2m1 + 2m2 = dim M = 2l− 2. Therefore m2 = l−m− 1, where l = kδ(m) by the theorem of Atiyah, Bott and Shapiro. The isoparametric hypersurfaces resulting from this are called isoparametric hypersurfaces of FKM-type. The following is a table of the multiplicities of the principal curvatures of the FKM- hypersurfaces for multiplicities (m1,m2) = (m, kδ(m)−m− 1) for small values of m. Of course, the multiplicity m2 must be positive in order for this construction to lead to an isoparametric hypersurface with four principal curvatures. In the table below, the cases where m2 ≤ 0 are denoted by a dash. δ(m)\| 1 2 4 4 8 8 8 8 16 32 · · · k 1 − − − − (5, 2) (6, 1) − − (9, 6) (10, 21) · · · 2 − (2, 1) (3, 4) (4, 3) (5, 10) (6, 9) (7, 8) (8, 7) (9, 22) (10, 53) · · · 3 (1, 1) (2, 3) (3, 8) (4, 7) (5, 18) (6, 17) (7, 16) (8, 15) (9, 38) (10, 85) · · · 4 (1, 2) (2, 5) (3, 12) (4, 11) (5, 26) (6, 25) (7, 24) (8, 23) (9, 54) · · · · 5 (1, 3) (2, 7) (3, 16) (4, 15) (5, 34) (6, 33) (7, 32) (8, 31) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Multiplicities of the principal curvatures of FKM-hypersurfaces If m ≡ 0 (mod 4) and l = kδ(m), this construction yields [k/2]+1 incongruent isoparametric families having the same multiplicities. Furthermore, the families with multiplicities (2, 1), (6, 1), (5, 2) and one of the (4, 3)-families are congruent to those with multiplicities (1, 2), (1, 6), (2, 5) and (3, 4), respectively, and these are the only coincidences under congruence among the FKM-hypersurfaces [58]. Many of the FKM examples are necessarily inhomogeneous, because their multiplicities do not agree with the multiplicities of any hypersurface on the list of Takagi and Takahashi. However, Ferus, Karcher and Münzner also gave a geometric proof of the inhomogeneity of many of their examples through an examination of the second fundamental forms of the focal submanifolds. Later Q.-M. Wang [165] proved many results concerning the topology of the FKM examples and Wu [169] showed that there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces with four distinct principal curvatures. All known examples of isoparametric hypersurfaces with four principal curvatures are of FKM-type with the exception of two homogeneous families, having multiplicities (2, 2) and (4, 5). Beginning with Münzner [93, 94], many mathematicians, including Abresch [1], Grove and Halperin [61], Tang [146] and Fang [53, 54], found restrictions on the multiplicities of the principal curvatures of an isoparametric hypersurface with four principal curvatures. This series of results culminated with the paper of Stolz [140], who proved that the multiplicities of an isoparametric hypersurface with four principal curvatures must be the same as those in the known examples of FKM-type or the two homogeneous exceptions. All of these papers are primarily topological in nature, based on Münzner’s result that an isoparametric hypersurface separates Sn into two disk bundles over the two focal submanifolds. In particular, the proof of Stolz is homotopy theoretic, and the main tools used are the Hopf invariant and the EHP-sequence. In fact, Stolz proved his result under the more general assumption that M is a compact, connected proper Dupin (not Isoparametric and Dupin Hypersurfaces 9 necessarily isoparametric) hypersurface embedded in Sn. Note that Thorbergsson [155] had ear- lier shown that a compact, connected proper Dupin hypersurface M ⊂ Sn also separates Sn into two disk bundles over the first focal submanifolds on either side of M , as in the isoparametric case. This will be discussed in more detail in the next section. Cecil, Chi and Jensen [29] then showed that if the multiplicities (m1,m2) of an isoparametric hypersurface M ⊂ Sn with four principal curvatures satisfy m2 ≥ 2m1 − 1, then M must be of FKM-type. Previously, Takagi [142] had shown that if one of the multiplicities m1 = 1, then M is homogeneous and of FKM-type. Ozeki and Takeuchi [108] next showed that if m1 = 2, then M is homogeneous and of FKM-type, except in the case of multiplicities (2, 2), in which case M must be the homogeneous example of Cartan. Taken together with these results of Takagi and Ozeki–Takeuchi, the theorem of Cecil, Chi and Jensen classifies isoparametric hypersurfaces with four principal curvatures for all possible pairs of multiplicities except for four cases, the homogeneous pair (4,5), and the FKM pairs (3,4), (6,9) and (7,8), for which the classification of isoparametric hypersurfaces remains an important open problem. We now briefly outline the proof of this result by Cecil, Chi and Jensen. In Sections 8–9 of [29], Cecil, Chi and Jensen use Cartan’s method of moving frames to find necessary and sufficient conditions (equations (8.1)–(8.4) of [29]) for the codimension m1 + 1 focal submanifold M+ of an isoparametric hypersurface M with four principal curvatures and multiplicities (m1,m2) to be a Clifford–Stiefel manifold V2(Cm1−1). (Later Chi [40] gave a different proof of the fact that equations (8.1)–(8.4) of [29] are necessary and sufficient to show that M+ is a Clifford–Stiefel manifold.) These necessary and sufficient conditions involve the shape operators Aη of M+, where η is a unit normal vector to M+ at a point x ∈ M+. These shape operators are isospectral in that every Aη at every point x ∈ M+ has the same eigenvalues −1, 0, 1, with respective multiplicities m2, m1, m2. If η is a unit normal vector to M+ at a point x ∈ M+, then the point η is also in M+ by Münzner’s results, since it lies at a distance π/2 along the normal geodesic to M+ beginning at the point x in the direction η. The shape operators corresponding to an orthonormal basis of normal vectors to M+ at the point x determine a family of m1 + 1 homogeneous polynomials. Similarly, the shape operators corresponding to an orthonormal basis of normal vectors to M+ at the point η determine a family of m1 +1 homogeneous polynomials. In Section 10 of [29], Cecil, Chi and Jensen show that these two families of polynomials have the same zero set in projective space by use of a formulation of the Cartan–Münzner polynomial due to Ozeki and Takeuchi [107]. Finally, in Sections 11–13 of [29], the authors employ techniques from algebraic geometry to show that the fact that these two sets of polynomials have the same zero set leads to a proof that the necessary and sufficient conditions for M+ to be a Clifford– Stiefel manifold are satisfied if m2 ≥ 2m1−1. This completes the proof that M is of FKM-type, since M is a tube of constant radius over the Clifford–Stiefel manifold M+. After this, Immervoll [72] gave a different proof of the theorem of Cecil, Chi and Jensen using isoparametric triple systems. The use of triple systems to study isoparametric hypersurfaces was introduced in a series of papers in the 1980’s by Dorfmeister and Neher [45, 46, 47, 48, 49, 50]. In the case of an isoparametric hypersurface with six principal curvatures, Münzner showed that all of the principal curvatures must have the same multiplicity m, and Abresch [1] showed that m must equal 1 or 2. By the classification of homogeneous isoparametric hypersurfaces due to Takagi and Takahashi [143], there is only one homogeneous family in each case up to congruence. In the case of multiplicity m = 1, Dorfmeister and Neher [49] showed that an isoparametric hypersurface must be homogeneous, thereby completely classifying that case. The proof of Dorfmeister and Neher is quite algebraic in nature, and recently Miyaoka [89] has given a shorter, more geometric proof of this result. Miyaoka [88] also gave a geometric description of the case m = 1, showing that a homogeneous isoparametric hypersurface M6 in S7 can be obtained as the inverse image under the Hopf 10 T.E. Cecil fibration h : S7 → S4 of an isoparametric hypersurface with three principal curvatures of multiplicity