aa r X i v : . [ m a t h . DG ] J a n Compact Dupin Hypersurfaces
Thomas E. CecilJanuary 15, 2021
Abstract
A hypersurface M in R n is said to be Dupin if along each cur-vature surface, the corresponding principal curvature is constant. ADupin hypersurface is said to be proper Dupin if the number of distinctprincipal curvatures is constant on M , i.e., each continuous principalcurvature function has constant multiplicity on M . These conditionsare preserved by stereographic projection, so this theory is essentiallythe same for hypersurfaces in R n or S n . The theory of compact properDupin hypersurfaces in S n is closely related to the theory of isopara-metric hypersurfaces in S n , and many important results in this fieldconcern relations between these two classes of hypersurfaces. Thisproblem was formulated in 1985 in a conjecture of Cecil and Ryan[17, p. 184], which states that every compact, connected proper Dupinhypersurface M ⊂ S n is equivalent to an isoparametric hypersurfacein S n by a Lie sphere transformation. This paper gives a survey ofprogress on this conjecture and related developments. This paper is a survey of the main results in the theory of compact, properDupin hypersurfaces in Euclidean space R n , which began with the study ofthe cyclides of Dupin in R in a book by Dupin [25] in 1822. This theoryis closely related to the well-known theory of isoparametric hypersurfaces in S n , i.e., hypersurfaces with constant principal curvatures in S n , introducedby E. Cartan [2]–[5] and developed by many mathematicians (see [7], [23],[65], [18, pp. 85–184] for surveys). 1n fact, many results in this field concern conditions under which a com-pact proper Dupin hypersurface is equivalent to an isoparametric hypersur-face in S n by a M¨obius transformation or a Lie sphere transformation. Thisproblem was formulated in 1985 in a conjecture of Cecil and Ryan [17, p. 184](Conjecture 5.1 below), which states that every compact, connected properDupin hypersurface is equivalent to an isoparametric hypersurface in a sphereby a Lie sphere transformation. This paper gives a survey of progress on thisconjecture and related developments.The theory of Dupin hypersurfaces is essentially the same in the twoambient spaces R n and S n , since the Dupin property is preserved by stereo-graphic projection τ : S n − { P } → R n with pole P ∈ S n , and its inverse mapfrom R n into S n (see, for example, [17, pp. 132–151], [18, pp. 28–30]). Ingeneral, we will use whichever ambient space is most convenient to explain acertain concept. Since much of the theory involves the relationship betweenDupin hypersurfaces and isoparametric hypersurfaces in spheres, we will firstformulate our definitions in terms of hypersurfaces in S n .Let f : M → S n be an immersed hypersurface, and let ξ be a locallydefined field of unit normals to f ( M ). A curvature surface of M is a smoothsubmanifold S ⊂ M such that for each point x ∈ S , the tangent space T x S is equal to a principal space of the shape operator A of M at x . Thisgeneralizes the classical notion of a line of curvature for a principal curvatureof multiplicity one.A hypersurface M is said to be Dupin if:(a) along each curvature surface, the corresponding principal curvature isconstant.Furthermore, a Dupin hypersurface M is called proper Dupin if, in additionto Condition (a), the following condition is satisfied:(b) the number g of distinct principal curvatures is constant on M .Clearly isoparametric hypersurfaces in S n are proper Dupin, as are thosehypersurfaces in R n obtained from isoparametric hypersurfaces in S n viastereographic projection (see, for example, [18, pp. 28–30]). For example,the well-known ring cyclides of Dupin in R are obtained in