one in S4. Miyaoka also showed that the two focal submanifolds of M6 are not congruent, even though they are lifts under h−1 of congruent Veronese surfaces in S4. Thus, these focal submanifolds are two non-congruent minimal homogeneous embeddings of RP2×S3 in S7. Peng and Hou [111] gave explicit forms for the Cartan–Münzner polynomials of degree six for the homogeneous isoparametric hypersurfaces with g = 6, and Fang [55] proved several results concerning the topology of isoparametric and compact proper Dupin hypersurfaces with six principal curvatures. The classification of isoparametric hypersurfaces with six principal curvatures of multiplicity m = 2 is part of Problem 34 on Yau’s [170] list of important open problems in geometry, and it remains an open problem. It has long been conjectured that the one homogeneous family in the case g = 6, m = 2, is the only isoparametric family in this case, but this conjecture has resisted proof for a long time. The approach that Miyaoka [89] used in the case m = 1 shows promise of possibly leading to a proof of this conjecture, but so far a complete proof has not been published. In the 1980’s, the notion of an isoparametric hypersurface was extended to submanifolds of codimension greater than one in Sn by several authors independently. (See Carter and West [17, 166], Harle [64], Strübing [141] and Terng [148].) An immersed submanifold φ : V → Rn (or Sn) is defined to be isoparametric if its normal bundle N(V ) is flat, and if for any locally defined normal field ξ which is parallel with respect to the normal connection ∇⊥, the eigenval- ues of the shape operator Aξ are constant. This theory was then developed extensively by many authors over the next decade, especially in the papers of Terng [148], Palais and Terng [109], and Hsiang, Palais and Terng [68] (see also the book [110]). Finally, Thorbergsson [156] used the theory of Tits buildings to show that all irreducible isoparametric submanifolds of codimension greater than one in Sn are homogeneous, and therefore they are principal orbits of isotropy representations of symmetric spaces, also known as generalized flag manifolds or standard em- beddings of R-spaces. (See Bott and Samelson [7], Takeuchi and Kobayashi [145], Dadok [42], Hahn [63] or the book by Berndt, Console and Olmos [6].) Later Olmos [105] and Heintze and Liu [65] gave alternate proofs of Thorbergsson’s result. Heintze, Olmos and Thorbergsson [66] defined a submanifold φ : V → Rn (or Sn) to have constant principal curvatures if for any smooth curve γ on V and any parallel normal vector field ξ(t) along γ, the shape operator Aξ(t) has constant eigenvalues along γ. If the normal bund- le N(M) is flat, then having constant principal curvatures is equivalent to being isoparametric. They then showed that a submanifold with constant principal curvatures is either isoparametric or a focal submanifold of an isoparametric submanifold. The excellent survey article of Thor- bergsson [157] gives a more detailed account of the theory of isoparametric submanifolds of codimension greater than one in Sn. An immersion φ : V → Rn of a compact, connected manifold V into Rn is said to be taut (see, for example, [13] or [38]) if every Morse function of the form Lp : V → R, where Lp(x) = \|p − φ(x)\|2, for p ∈ Rn, has the minimum number of critical points required by the Morse inequalities using Z2-homology, i.e., it is a perfect Morse function. Tautness can also be studied for submanifolds of Sn using spherical distance functions instead of Euclidean distance functions, and tautness is invariant under stereographic projection and its inverse map. Isoparametric submanifolds (of any codimension) and their focal submanifolds are all taut submanifolds of Sn. This was proven by Cecil and Ryan [37] for hypersurfaces and by Hsiang, Palais and Terng [68] for isoparametric submanifolds of codimension greater than one in Sn, and this was an important fact in the general development of the theory. Carter and West [14, 15, 16, 18], introduced the notion of totally focal submanifolds and studied its relationship to the isoparametric property. A submanifold φ : V → Rn is said to Isoparametric and Dupin Hypersurfaces 11 be totally focal if the critical points of every Euclidean distance function Lp on V are either all non-degenerate or all degenerate. An isoparametric submanifold in Sn is totally focal, and the main result of [18] is that a totally focal submanifold must be isoparametric. However, Terng and Thorbergsson [154, p. 197] have noted that there is a gap in the proof of this assertion, specifically in the proof of Theorem 5.1 of [18]. Wu [168] and Zhao [171] generalized the theory of isoparametric submanifolds of codimension greater than one to submanifolds of hyperbolic space, and Verhóczki [159] developed a theory of isoparametric submanifolds for Riemannian manifolds which do not have constant curvature. West [167] and Mullen [92] formulated a theory of isoparametric systems on symmetric spaces, and Terng and Thorbergsson [153] studied compact isoparametric submanifolds of symmetric spaces using the related notion of equifocal submanifolds. Tang [147] then did a thorough study of the possible multiplicities of the focal points of equifocal hypersurfaces in symmetric spaces. In a generalization of Thorbergsson’s result for submanifolds of Sn, Christ [41] proved that a complete connected irreducible equifocal submanifold of codimension greater than one in a simply connected compact symmetric space is homogeneous. Finally, a promising recent generalization of the theory of isoparametric submanifolds is the theory of singular Riemannian foliations admitting sections (see Alexandrino [2], Töben [158], and Lytchak and Thorbergs- son [80]). In a later paper, Terng and Thorbergsson [154] gave a definition of tautness for submanifolds of arbitrary complete Riemannian manifolds, and they discussed the notions of isoparamet- ric, equifocal and Dupin submanifolds in that setting. In a related development, Carter and Şentürk [12] studied the space of immersions parallel to a given immersion whose normal bundle has trivial holonomy group. Terng [151] considered isoparametric submanifolds in infinite-dimensional Hilbert spaces and generalized many results from the finite-dimensional case to that setting. Pinkall and Thorbergs- son [118] then gave more examples of such submanifolds, and Heintze and Liu [65] generalized the finite-dimensional homogeneity result of Thorbergsson [156] to the infinite-dimensional case. Nomizu [104] began the study of isoparametric hypersurfaces in pseudo-Riemannian space forms by proving a generalization of Cartan’s identity for space-like hypersurfaces in a Lorentzian space form M̃n 1 (c) of constant sectional curvature c. As a consequence of this identity, No- mizu showed that a space-like isoparametric hypersurface in M̃n 1 (c) can have at most two distinct principal curvatures if c ≥ 0. Recently, Li and Xie [77] have shown that this con- clusion also holds for space-like isoparametric hypersurfaces in M̃n 1 (c) for c < 0. Magid [81] studied isoparametric hypersurfaces in Lorentz space whose shape operator is not diagonaliz- able, and Hahn [62] contributed an extensive study of isoparametric hypersurfaces in pseudo- Riemannian space forms of arbitrary signatures. Recently Geatti and Gorodski [60] have extended this theory further by showing that a polar orthogonal representation of a con- nected real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. Working in a different direction, Niebergall and Ryan [97, 98, 99, 100] developed a theory of isoparametric and Dupin hypersurfaces in affine differential geometry. There is also an extensive theory of real hypersurfaces with constant principal curvatures in complex space forms that is closely related to the theory of isoparametric hypersurfaces in spheres. See the survey articles of Niebergall and Ryan [101] and Berndt [5] for more detail. In applications to Riemannian geometry, Solomon [131, 132, 133] found results concerning the spectrum of the Laplacian of isoparametric hypersurfaces in Sn with three or four prin- cipal curvatures, and Eschenburg and Schroeder [52] studied the behavior of the Tits metric on isoparametric hypersurfaces. Finally, Ferapontov [56, 57] studied the relationship between isoparametric and Dupin hypersurfaces and Hamiltonian systems of hydrodynamic type and listed several open research problems in that context. 12 T.E. Cecil 3 Dupin hypersurfaces As noted in the introduction, the theory of proper Dupin hypersurfaces is closely related to that of isoparametric hypersurfaces. In contrast to the situation for isoparametric hypersurfaces, however, there are both local and global aspects to the theory of proper Dupin hypersurfaces with quite different results concerning the number of distinct principal curvatures and their multiplicities, for example. In this section, we survey the primary results in this field. For the sake of completeness, we will begin with a formal definition of the Dupin and proper Dupin properties. Via stereographic projection τ : Sn − {P} → Rn with pole P ∈ Sn and its inverse map σ from Rn into Sn, the theory of Dupin hypersurfaces is essentially the same for hypersurfaces in Rn or Sn (see, for example, [38, pp. 132–151]), and we will use whichever ambient space is most convenient for the discussion at hand. Since we have been dealing with isoparametric hypersurfaces in spheres, we will formulate our definitions here in terms of hypersurfaces in Sn. Let f : M → Sn be an immersed hypersurface, and let ξ be a locally defined field of unit normals to f(M). A curvature surface of M is a smooth submanifold S such that for each point x ∈ S, the tangent space TxS is equal to a principal space of the shape operator A of M at x. The hypersurface M is said to be Dupin if: (a) along each curvature surface, the corresponding principal curvature is constant. The hypersurface M is called proper Dupin if, in addition to Condition (a), the following condi- tion is satisfied: (b) the number g of distinct principal curvatures is constant on M . An obvious and important class of proper Dupin hypersurfaces are the isoparametric hypersur- faces in Sn, and those hypersurfaces in Rn obtained from isoparametric hypersurfaces in Sn via stereographic projection. For example, the well-known ring cyclides of Dupin in R3 are obtained from a standard product torus S1(r) × S1(s) ⊂ S3, r2 + s2 = 1, in this way. These examples will be discussed in more detail later in this section. We begin by mentioning several well-known basic facts about Dupin hypersurfaces which are proven in Section 2.4 of [38], for example. As above, let f : M → Sn be an immersed hypersurface, and let ξ be a locally defined field of unit normals to f(M). First by the Co- dazzi equation, Condition (a) is automatically satisfied on a curvature surface S of dimen- sion greater than one. Second, Condition (b) is equivalent to requiring that each continuous principal curvature have constant multiplicity on M . Further, the number of distinct prin- cipal curvatures is locally constant on a dense open subset of any hypersurface in Sn (see Singley [130]). Next, if a continuous principal curvature function µ has constant multiplicity m on a con- nected open subset U ⊂ M , then µ is a smooth function, and the distribution Tµ of principal vectors corresponding to µ is a smooth foliation whose leaves are the curvature surfaces cor- responding to µ. The principal curvature µ is constant along each of its curvature surfaces in U if and only if these curvature surfaces are open subsets of m-dimensional great or small spheres in Sn. Suppose that µ = cot θ, for 0 < θ < π, where θ is a smooth function on U . The corresponding focal map fµ which maps x ∈ M to the focal point fµ(x) is given by the formula, fµ(x) = cos θ(x) f(x) + sin θ(x) ξ(x). (12) The principal curvature µ also determines a second focal map, whose image is antipodal to the image of fµ, obtained by replacing θ by θ + π in equation (12). The principal curvature function µ is constant along each of its curvature surfaces in U if and only if the focal map fµ Isoparametric and Dupin Hypersurfaces 13 factors through an immersion of the (n− 1−m)-dimensional space of leaves U/Tµ into Sn, and thus fµ(U) is an (n− 1−m)-dimensional submanifold of Sn. The curvature sphere Kµ(x) corresponding to the principal curvature µ at a point x ∈ U is the hypersphere in Sn through f(x) centered at the focal point fµ(x). Thus, Kµ(x) is tangent to f(M) at f(x). The principal curvature µ is constant along each of its curvature surfaces if and only if the curvature sphere map Kµ is constant along each of these curvature surfaces. Thus, on an open subset U on which Condition (b) holds, Condition (a) is equivalent to requir- ing that each curvature surface in each principal foliation be an open subset of a metric sphere in Sn of dimension equal to the multiplicity of the corresponding principal curvature. Condi- tion (a) is also equivalent to the condition that along each curvature surface, the corresponding curvature sphere map is constant. Finally, on U , Condition (a) is equivalent to requiring that for each principal curvature µ, the image of the focal map fµ is a smooth submanifold of Sn of codimension m + 1, where m is the multiplicity of µ. One consequence of these results is that like isoparametric hypersurfaces, proper Duper hy- persurfaces are algebraic. For simplicity, we will formulate this result in terms of hypersurfaces of Rn. The theorem states that a connected proper Dupin hypersurface f : M → Rn must be contained in a connected component of an irreducible algebraic subset of Rn of dimension n− 1. This result was widely thought to be true in the 1980’s, and indeed Pinkall [113] sent the author a letter in 1984 that contained a sketch of a proof of this result. However, a proof was not published until recently by Cecil, Chi and Jensen [32], who used methods of real algebraic geometry to give a complete proof based on Pinkall’s sketch. The proof makes use of the various principal foliations whose leaves are open subsets of spheres to construct an analytic algebraic parametrization of a neighborhood of f(x) for each point x ∈ M . In contrast to the situation for isoparametric hypersurfaces, however, a connected proper Dupin hypersurface in Sn does not necessarily lie in a compact connected proper Dupin hypersurface, as discussed later in this section. The algebraicity, and hence analyticity, of proper Dupin hypersurfaces was useful in clarifying certain fine points in the recent paper [32] of Cecil, Chi and Jensen on proper Dupin hypersurfaces with four principal curvatures. The definition of Dupin can be extended to submanifolds of codimension greater than one as follows. Let φ : V → Rn (or Sn) be a submanifold of codimension greater than one, and let Bn−1 denote the unit normal bundle of φ(V ). A curvature surface (see Reckziegel [121]) is a connected submanifold S ⊂ V for which there is a parallel section η : S → Bn−1 such that for each x ∈ S, the tangent space TxS is equal to some smooth eigenspace of the shape operator Aη. The submanifold φ(V ) is said to be Dupin if along each curvature surface, the corresponding principal curvature of Aη is constant, and a Dupin submanifold is proper Dupin if the number of distinct principal curvatures is constant on the unit normal bundle Bn−1. An isoparametric submanifold of codimension greater than one is always Dupin, but it may not be proper Dupin. (See [152, pp. 464–469] for more detail on this point.) Pinkall [114, p. 439] proved that every extrinsically symmetric submanifold of a real space