this way from astandard product torus S ( r ) × S ( s ) in S , where r + s = 1.We now mention several basic facts about Dupin hypersurfaces (see, forexample, [18, pp. 9–35] for proofs). Let f : M → S n be an immersed2ypersurface, and let ξ be a locally defined field of unit normals to f ( M ).Using the Codazzi equation, one can show that Condition (a) above is alwayssatisfied on a curvature surface S of dimension greater than one. However,Condition (a) is not necessarily satisfied on a curvature surface of dimensionone (i.e., a line of curvature). Second, Condition (b) is equivalent to requiringthat each continuous principal curvature function has constant multiplicityon M . Further, for any hypersurface M in S n , there exists a dense opensubset of M on which the number of distinct principal curvatures is locallyconstant (see, for example, Singley [57]).Next, it also follows from the Codazzi equation that if a continuous prin-cipal curvature function µ has constant multiplicity m on a connected opensubset U ⊂ M , then µ is a smooth function, and the distribution T µ ofprincipal spaces corresponding to µ is a smooth foliation whose leaves arethe curvature surfaces corresponding to µ on U . This principal curvaturefunction µ is constant along each of its curvature surfaces in U if and onlyif these curvature surfaces are open subsets of m -dimensional great or smallspheres in S n .Suppose that µ = cot θ , for 0 < θ < π , where θ is a smooth function on U . The corresponding focal map f µ which maps x ∈ M to the focal point f µ ( x ) is given by the formula, f µ ( x ) = cos θ ( x ) f ( x ) + sin θ ( x ) ξ ( x ) . (1)The principal curvature µ also determines a second focal map obtained byreplacing θ by θ + π in equation (1). The image of this second focal map isantipodal to the image of f µ . The principal curvature function µ is constantalong each of its curvature surfaces in U if and only if the focal map f µ factorsthrough an immersion of the ( n − − m )-dimensional space of leaves U/T µ into S n , and so f µ ( U ) is an ( n − − m )-dimensional submanifold of S n .By definition, the curvature sphere K µ ( x ) corresponding to the principalcurvature µ at a point x ∈ U is the hypersphere in S n through f ( x ) withone of its centers at the focal point f µ ( x ), and the other center at the focalpoint antipodal to f µ ( x ). Thus, K µ ( x ) is tangent to f ( M ) at f ( x ). It iseasy to show that the principal curvature map µ is constant along each ofits curvature surfaces if and only if the curvature sphere map K µ is constantalong each of these same curvature surfaces.In summary, on an open subset U on which Condition (b) holds, Condi-tion (a) is equivalent to requiring that each curvature surface in each principal3oliation be an open subset of a metric sphere in S n of dimension equal to themultiplicity of the corresponding principal curvature. Condition (a) is alsoequivalent to requiring that along each curvature surface, the correspondingcurvature sphere map is constant. Finally, Condition (a) is equivalent to re-quiring that for each principal curvature µ , the image of the focal map f µ is asmooth submanifold of S n of codimension m + 1, where m is the multiplicityof µ .There exist many examples of Dupin hypersurfaces that are not properDupin, because the number of distinct principal curvatures is not constanton the hypersurface. This also results in curvature surfaces that are notleaves of a principal foliation. The following example due to Pinkall [50] of atube M in R of constant radius over a torus of revolution T ⊂ R ⊂ R illustrates these points. This description of Pinkall’s example is taken fromthe book [6, p. 69]. Example 1.1.
A Dupin hypersurface that is not proper Dupin.
Let T be a torus of revolution in R , and embed R into R = R × R . Let η be a field of unit normals to T in R . Let M be a tube of sufficiently smallradius ε > T in R , so that M is a compact smooth embeddedhypersurface in R . The normal space to T in R at a point x ∈ T isspanned by η ( x ) and e = (0 , , , A η of T has twodistinct principal curvatures at each point of T , while the shape operator A e of T is identically zero. Thus the shape operator A ζ for the normal ζ = cos θ η ( x ) + sin θ e at a point x ∈ T is given by A ζ = cos θ A η ( x ) . From the formulas for the principal curvatures of a tube (see Cecil–Ryan[18, p. 17]), one finds that at all points of M where x = ± ε , there arethree distinct principal curvatures of multiplicity one, which are constantalong their corresponding lines of curvature (curvature surfaces of dimensionone). However, on the two tori, T × {± ε } , the principal curvature κ = 0has multiplicity two. These two tori are curvature surfaces for this principalcurvature, since the principal space corresponding to κ is tangent to eachtorus at every point. M is a Dupin hypersurface in R , but it is not properDupin, since the number of distinct principal curvatures is not constant on4 . The two tori T × {± ε } are curvature surfaces, but they are not leavesof a principal foliation on M .One consequence of the results mentioned above is that proper Dupinhypersurfaces are algebraic, as is the case with isoparametric hypersurfaces,as shown by M¨unzner [41]–[42]. This result is most easily formulated forhypersurfaces in R n . It states that a connected proper Dupin hypersurface f : M → R n must be contained in a connected component of an irreduciblealgebraic subset of R n of dimension n −
1. Pinkall [48] sent the author a letterin 1984 that contained a sketch of a proof of this result, but he did not publisha proof. In 2008, Cecil, Chi and Jensen [13] used methods of real algebraicgeometry to give a proof of this result based on Pinkall’s sketch. The proofmakes use of the various principal foliations whose leaves are open subsets ofspheres to construct an analytic algebraic parametrization of a neighborhoodof f ( x ) for each point x ∈ M . In contrast to the situation for isoparametrichypersurfaces, however, a connected proper Dupin hypersurface in S n doesnot necessarily lie in a compact connected proper Dupin hypersurface.The definition of Dupin can be extended to submanifolds of codimensiongreater than one as follows. Let φ : V → R n (or S n ) be a submanifold ofcodimension greater than one, and let B n − denote the unit normal bundleof φ ( V ). In this case, a curvature surface (see Reckziegel [53]) is definedto be a connected submanifold S ⊂ V for which there is a parallel section η : S → B n − such that for each x ∈ S , the tangent space T x S is equal tosome smooth eigenspace of the shape operator A η . The submanifold φ ( V ) issaid to be Dupin if along each curvature surface, the corresponding principalcurvature of A η is constant. A Dupin submanifold is proper Dupin if thenumber of distinct principal curvatures is constant on the unit normal bundle B n − .Terng [62] generalized the notion of an isoparametric hypersurface to sub-manifolds of codimension greater than one, and she proved that an isopara-metric submanifold of codimension greater than one is always Dupin, but itmay not be proper Dupin [63, pp. 464–469]. In related results, Pinkall [50, p.439] proved that every extrinsically symmetric submanifold of a real spaceform is Dupin. Then Takeuchi [61] determined which of these are properDupin. 5 Submanifolds in Lie sphere geometry
Many of the important results described in this paper concern finding condi-tions under which a compact proper Dupin hypersurface in S n is equivalentby a Lie sphere transformation to an isoparametric hypersurface in S n . Tomake this precise, we now give a brief description of the method for studyinghypersurfaces in R n or S n within the context of Lie sphere geometry (intro-duced by Lie [32]). The reader is referred to the papers of Pinkall [50], Chern[19], Cecil and Chern [8]–[9], Cecil and Jensen [14], or the books of Cecil [6],or Jensen, Musso and Nicolodi [30], for more detail.Lie sphere geometry is situated in real projective space P n , so we nowbriefly review some concepts and notation from projective geometry. Wedefine an equivalence relation on R n +1 − { } by setting x ≃ y if x = ty forsome nonzero real number t . We denote the equivalence class determined bya vector x by [ x ]. Projective space P n is the set of such equivalence classes,and it can naturally be identified with the space of all lines through the originin R n +1 . The rectangular coordinates ( x , . . . , x n +1 ) are called homogeneouscoordinates of the point [ x ] in P n , and they are only determined up to anonzero scalar multiple.A Lie sphere in S n is an oriented hypersphere or a point sphere in S n .The set of all Lie spheres is in bijective correspondence with the set of allpoints [ x ] = [( x , . . . , x n +3 )] in projective space P n +2 that lie on the quadrichypersurface Q n +1 determined by the equation h x, x i = 0, where h x, y i = − x y + x y + · · · + x n +2 y n +2 − x n +3 y n +3 (2)is a bilinear form of signature ( n + 1 ,