form is Dupin. Takeuchi [144] then determined which of these are proper Dupin. Pinkall [114, 115] situated the study of Dupin hypersurfaces in the context of Lie sphere geometry, and this approach has proven to be very useful in subsequent research in the field. In particular, Dupin submanifolds of codimension greater than one in Sn can be studied nat- urally in this setting. Here we give a brief outline of this approach to Dupin hypersurfaces in Sn. (See Pinkall [115], Chern [39], Cecil and Chern [27], or the book [26] for more de- tail.) A Lie sphere is an oriented hypersphere or a point sphere (zero radius) in Sn. The set of all Lie spheres in Sn is in bijective correspondence with the set of points [x] = [(x1, . . . , xn+3)] (ho- mogeneous coordinates) in real projective space Pn+2 that lie on the quadric hypersurface Qn+1 14 T.E. Cecil (the Lie quadric) determined by the equation 〈x, x〉 = 0, where 〈x, y〉 = −x1y1 + x2y2 + · · ·+ xn+2yn+2 − xn+3yn+3 (13) is a bilinear form of signature (n + 1, 2) on the indefinite inner product space Rn+3 2 . Here the sphere Sn is identified with the unit sphere in Rn+1 ⊂ Rn+3 2 , where Rn+1 is spanned by the standard basis vectors {e2, . . . , en+2} in Rn+3 2 . Specifically, the oriented hypersphere with center p ∈ Sn and signed radius ρ (the sign denotes the orientation) corresponds to the point [(cos ρ, p, sin ρ)] in Qn+1. Point spheres p in Sn correspond to those points [(1, p, 0)] in Qn+1 with radius ρ = 0. The Lie quadric Qn+1 contains projective lines but no linear subspaces of Pn+2 of higher dimension. If [x] and [y] are points in Qn+1, the line [x, y] determined by [x] and [y] lies on Qn+1 if and only if 〈x, y〉 = 0. This condition means that the hyperspheres in Sn corresponding to [x] and [y] are in oriented contact, i.e., they are tangent and have the same orientation at the point of contact. For a point sphere and an oriented sphere, oriented contact means that the point lies on the sphere. A Lie sphere transformation is a projective transformation of Pn+2 that maps Qn+1 to itself. In terms of the geometry of Sn, a Lie sphere transformation is a diffeomorphism on the space of Lie spheres that preserves oriented contact of spheres (see Pinkall [115, p. 431]), since it takes lines on Qn+1 to lines on Qn+1. The group of Lie sphere transformations is isomorphic to O(n+1, 2)/{±I}, where O(n+1, 2) is the orthogonal group for the metric in equation (13). A Lie sphere transformation that takes point spheres to point spheres is a Möbius transformation, i.e., it is induced by a conformal diffeomorphism of Sn, and the set of all Möbius transformations is a subgroup of the Lie sphere group. An example of a Lie sphere transformation that is not a Möbius transformation is a parallel transformations Pt, which fixes the center of each Lie sphere but adds t to its signed radius. The group of Lie sphere transformations is generated by the union of the Möbius group and the group of all parallel transformations. The manifold Λ2n−1 of projective lines on Qn+1 has a contact structure, i.e., a globally defined 1-form ω such that ω ∧ dωn−1 never vanishes on Λ2n−1. The condition ω = 0 defines a codimension one distribution D on Λ2n−1 which has integral submanifolds of dimension n− 1 but none of higher dimension. A Legendre submanifold is one of these integral submanifolds of maximal dimension, i.e., an immersion λ : Mn−1 → Λ2n−1 such that λ∗ω = 0. An oriented hypersurface f : Mn−1 → Sn with field of unit normals ξ : Mn−1 → Sn naturally induces a Legendre submanifold λ = [k1, k2] (the line determined by the points [k1] and [k2] in Qn+1), where k1 = (1, f, 0), k2 = (0, ξ, 1). (14) For each x ∈ Mn−1, [k1(x)] is the unique point sphere and [k2(x)] is the unique great sphere in the parabolic pencil of spheres in Sn corresponding to the points on the line λ(x). Similarly, an immersed submanifold φ : V → Sn of codimension greater than one induces a Legendre submanifold whose domain is the bundle Bn−1 of unit normal vectors to φ(V ). In each case, λ is called the Legendre lift of the submanifold in Sn. Note that the space of Legendre submanifolds is larger than the space of Legendre lifts of submanifolds of Sn, since the point sphere map of an arbitrary Legendre submanifold may not have constant rank as a map into Sn. A Lie sphere transformation β maps lines on Qn+1 to lines on Qn+1, so it naturally induces a map β̃ from Λ2n−1 to itself. If λ is a Legendre submanifold, then β̃λ is also a Legendre submanifold, which is denoted βλ for short. These two Legendre submanifolds are said to be Lie equivalent. We will also say that two submanifolds of Sn are Lie equivalent, if their corresponding Legendre lifts are Lie equivalent. If β is a Möbius transformation, then the two Legendre submanifolds are said to be Möbius equivalent. Finally, if β is the parallel transformation Pt Isoparametric and Dupin Hypersurfaces 15 and λ is the Legendre submanifold induced by an oriented hypersurface f : M → Sn, then Ptλ is the Legendre lift of the parallel hypersurface f−t at oriented distance −t from f (see, for example, [26, p. 67]). To define the Dupin property for a Legendre submanifold λ : Mn−1 → Λ2n−1, one replaces the principal curvature function on Mn−1 in Conditions (a) and (b) above with the corresponding curvature sphere map K : Mn−1 → Qn+1, which is naturally defined in this setting. One can easily show that a Lie sphere transformation β maps curvatures spheres of λ to curvature spheres of βλ, and that Conditions (a) and (b) are preserved by β (see [26, pp. 67–70]). Thus, both the Dupin and proper Dupin properties are invariant under Lie sphere transformations. Pinkall [115] introduced four standard constructions for obtaining a proper Dupin hypersur- face in Rn+m from a proper Dupin hypersurface in Rn. We first describe Pinkall’s constructions in the case m = 1. Start with a Dupin hypersurface Wn−1 in Rn and then consider Rn as the linear subspace Rn × {0} in Rn+1. The following constructions yield a Dupin hypersurface Mn in Rn+1. (1) Let Mn be the cylinder Wn−1 ×R in Rn+1. (2) Let Mn be the hypersurface in Rn+1 obtained by rotating Wn−1 around an axis Rn−1 ⊂ Rn. (3) Project Wn−1 stereographically onto a hypersurface V n−1 ⊂ Sn ⊂ Rn+1. Let Mn be the cone over V n−1 in Rn+1. (4) Let Mn be a tube in Rn+1 around Wn−1. Even though Pinkall gave these four constructions, he noted that the tube construction and the cone construction are Lie equivalent [115, p. 438], and therefore in the context of Lie sphere geometry, it is sufficient to deal only with the tube, cylinder and surface of revolution construc- tions. In general, these constructions introduce a new principal curvature of multiplicity one which is easily seen to be constant along its lines of curvature. The other principal curvatures are determined by the principal curvatures of Wn−1, and the Dupin property is preserved for these principal curvatures (see [22] or [26, pp. 127–141] for a full discussion of these constructions in the setting of Lie sphere geometry). Thus, if Wn−1 is a proper Dupin hypersurface in Rn with g distinct principal curvatures, then in general, Mn is a proper Dupin hypersurface in Rn+1 with g + 1 distinct principal curvatures. However, this is not always the case, as we will explain below. Pinkall also pointed out that these constructions can easily be generalized to produce a new principal curvature of multiplicity m > 1 by considering the constructions in Rn ×Rm instead of Rn ×R. Let us now examine the possible problems encountered in trying to obtain a proper Dupin hypersurface by each of the three constructions. In all cases, suppose that Wn−1 is a proper Dupin hypersurface in Rn with g distinct principal curvatures. In the cylinder construction, the new principal curvature of Mn is identically zero while the other principal curvatures of Mn