2) on the indefinite inner product space R n +32 . The quadric Q n +1 is called the Lie quadric .The specific correspondence is as follows. We identify S n with with theunit sphere in R n +1 ⊂ R n +32 , where R n +1 is spanned by the standard basisvectors { e , . . . , e n +2 } in R n +32 . Then the oriented hypersphere with center p ∈ S n and signed radius ρ corresponds to the point in Q n +1 with homoge-neous coordinates, (cos ρ, p, sin ρ ) , (3)where − π < ρ < π .We can designate the orientation of the sphere by the sign of ρ as follows.A positive radius ρ in (3) corresponds to the orientation of the sphere givenby the field of unit normals which are tangent vectors to geodesics in S n going6rom − p to p , and a negative radius corresponds to the opposite orientation.Each oriented sphere can be considered in two ways, with center p and signedradius ρ, − π < ρ < π , or with center − p and the appropriate signed radius ρ ± π . Point spheres p in S n correspond to those points [(1 , p, Q n +1 with radius ρ = 0.Due to the signature of the metric h , i , the Lie quadric Q n +1 containsprojective lines but no linear subspaces of P n +2 of higher dimension (see, forexample, [6, p. 21]). A straightforward calculation shows that if [ x ] and [ y ]are two points on the quadric, then the line [ x, y ] lies on Q n +1 if and only if h x, y i = 0. Geometrically, this condition means that the hyperspheres in S n corresponding to [ x ] and [ y ] are in oriented contact, i.e., they are tangent toeach other and have the same orientation at the point of contact. For a pointsphere and an oriented sphere, oriented contact means that the point lies onthe sphere. The 1-parameter family of Lie spheres in S n corresponding tothe points on a line on the Lie quadric is called a parabolic pencil of spheres .If we wish to work in R n , the set of Lie spheres consists of all orientedhyperspheres, oriented hyperplanes, and point spheres in R n ∪ {∞} . As inthe spherical case, we can find a bijective correspondence between the setof all Lie spheres and the set of points on Q n +1 , and the notions of orientedcontact and parabolic pencils of Lie spheres are defined in a natural way (see,for example, [6, pp. 14–23]).A Lie sphere transformation is a projective transformation of P n +2 whichmaps the Lie quadric Q n +1 to itself. In terms of the geometry of R n (or S n ),a Lie sphere transformation maps Lie spheres to Lie spheres. Furthermore,since a Lie sphere transformation maps lines on Q n +1 to lines on Q n +1 , a Liesphere transformation preserves oriented contact of Lie spheres (see Pinkall[50, p. 431] or [6, pp. 25–30]).The group of Lie sphere transformations is isomorphic to O ( n +1 , / {± I } ,where O ( n +1 ,
2) is the orthogonal group for the metric in equation (2). A Liesphere transformation that takes point spheres to point spheres is a
M¨obiustransformation , i.e., it is induced by a conformal diffeomorphism of S n , andthe set of all M¨obius transformations is a subgroup of the Lie sphere group.An example of a Lie sphere transformation that is not a M¨obius transforma-tion is a parallel transformations P t , which fixes the center of each Lie spherebut adds t to its signed radius (see [6, pp. 25–49]).The (2 n − n − of projective lines on the quadric Q n +1 has a contact structure, i.e., a 1-form ω such that ω ∧ ( dω ) n − does notvanish on Λ n − . The condition ω = 0 defines a codimension one distribution7 on Λ n − which has integral submanifolds of dimension n −
1, but none ofhigher dimension. Such an integral submanifold λ : M n − → Λ n − of D ofdimension n − Legendre submanifold (see [6, pp. 51–64]).An oriented hypersurface f : M n − → S n with field of unit normals ξ : M n − → S n naturally induces a Legendre submanifold λ = [ k , k ], where k = (1 , f, , k = (0 , ξ, , (4)in homogeneous coordinates. For each x ∈ M n − , [ k ( x )] is the unique pointsphere and [ k ( x )] is the unique great sphere in the parabolic pencil of spheresin S n corresponding to the points on the line λ ( x ). Similarly, an immersedsubmanifold φ : V → S n of codimension greater than one induces a Legendresubmanifold whose domain is the bundle B n − of unit normal vectors to φ ( V ).In each case, λ is called the Legendre lift of the submanifold in S n . In a similarway, a submanifold of R n naturally induces a Legendre submanifold.If β is a Lie sphere transformation, then β maps lines on Q n +1 to lines on Q n +1 , so it naturally induces a map ˜ β from Λ n − to itself. If λ is a Legendresubmanifold, then one can show that ˜ βλ is also a Legendre submanifold,which is denoted βλ for short. These two Legendre submanifolds are said tobe Lie equivalent . We will also say that two submanifolds of S n or R n areLie equivalent, if their corresponding Legendre lifts are Lie equivalent.If β is a M¨obius transformation, then the two Legendre submanifolds aresaid to be M¨obius equivalent . Finally, if β is the parallel transformation P t and λ is the Legendre lift of an oriented hypersurface f : M → S n , then P t λ is the Legendre lift of the parallel hypersurface f − t at oriented distance − t from f (see, for example, [6, p.67]).It is easy to generalize the definitions of Dupin and proper Dupin hyper-surfaces in S n to the class of Legendre submanifolds in Lie sphere geometry.We simply replace the notion of a principal curvature (which is not Lie invari-ant) with the notion of a curvature sphere (which is Lie invariant). We thensay that a Legendre submanifold λ : M n − → Λ n − is a Dupin submanifold if:(a) along each curvature surface, the corresponding curvature sphere mapis constant.Furthermore, a Dupin submanifold λ is called proper Dupin if, in addition toCondition (a), the following condition is satisfied:8b) the number g of distinct curvature spheres is constant on M .One can easily show that a Lie sphere transformation β maps curvaturespheres of λ to curvature spheres of βλ , and that Conditions (a) and (b) arepreserved by β (see [6, pp.67–70]). Thus, both the Dupin and proper Dupinproperties are invariant under Lie sphere transformations. In this section, we discuss Pinkall’s [50] method for constructing local exam-ples of proper Dupin hypersurfaces in Euclidean space that have an arbitrarynumber of distinct principal curvatures with any given multiplicities. Specifi-cally, these are constructions for obtaining a Dupin hypersurface W in R n + m from a Dupin hypersurface M in R n . We first describe these constructionsin the case m = 1 as follows.Begin with a Dupin hypersurface M n − in R n and then consider R n asthe linear subspace R n × { } in R n +1 . The following constructions yield aDupin hypersurface W n in R n +1 . i. W n is a cylinder M n − × R in R n +1 .ii. W n is the hypersurface in R n +1 obtained by rotating M n − around an axis R n − ⊂ R n . (5) iii. W n is a tube of constant radius around M n − in R n +1 .iv. Project M n − stereographically onto a hypersurface V n − ⊂ S n in R n +1 . W n is the cone over V n − in R n +1 . In general, these constructions introduce a new principal curvature of mul-tiplicity one on W n which is constant along its lines of curvature. The otherprincipal curvatures are determined by the principal curvatures of M n − ,and the Dupin property is preserved for these principal curvatures. Thus, if M n − is a proper Dupin hypersurface in R n with g distinct principal curva-tures, then in general, W n is a proper Dupin hypersurface in R n +1 with g + 1distinct principal curvatures. However, this is not always the case, as thereare cases where the number of distinct principal curvatures of W n does notequal g + 1 at some points, as we will discuss after the proof of Theorem 3.1below. These constructions can be generalized to produce a new principal9urvature of multiplicity m by considering R n as a subset of R n × R m ratherthan R n × R .Although Pinkall gave these four constructions, he showed [50, p. 438]that the cone construction is redundant, since it is Lie equivalent to the tubeconstruction (see also [6, p.144]). Thus, we often work with just the cylinder,surface of revolution, and tube constructions, as in [6].Using these constructions, Pinkall [50] showed how to produce a properDupin hypersurface in Euclidean space with an arbitrary number of distinctprincipal curvatures, each with any given multiplicity as follows. Theorem 3.1. (Pinkall, 1985) Given positive integers m , . . . , m g with m + · · · + m g = n − , there exists a proper Dupin hypersurface in R n with g distinct principal cur-vatures having respective multiplicities m , . . . , m g .Proof. The proof is by an inductive construction, which will be clear oncewe do the first few examples. To begin, note that a usual torus of revolutionin R is a proper Dupin hypersurface with two principal