are equal to those of Wn−1. Thus if one of the principal curvatures κ of Wn−1 is zero at some points but not identically zero, then the number of distinct principal curvatures is not constant on Mn, and Mn is Dupin but not proper Dupin. If Mn is a tube in Rn+1 of radius ε over Wn−1, then there are exactly two distinct principal curvatures at the points in the set Wn−1 × {±ε} in Mn, regardless of the number of distinct principal curvatures on Wn−1. Thus, Mn is not a proper Dupin hypersurface unless the original hypersurface Wn−1 is totally umbilic, i.e., it has only one distinct principal curvature at each point. For the surface of revolution construction, if the focal point corresponding to a principal curvature κ at a point x of the profile submanifold Wn−1 lies on the axis of revolution Rn−1, 16 T.E. Cecil then the principal curvature of Mn at x determined by κ is equal to the new principal curvature of Mn resulting from the surface of revolution construction. Thus, if the focal point of Wn−1 corresponding to κ lies on the axis of revolution for some but not all points of Wn−1, then Mn is not proper Dupin. Another problem with these constructions is that they may not yield an immersed hyper- surface in Rn+1. In the tube construction, if the radius of the tube is the reciprocal of one of the principal curvatures of Wn−1 at some point, then the constructed object has a singularity. For the surface of revolution construction, a singularity occurs if the profile submanifold Wn−1 intersects the axis of revolution. These problems can be resolved by working in the context of Lie sphere geometry (see [26, pp. 127–141]). As noted above, these constructions can be generalized to the setting of Lie sphere geometry by considering Legendre lifts of hypersurfaces in Euclidean space. In that context, a proper Dupin submanifold λ : Mn−1 → Λ2n−1 is said to be reducible if it is is locally Lie equivalent to the Legendre lift of a hypersurface in Rn obtained by one of Pinkall’s constructions. Pinkall [115] found a useful characterization of reducibility in the context of Lie sphere ge- ometry when he proved that a proper Dupin submanifold λ : Mn−1 → Λ2n−1 is reducible if and only if the image of one its curvature sphere maps K lies in a linear subspace of codimension two in Pn+2. Much of the recent research in the field has focused on the classification of irreducible proper Dupin hypersurfaces. The following example (see [26, pp. 132–133] for more detail) shows that Pinkall’s construc- tions can produce a proper Dupin submanifold with the same number of distinct curvature spheres as the original proper Dupin submanifold, rather than increasing the number of distinct curvature spheres by one. Let V 2 be a Veronese surface embedded in the unit sphere S4 ⊂ R5. Let N3 be a tube of radius ε, for 0 < ε < π/3, over V 2 in S4. Then N3 is an isoparametric hypersurface S4 with three distinct principal curvatures, and N3 is an irreducible proper Dupin hypersurface, since Cecil, Chi and Jensen [30] proved that a compact, connected proper Dupin hypersurface Mn−1 ⊂ Sn with g ≥ 3 principal curvatures is irreducible. N3 is not the result of the tube construction, because the Veronese surface is substantial (does not lie in a hyperplane) in R5. Now embed R5 as a hyperplane through the origin in R6, and let e6 be a unit normal vector to R5 in R6. The Veronese surface V 2 is a codimension three submanifold of the unit sphere S5 ⊂ R6. A tube M4 over V 2 in S5 is a reducible Dupin hypersurface, because the family of curvatures spheres coming from the tube construction lies in a linear space of codimension two in P7, since these curvature spheres all have the same radius and their centers all lie in the space R5. This tube M4 is Dupin but not proper Dupin. At points of the tube M4 coming from points of the form (x,±e6) in the unit normal bundle B4 of V 2 in S5, there are two distinct principal curvatures, both of multiplicity two. At the other points of M4, there are three distinct principal curvatures, one of multiplicity two, and the other two of multiplicity one. Thus, the open dense subset U of M4 on which there are three principal curvatures is a reducible proper Dupin hypersurface, but M4 itself is not proper Dupin. The number g = 3 of distinct principal curvatures (or curvature spheres) on U is the same as the number of distinct curvature spheres of the Legendre lift λ of the original submanifold V 2 ⊂ S4, since the point sphere map of λ is a curvature sphere map, due to the fact that V 2 has codimension greater than one in S4. The other two curvature spheres of λ are determined by the two principal curvatures of the surface V 2. This example is important for illustrating some of the subtleties involved in studying the concept of reducibility. For example, Dajczer, Florit and Tojeiro [43] studied reducibility in the context of Euclidean submanifold theory. They formulated a concept of weak reducibility for proper Dupin submanifolds that have a flat normal bundle. In particular, they defined a proper Dupin hypersurface f : Mn−1 → Rn (or Sn) to be weakly reducible if, for some principal cur- vature κi with corresponding principal space Ti, the orthogonal complement T⊥i is integrable. Isoparametric and Dupin Hypersurfaces 17 Dajczer, Florit and Tojeiro showed that if a proper Dupin hypersurface f : Mn−1 → Rn is Lie equivalent to a proper Dupin hypersurface with g + 1 distinct principal curvatures that is obtained from a proper Dupin hypersurface with g distinct principal curvatures by one of the standard constructions, then f is weakly reducible. Thus, reducible implies weakly reducible for such hypersurfaces. However, one can show that the open set U with three principal curvatures in the example above is reducible but not weakly reducible, because none of the orthogonal comple- ments of the principal spaces is integrable. Note that U is not constructed from a proper Dupin submanifold with two curvature spheres, but rather one from one with three curvature spheres. Cecil and Jensen [33, 34] defined a proper Dupin hypersurface Mn−1 in Rn to be locally irreducible if Mn−1 does not contain a reducible open subset. A locally irreducible proper Dupin hypersurface is obviously irreducible, and using the analyticity of proper Dupin hypersurfaces, Cecil, Chi and Jensen [30] proved conversely that an irreducible proper Dupin hypersurface is locally irreducible. Thus, the two concepts are equivalent. Using his constructions listed above, Pinkall [115] (see also [26, p. 126]) demonstrated that proper Dupin hypersurfaces are plentiful. Specifically, he showed that given positive integers m1, . . . ,mg with m1 + · · ·+ mg = n− 1, there exists a proper Dupin hypersurface Mn−1 in Rn with g distinct principal curvatures having respective multiplicities m1, . . . ,mg. The proper Dupin hypersurfaces that Pinkall constructs in his proof are all reducible, and in general, they cannot be completed to a compact proper Dupin hypersurface, because of the difficulties dis- cussed above. In fact, compact proper Dupin hypersurfaces are much more rare, as we will describe below. Recall from Section 2 that an immersion φ : V → Rn of a compact, connected manifold V into Rn is taut if every non-degenerate Euclidean distance function Lp, p ∈ Rn, has the minimum number of critical points on V required by the Morse inequalities using Z2-homology. Tautness can also be studied for submanifolds of Sn using spherical distance functions. A taut submanifold of Sn (or Rn) must be Dupin, although not necessarily proper Dupin. This was first shown for compact surfaces in R3 by Banchoff [4] (see also [21] for the non- compact case), then by Miyaoka [85] for hypersurfaces, and independently by Pinkall [116] for submanifolds of arbitrary codimension. Conversely, Thorbergsson [155] proved that a compact, connected