curvatures. To con-struct a proper Dupin hypersurface M in R with three principal curvaturesof multiplicity one, begin with an open subset U of a torus of revolution in R on which neither principal curvature vanishes. Take M to be the cylinder U × R in R × R = R . Then M has three distinct principal curvatures ateach point, one of which is identically zero. These are clearly constant alongtheir corresponding 1-dimensional curvature surfaces (lines of curvature).To get a proper Dupin hypersurface in R with three principal curvatureshaving respective multiplicities m = m = 1, m = 2, one simply takes thecylinder U × R ⊂ R × R = R , where U is the open subset of the torus defined above. To obtain a properDupin hypersurface M in R with four principal curvatures of multiplicityone, first invert the hypersurface M defined above in a 3-sphere in R chosenso that the image of M contains an open subset W on which no principalcurvature vanishes. The hypersurface W is proper Dupin, since the properDupin property is preserved by M¨obius transformations. Now take M tobe the cylinder W × R in R × R = R . Then M is a proper Dupinhypersurface in R with four principal curvatures of multiplicity one.10n general, there are problems in trying to produce compact proper Dupinhypersurfaces by using the constructions in equation (5). We now examinesome of the problems involved with the the cylinder, surface of revolution,and tube constructions individually (see [6, pp. 127–141] for more details).For the cylinder construction (5i), the new principal curvature of W n is identically zero, while the other principal curvatures of W n are equal tothose of M n − . Thus, if one of the principal curvatures µ of M n − is zero atsome points but not identically zero, then the number of distinct principalcurvatures is not constant on W n , and so W n is Dupin but not proper Dupin.For the surface of revolution construction (5ii), if the focal point corre-sponding to a principal curvature µ at a point x of the profile submanifold M n − lies on the axis of revolution R n − , then the principal curvature of W n at x determined by µ is equal to the new principal curvature of W n resultingfrom the surface of revolution construction. Thus, if the focal point of M n − corresponding to µ lies on the axis of revolution for some but not all pointsof M n − , then W n is not proper Dupin.If W n is a tube (5iii) in R n +1 of radius ǫ over M n − , then there are exactlytwo distinct principal curvatures at the points in the set M n − × {± ǫ } in W n ,regardless of the number of distinct principal curvatures on M n − . Thus, W n is not a proper Dupin hypersurface unless the original hypersurface M n − istotally umbilic, i.e., it has only one distinct principal curvature at each point.Another problem with these constructions is that they may not yieldan immersed hypersurface in R n +1 . In the tube construction, if the radiusof the tube is the reciprocal of one of the principal curvatures of M n − atsome point, then the constructed object has a singularity. For the surface ofrevolution construction, a singularity occurs if the profile submanifold M n − intersects the axis of revolution.Many of the issues mentioned in the preceding paragraphs can be resolvedby working in the context of Lie sphere geometry and considering Legendrelifts of hypersurfaces in Euclidean space (see [6, pp.127–141]). In that con-text, a proper Dupin submanifold λ : M n − → Λ n − is said to be reducible if it is is locally Lie equivalent to the Legendre lift of a hypersurface in R n obtained by one of Pinkall’s constructions in equation (5).Pinkall [50] found a useful characterization of reducibility in the contextof Lie sphere geometry when he proved that a proper Dupin submanifold λ : M n − → Λ n − is reducible if and only if the image of one its curvaturesphere maps K lies in a linear subspace of codimension two in P n +2 (see also[6, pp. 141–148]). 11ne can obtain a reducible compact proper Dupin hypersurface with twoprincipal curvatures by revolving a circle C in R about an axis R ⊂ R that is disjoint from C to obtain a torus of revolution. However, Cecil, Chiand Jensen [11] (see also [6, pp. 146–147]) showed that every compact properDupin hypersurface with more than two principal curvatures is irreducible. Theorem 3.2. (Cecil-Chi-Jensen, 2007) If M n − ⊂ R n is a compact, con-nected proper Dupin hypersurface with g ≥ principal curvatures, then M n − is irreducible. The proof uses known facts about the topology of a compact proper Dupinhypersurface and the topology of a compact hypersurface obtained by one ofPinkall’s constructions (see [11] or [6, pp. 146–148] for a complete proof).