proper Dupin hypersurface Mn−1 ⊂ Sn (or Rn) with g distinct principal curvatures is taut. Thorbergsson used the principal foliations on Mn−1 to construct concrete Z2-cycles in Mn−1 which show that all critical points of non-degenerate distance functions Lp are of linking type (see Morse and Cairns [91, p. 258]). Therefore such distance functions have the minimum number β(Mn−1) = 2g critical points, where β(Mn−1) is the sum of the Z2-Betti numbers of Mn−1. Pinkall [116] later extended Thorbergsson’s result to compact submanifolds of higher codimension for which the number of distinct principal curvatures is constant on the unit normal bundle. Note that unlike the case for isoparametric hypersurfaces, the focal submanifolds of a com- pact, connected proper Dupin hypersurface Mn−1 need not be taut. For example, for a ring cyclide of Dupin M2 ⊂ R3 obtained by inverting a torus of revolution in a sphere, one of the focal submanifolds is an ellipse, which is not a taut embedding of S1. Ozawa [106] proved that if V is a taut compact submanifold of Rn (or Sn), then every connected component S of a critical set of a distance function Lp on V is a smooth submani- fold of V , which is non-degenerate as a critical submanifold in the sense of Morse–Bott critical point theory (see [7]), and S itself is taut in Rn. Using Ozawa’s result, one can prove (see [25, p. 154]) that if Mn−1 ⊂ Rn is a taut compact, connected hypersurface, then given any principal space Tµ at any point x ∈ Mn−1, there is a curvature surface S through x whose tangent space is equal to Tµ, and µ is constant along S. Thus Mn−1 is Dupin in this strong sense of having a curvature surface tangent to every principal space (this is not assumed as part of Condition (a) in the definition of a Dupin hypersurface). An important open ques- 18 T.E. Cecil tion is whether the converse of this theorem is true. That is, if Mn−1 ⊂ Rn is a compact, connected non-proper Dupin hypersurface with the property that given any principal space Tµ at any point x ∈ Mn−1, there is a curvature surface S through x whose tangent space is equal to Tµ, must Mn−1 be taut? Thorbergsson’s proof that a compact proper Dupin hy- persurface must be taut relies on the fact that all the curvature surfaces are spheres. In the non-proper Dupin case, Ozawa’s work implies that there are some curvature surfaces that are not spheres. Terng defined a Dupin submanifold V of arbitrary codimension to have constant multiplici- ties if the multiplicities of the principal curvatures of any parallel normal field ξ(t) along any piecewise smooth curve on V are constant. Terng [150], [152, p. 467] then proved that a compact, connected Dupin submanifold with constant multiplicities is taut. Using tautness, Thorbergsson [155] proved that a compact, connected proper Dupin hyper- surface Mn−1 ⊂ Sn separates Sn into two disk bundles over the first focal submanifolds on either side of Mn−1, as in the case of isoparametric hypersurfaces. Thus, Münzner’s restriction that the number of distinct principal curvatures must be 1, 2, 3, 4 or 6 holds for compact, connected proper Dupin hypersurfaces Mn−1 ⊂ Sn also. Furthermore, the restrictions on the possible multiplicities of the principal curvatures due to Stolz [140] in the case g = 4 and Grove and Halperin [61] in the case g = 6 still hold. These results have led to a pursuit of classification results for compact proper Dupin hypersurfaces based on the number g of distinct principal curvatures. In the case g = 1, it is well known that a compact, connected proper Dupin hypersur- face Mn−1 ⊂ Sn must be a great or small hypersphere. In the case g = 2, Cecil and Ryan [35] (see also [36] or [38, p. 168]) showed that Mn−1 must be Möbius equivalent to an isoparametric hypersurface, i.e., a standard product of spheres Sk(r) × Sn−k−1(s) ⊂ Sn, r2 + s2 = 1. In the case g = 3, Miyaoka [84] proved that a compact, connected proper Dupin hypersurface Mn−1 ⊂ Sn must be Lie equivalent to an isoparametric hypersurface, although not necessarily Möbius equivalent. These results together with Thorbergsson’s restriction on the number g of distinct principal curvatures led to the widely held conjecture [38, p. 184] that every compact, connected proper Dupin hypersurface Mn−1 ⊂ Sn is Lie equivalent to an isoparametric hypersurface in a sphere. However, in 1988 Pinkall and Thorbergsson [117], and Miyaoka and Ozawa [90] gave two different methods for producing counterexamples to this conjecture with four principal curvatures. The method of Miyaoka and Ozawa also yields counterexamples to the conjecture in the case of six principal curvatures. Both of these constructions involve a consideration of the Lie curvatures of a proper Dupin hypersurface. If Mn−1 ⊂ Sn is a proper Dupin hypersurface with g ≥ 4 distinct principal curvatures, then the cross-ratios of the principal curvatures taken four at a time are called the Lie curvatures of Mn−1. These Lie curvatures were first studied by Miyaoka [86] who proved that they are invariant under Lie sphere transformations. This is actually quite easy to see in the context of projective geometry, since each Lie curvature is the cross-ratio of four points (corresponding to curvature spheres) on a projective line in Pn+2. A Lie sphere transformation β is a projective transformation and it maps curvature spheres of a Legendre submanifold λ to curvature spheres of the Legendre submanifold βλ, so it preserves the cross-ratios of the curvature spheres, and therefore it preserves the cross-ratios of the principal curvatures (see also [26, p. 75]). The counterexamples of Pinkall and Thorbergsson [117] (see also [26, pp. 112–117]) to the conjecture are obtained by modifying the isoparametric hypersurfaces of FKM-type discussed in the previous section. Recall that the Clifford–Stiefel manifold V2(Cm−1) is a submanifold of S2l−1 ⊂ R2l = Rl × Rl of codimension m + 1. Given positive real numbers α and β with α2 + β2 = 1, where α 6= 1/ √ 2, β 6= 1/ √ 2, Pinkall and Thorbergsson define a linear map Isoparametric and Dupin Hypersurfaces 19 Tα,β : R2l → R2l, by Tα,β(u, v) = √ 2 (αu, βv). Then for (u, v) ∈ V2(Cm−1), we have \|Tα,β(u, v)\|2 = 2 ( α2(u · u) + β2(v · v) ) = 2 ( α2 2 + β2 2 ) = 1, and thus the image Tα,βV2(Cm−1) is a submanifold of S2l−1 of codimension m + 1 also. Pinkall and Thorbergsson prove that a tube over Tα,βV2(Cm−1) in S2l−1 is a compact, connected proper Dupin hypersurface with four principal curvatures that does not have constant Lie curvature, and therefore it is not Lie equivalent to an isoparametric hypersurface. The construction of counterexamples to the conjecture due to Miyaoka and Ozawa [90] (see also [26, pp. 117–123]) is based on the Hopf fibration h : S7 → S4. Miyaoka and Ozawa first show that if M is a taut compact submanifold of S4, then h−1(M) is a taut compact submanifold of S7. Using this, they next show that if M is a proper Dupin hypersurface in S4 with g distinct principal curvatures, then h−1(M) is a proper Dupin hypersurface in S7 with 2g principal curvatures. To complete the argument, they show that if a hypersurface M ⊂ S4 is proper Dupin but not isoparametric, then the Lie curvatures of h−1(M) are not constant, and therefore h−1(M) is not Lie equivalent to an isoparametric hypersurface in S7. For g = 2 or 3, respectively, this gives a compact proper Dupin hypersurface with 4 or 6 principal curvatures, respectively, in S7, which is not Lie equivalent to an isoparametric hyper- surface. These counterexamples are both based on the fact that the constructed proper Dupin hyper- surface does not have constant Lie curvatures. This leads to a revision of the conjecture [31, p. 52] which states that every compact, connected proper Dupin hypersurface Mn−1 ⊂ Sn with g = 4 or 6 principal curvatures and constant Lie curvatures is Lie equivalent to an isoparametric hypersurface in a sphere. This revised conjecture is still an important open problem, although it has has been shown to be true in some cases, as we now describe. Miyaoka [86, 87] began by showing that if some additional assumptions are made regarding the intersections of the leaves of the various principal foliations, then this revised conjecture is true in both cases g = 4 and g = 6. Thus far, however, it has not been shown that Miyaoka’s additional assumptions are satisfied in general. In the case g = 6, this work of Miyaoka is the only known progress towards proving the revised conjecture. If Mn−1 ⊂ Sn is a compact, connected proper Dupin hypersurface with four principal cur- vatures having multiplicities m1, m2, m3, m4, then the multiplicities must satisfy m1 = m2, m3 = m4, when the principal curvatures are appropriately ordered, by the work of Thor- bergsson [155] and Münzner [93, 94]. Cecil, Chi and Jensen [30] have recently shown that if Mn−1 ⊂ Sn is a compact, connected proper Dupin hypersurface with four principal curvatures having multiplicities m1 = m2 ≥ 1, m3 = m4 = 1, and constant Lie curvature, then Mn−1 is Lie equivalent to an isoparametric hypersurface. This result actually follows from a local clas- sification of irreducible proper Dupin hypersurfaces with four principal curvatures [30], which we will discuss below. Thus, the full revised conjecture in the case g = 4 would be proven if one could remove the assumption m3 = m4 = 1, but so far this has not been done. In his early work on the subject, Pinkall [112, 114, 115], proved some important local clas- sification results concerning proper Dupin hypersurfaces, and this approach has been extended to more general settings by using the notion of irreducibility. Although it has not been shown that irreducibility places any restriction on the number g of distinct principal curvatures other than g ≥ 3, most of the known results for irreducible proper Dupin hypersurfaces have been obtained in the cases g = 3 or 4. As with compact proper Dupin hypersursurfaces, we will discuss the known local classification results for proper Dupin hypersurfaces of Sn based on the number g of distinct principal curva- tures. It is well known, that a connected proper Dupin hypersurface in Sn with g = 1 (totally umbilic) distinct principal curvature must be an open subset of a hypersphere. 20 T.E. Cecil In the case g = 2, Pinkall [114] obtained a complete local classification which we now describe in detail. As a generalization of the well-known cyclides of Dupin in R3, Pinkall defined a cyclide of Dupin of characteristic (p, q) to be a proper Dupin hypersurface in Sn (or Rn) with two distinct principal curvatures of respective multiplicities p and q. An example of a cyclide of Dupin of characteristic (p, q) is the standard product of spheres, Sp(r)×Sq(s) ⊂ Sn, r2 +s2 = 1, where n = p + q + 1, as in equation (5). This is an isoparametric hypersurface in Sn with two principal curvatures, as discussed earlier. If one varies the value of r in equation (5) between 0 and 1, one obtains an isoparametric family of hypersurfaces with two principal curvatures. The hypersurfaces in this family are Lie equivalent by parallel transformation, but they are not Möbius equivalent for different values of r. Pinkall proved that every connected cyclide of Dupin is contained in a unique compact, connected cyclide, and any two cyclides of the same characteristic (p, q) are locally Lie equivalent. Thus, every cyclide of Dupin of characteristic (p, q) is locally Lie equivalent to a standard product of spheres, as in equation (5). Using Pinkall’s Lie geometric classification one can derive the following Möbius geometric classification of the cyclides of Dupin [24] (see also [26, p. 151]). Every connected cyclide Mn−1 of characteristic (p, q) in Rn is Möbius equivalent to an open subset of a hypersurface of revolution obtained by revolving a q-sphere Sq ⊂ Rq+1 ⊂ Rn about an axis of revolution Rq ⊂ Rq+1 or a p-sphere Sp ⊂ Rp+1 ⊂ Rn about an axis Rp ⊂ Rp+1. Further, two such hypersurfaces of revolution are Möbius equivalent if and only if they have the same value of ρ = r/a, where r is the radius of the profile sphere Sq and a > 0 is the distance from the center of Sq to the axis of revolution. In this theorem, the profile sphere is allowed to intersect the axis of revolution, which results in Euclidean singularities. However, in the context of Lie sphere geometry, the corresponding Legendre map is an immersion. The classical cyclides of Dupin in R3 were first studied by Dupin [51] in 1822 and later by many prominent nineteenth century mathematicians, including Liouville [79], Cayley [19], and Maxwell [82], whose paper contains stereoscopic figures of the various types of cyclides. (See Lilienthal [78] for an account of the nineteenth century work on the cyclides.) More recent descriptions of the classical cyclides are contained in the books of Fladt and Baur [59, pp. 354– 379], Cecil and Ryan [38, pp. 151–166], and [26, pp. 148–159]. The classical cyclides are the only surfaces in R3 with two principal curvatures at each point such that all lines of curvature in both families are circles or straight lines. This is just the proper Dupin condition, of course. Using exterior differential systems, Ivey [73] showed that any surface in R3 containing two orthogonal families of circles is a cyclide of Dupin. The classical cyclides have also been used in computer aided geometric design of surfaces. See, for example, the papers of Degen [44], Pratt [119, 120], Srinivas and Dutta [136, 137, 138, 139], and Schrott and Odehnal [127]. Pinkall began the study of proper Dupin hypersurfaces with three distinct principal curvatures in his dissertation [112], published as a paper [115] (see also [28] or [26, pp. 168–190]). Pinkall found a complete local classification up to Lie equivalence for Dupin hypersurfaces with three principal curvatures in R4. He proved that any two irreducible proper Dupin hypersurfaces with g = 3 in R4 are locally Lie equivalent, each being Lie equivalent to an open subset of Cartan’s isoparametric hypersurface in S4. For reducible proper Dupin hypersurfaces with g = 3 in R4, Pinkall showed that there is a 1-parameter family of Lie equivalence classes. Niebergall [95] next proved that every connected proper Dupin hypersurface in R5 with three principal curvatures is reducible. Then Cecil and Jensen [33] proved that if Mn−1 is an irre- ducible, connected proper Dupin hypersurface in Sn with three distinct principal curvatures of multiplicities m1, m2, m3, then m1 = m2 = m3 = m, and Mn−1 is Lie equivalent to an isoparametric hypersurface in Sn. It then follows from Cartan’s classification of isoparametric hypersurfaces with g = 3 mentioned in Section 2 that m = 1, 2, 4 or 8. Note that in the original Isoparametric and Dupin Hypersurfaces 21 paper [33], this result was proven under the assumption that Mn−1 is locally irreducible. How- ever, as noted above, local irreducibility has now been shown to be equivalent to irreducibility. The proof of this result of Cecil and Jensen is accomplished by using Cartan’s method of moving frames in the context of Lie sphere geometry. A key tool in this context is the result (see [23] or [26, p. 77]) that a Legendre submanifold λ : Mn−1 → Λ2n−1 with g distinct curvature spheres K1, . . . ,Kg at each point is Lie equivalent to the Legendre lift of an isoparametric hypersurface in Sn if and only if there exist g points P1, . . . , Pg on a timelike line in Pn+2 such that 〈Ki, Pi〉 = 0, for 1 ≤ i ≤ g. An open problem is the classification of reducible Dupin hypersurfaces with three principal curvatures up to Lie equivalence. As noted above, Pinkall [115] found such a classification in the case of M3 ⊂ R4. It may be possible to generalize Pinkall’s result to higher dimensions using