Remark 3.1.
From Theorem 3.2, we see that one approach to obtainingclassifications of compact proper Dupin hypersurfaces with more than twoprincipal curvatures is by assuming that the hypersurface is irreducible andthen working locally in the context of Lie sphere geometry using the methodof moving frames. This approach has been used successfully in the papers ofPinkall [47], [49]–[50], Cecil and Chern [9], Cecil and Jensen [14]–[15], andCecil, Chi and Jensen [11]. We will discuss this in more detail in Section 5(see also [6, pp. 168–190]).
In this section, we briefly review the theory of isoparametric hypersurfaces inreal space forms. This leads to many important examples of compact properDupin hypersurfaces. Recall that an immersed hypersurface M in a realspace form, R n , S n , or real hyperbolic space H n , is said to be isoparametric if it has constant principal curvatures.An isoparametric hypersurface M in R n can have at most two distinctprincipal curvatures, and M must be an open subset of a hyperplane, hyper-sphere or a spherical cylinder S k × R n − k − . This was shown by Levi–Civita[31] for n = 3 and by B. Segre [55] for arbitrary n .E. Cartan [2]–[5] began the study of isoparametric hypersurfaces in theother space forms in a series of four papers in the 1930’s. In hyperbolicspace H n , he showed that an isoparametric hypersurface can have at most12wo distinct principal curvatures, and it is either totally umbilic or else astandard product S k × H n − k − in H n (see also Ryan [54, pp. 252–253] orCecil-Ryan [18, pp. 97–98]).In the sphere S n , however, Cartan showed that there are many morepossibilities. He found examples of isoparametric hypersurfaces in S n with1 , , g ≤ g = 1, then the isoparametric hypersurface M is totally umbilic, andit must be a great or small sphere. If g = 2, then M must be a standardproduct of two spheres, S p ( r ) × S n − p − ( s ) ⊂ S n , r + s = 1 , where 1 ≤ p ≤ n − g = 3, Cartan [3] showed that all the principal curvaturesmust have the same multiplicity m = 1 , , FP into S m +1 , where F is the division algebra R , C , H (quaternions), O (Cayley numbers), for m = 1 , , , , respectively (seealso Cecil-Ryan [18, pp. 151–155]). Thus, up to congruence, there is onlyone such family for each value of m .Cartan’s theory was further developed by Nomizu [43]–[44], Takagi andTakahashi [60], Ozeki and Takeuchi [45]–[46], and by M¨unzner [41]–[42], whoproved the following fundamental result (see also Chapter 3 of Cecil–Ryan[18] or the survey article by Thorbergsson [65]). Note that M¨unzner’s paperswere published in 1980–1981, although the first preprints of these papersappeared in the early 1970’s, and the results were used by other researchersshortly after that. Theorem 4.1. (M¨unzner) The number g of distinct principal curvatures ofa connected isoparametric hypersurface M ⊂ S n must be , , , or . M¨unzner first showed that every connected isoparametric hypersurface in S n is algebraic, and thus it is an open subset of a unique compact, connectedisoparametric hypersurface in S n . The proof of Theorem 4.1 then beginswith the assumption that M ⊂ S n is a compact, connected isoparametrichypersurface. The proof involves a lengthy, delicate computation using thecohomology rings of M and its two focal submanifolds in S n . The structure13f these cohomology rings is determined by the topological situation that M ⊂ S n divides S n into two ball bundles over the two focal submanifoldsof M in S n . These two focal submanifolds, M + and M − , lie in differentcomponents of the complement of M in S n .This topological situation has been used by various authors to determinethe possible multiplicities of the principal curvatures of an isoparametrichypersurface with g = 4 or g = 6 principal curvatures. This same topologi-cal situation also holds for compact, connected proper Dupin hypersurfacesembedded in S n , and thus the same restrictions on the number g of distinctprincipal curvatures and their multiplicities also hold on those hypersurfaces,as we will describe in Section 