the approach of [33]. Cecil, Chi and Jensen [30] proved that a compact, connected proper Dupin hypersurface Mn−1 ⊂ Sn with g ≥ 3 principal curvatures is irreducible. Thus, the examples above of Pinkall and Thorbergsson [117] and Miyaoka and Ozawa [90] of compact proper Dupin hypersurfaces with g = 4 and non-constant Lie curvature are irreducible proper Dupin hypersurfaces with g = 4 that are not Lie equivalent to an isoparametric hypersurface. However, it is still possible that every irreducible Dupin hypersurface with g = 4 and constant Lie curvature is Lie equivalent to an isoparametric hypersurface, and this has been shown under additional assumptions, as we now describe. The first result in this direction is due to Niebergall [96] who showed that a connected ir- reducible proper Dupin hypersurface M4 in S5 with four principal curvatures and constant Lie curvature is Lie equivalent to an isoparametric hypersurface under an additional assumption that in an appropriate moving frame, the covariant derivatives of certain naturally defined functions are zero. Cecil and Jensen [34] then showed that Niebergall’s additional assumptions are un- necessary, because these functions are always constant in the appropriate Lie frame. Thus, they proved that every connected irreducible proper Dupin hypersurface M4 in S5 with four principal curvatures and constant Lie curvature is Lie equivalent to an isoparametric hypersurface. This result was then generalized to higher dimensions by Cecil, Chi and Jensen [30] who proved that if Mn−1 is an irreducible, connected proper Dupin hypersurface in Sn with four principal curvatures having multiplicities m1 = m2 ≥ 1 and m3 = m4 = 1 and constant Lie curvature −1, then Mn−1 is Lie equivalent to an isoparametric hypersurface. Note that Münzner [93, 94] had shown earlier that if Mn−1 ⊂ Sn is an isoparametric hypersurface with four principal curvatures, then the multiplicities of the principal curvatures must satisfy m1 = m2 and m3 = m4, and the Lie curvature must equal −1, if the principal curvatures are appropriately ordered. These results lead to the following local conjecture [31, p. 53]: if Mn−1 is an irreducible, connected, proper Dupin hypersurface in Sn with four principal curvatures having multiplicities m1, m2, m3, m4, and constant Lie curvature c, then m1 = m2, m3 = m4, c = −1, and Mn−1 is Lie equivalent to an isoparametric hypersurface. Note that it has not yet been shown that irreducibility implies that m1 = m2 and m3 = m4, nor that c = −1. We remark that the hypothesis of irreducibility is definitely needed in the conjecture because one can construct reducible non-compact proper Dupin hypersurfaces with g = 4 and constant Lie curvature c, for every negative value of c, if the principal curvatures are appropriately ordered (see [23] or [26, pp. 80–82]). This construction yields examples where the multiplicities satisfy m1 = m2 and m3 = m4, and also examples where the multiplicities do not satisfy this restriction. These examples are all obtained as open subsets of a tube in Sn over an isoparametric hypersurface with three principal curvatures V k−1 ⊂ Sk ⊂ Sn, and they cannot be completed to be compact proper Dupin hypersurfaces. They are also reducible as Dupin hypersurfaces. 22 T.E. Cecil Using a different approach based on the theory of higher-dimensional Laplace invariants [74], Riveros and Tenenblat [123, 124] gave a local classification of proper Dupin hypersurfaces M4 in R5 with four distinct principal curvatures which are parametrized by lines of curvature. C.-P. Wang [160, 162] studied the Möbius geometry of submanifolds in Sn in a series of papers. Using Cartan’s method of moving frames, Wang found a complete set of Möbius invariants for surfaces in R3 without umbilic points [160] and for hypersurfaces in R4 with three principal curvatures at each point [161]. Then in [162], Wang defined a Möbius invariant metric g and second fundamental form B for submanifolds in Sn. Wang then proved that for hypersurfaces in Sn with n ≥ 4, the pair (g,B) forms a complete Möbius invariant system which determines the hypersurface up to Möbius transformations. In a related result, Riveros, Rodrigues and Tenenblat [122] proved that a proper Dupin hypersurface Mn ⊂ Rn+1, n ≥ 4, with n distinct principal curvatures and constant Möbius curvatures cannot be parametrized by lines of curvature. They also showed that up to Möbius transformations, there is a unique proper Dupin hypersurface M3 ⊂ R4 with three principal curvatures and constant Möbius curvature that is parametrized by lines of curvature. This M3 is a cone in R4 over a standard flat torus in the unit sphere S3 ⊂ R4. In [76], H. Li, Lui, Wang and Zhao introduced the concept of a Möbius isoparametric hy- persurface in a sphere Sn. They showed that a (Euclidean) isoparametric hypersurface is auto- matically Möbius isoparametric, whereas a Möbius isoparametric hypersurface must be proper Dupin. Later Rodrigues and Tenenblat [125] showed that if M ⊂ Sn is a hypersurface with a constant number g of distinct principal curvatures at each point, where g ≥ 3, then M is Möbius isoparametric if and only if M is Dupin with constant Möbius curvatures. Recently significant progress has been made in the classification of Möbius isoparametric hypersurfaces. First, H. Li, Lui, Wang and Zhao [76] showed that a connected Möbius isopara- metric hypersurface in Sn with two distinct principal curvatures is Möbius equivalent to an open subset of one of the following three types of hypersurfaces in Sn: (i) a standard product of spheres Sk(r)× Sn−k−1(s) ⊂ Sn, r2 + s2 = 1, (ii) the image under inverse stereographic projection from Rn → Sn − {P} of a standard cylinder Sk(1)×Rn−k−1 ⊂ Rn, (iii) the image under hyperbolic stereographic projection from Hn → Sn of a standard product Sk(r)×Hn−k−1( √ 1 + r2) ⊂ Hn. Later Hu and H. Li [70] classified Möbius isoparametric hypersurfaces in S4, and Hu, H. Li and Wang [71] classified Möbius isoparametric hypersurfaces in S5. Then Hu and D. Li [69] studied Möbius isoparametric hypersurfaces with three distinct principal curvatures in Sn and found a complete classification of such hypersurfaces in S6. Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 0405529. The author is grateful for several helpful comments in the reports of the referees. 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. 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publishDate	2008
publisher	Інститут математики НАН України
record_format	dspace
spelling	Cecil, T.E. 2019-02-19T12:58:42Z 2019-02-19T12:58:42Z 2008 Isoparametric and Dupin Hypersurfaces / T.E. Cecil // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 171 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53C40; 53C42; 53B25 https://nasplib.isofts.kiev.ua/handle/123456789/149015 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. This paper is a contribution to the Special Issue “Elie Cartan and Differential Geometry”. This material is based upon work supported by the National Science Foundation under Grant No. 0405529. The author is grateful for several helpful comments in the reports of the referees. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Isoparametric and Dupin Hypersurfaces Article published earlier
spellingShingle	Isoparametric and Dupin Hypersurfaces Cecil, T.E.
title	Isoparametric and Dupin Hypersurfaces
title_full	Isoparametric and Dupin Hypersurfaces
title_fullStr	Isoparametric and Dupin Hypersurfaces
title_full_unstemmed	Isoparametric and Dupin Hypersurfaces
title_short	Isoparametric and Dupin Hypersurfaces
title_sort	isoparametric and dupin hypersurfaces
url	https://nasplib.isofts.kiev.ua/handle/123456789/149015
work_keys_str_mv	AT cecilte isoparametricanddupinhypersurfaces

Isoparametric and Dupin Hypersurfaces

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