5.The classifications of isoparametric hypersurfaces in S n with g = 4 or g = 6 principal curvatures have recently been completed in a series of results,which we will now describe. In the case g = 4, M¨unzner proved that the prin-cipal curvatures can have at most two distinct multiplicities m , m . Ferus,Karcher and M¨unzner [27] then used representations of Clifford algebras toconstruct for every positive integer m an infinite series of isoparametrichypersurfaces with four principal curvatures having respective multiplicities( m , m ), where m is nondecreasing and unbounded in each series (see also[6, pp. 95–112], [18, pp. 162–180]). This class of FKM-type isoparamet-ric hypersurfaces contains all examples of isoparametric hypersurfaces withfour principal curvatures with the exception of two homogeneous examples,which have multiplicities (2 ,
2) and (4 , Clifford-Stiefel manifold , as describedby Pinkall and Thorbergsson [52] (see also [18, p. 174]).Stolz [58] then proved that the multiplicities of the principal curvaturesof an isoparametric hypersurface with four principal curvatures must be thesame as those of an isoparametric hypersurface of FKM-type or one of the twohomogeneous exceptions. Cecil, Chi and Jensen [10] next showed that if themultiplicities satisfy m ≥ m −
1, then the isoparametric hypersurface mustbe of FKM-type (see also Immervoll [29] for a different proof of this result).Taken together with known classification results of Takagi [59] for the case m = 1, and Ozeki and Takeuchi [45]–[46] for the case m = 2, this handlesall possible pairs of multiplicities except for four cases, the homogeneous pair(4 ,
5) and the FKM pairs (3 , , (6 ,
9) and (7 , g = 4. This showed thatevery isoparametric hypersurface with four principal curvatures is either of14KM-type or else a homogeneous example with multiplicities (2 ,
2) or (4 , g = 6 principal curvatures, M¨unzner [41]–[42] showed thatall the principal curvatures have the same multiplicity m , and Abresch [1]showed that m equals 1 or 2. Takagi and Takahashi [60] found homogeneousisoparametric families in both cases m = 1 and m = 2, and they showed thatup to congruence, there is only one homogeneous family of isoparametrichypersurfaces in each case.Dorfmeister and Neher [24] then proved that every isoparametric hyper-surface M ⊂ S with g = 6 principal curvatures of multiplicity m = 1 ishomogeneous. Their proof is very algebraic in nature, and later Miyaoka[37] and Siffert [56] gave alternative approaches to the proof in the case m = 1. Finally, Miyaoka [38]–[39] completed the classification of isopara-metric hypersurfaces with g = 6 principal curvatures by proving that everyisoparametric hypersurface with six principal curvatures of multiplicity twois homogeneous. In this section, we discuss results on the classification of compact properDupin hypersurfaces. Of course, compact isoparametric hypersurfaces in S n have constant principal curvatures, so they are obviously compact properDupin hypersurfaces in S n .The images of compact isoparametric hypersurfaces under stereographicprojection from S n − { P } to R n are compact proper Dupin hypersurfaces in R n (see, for example, [18, pp. 28–30]). In addition, since the proper Dupinproperty is invariant under Lie sphere transformations, any compact properhypersurface in S n or R n that is Lie equivalent to a compact isoparametrichypersurface in S n is a compact proper Dupin hypersurface. This gives alarge class of interesting examples of compact proper Dupin hypersurfaces.Since the 1970’s, an important research topic in this area has been to whatextent there are any other examples.Following M¨unzner’s work, Thorbergsson [64] proved the following the-orem which shows that the restriction in Theorem 4.1 on the number g ofdistinct principal curvatures also holds for a compact proper Dupin hypersur-face embedded in S n . This is in stark contrast to Pinkall’s Theorem 3.1 whichstates that there are no restrictions on the number of distinct principal cur-vatures or their multiplicities for non-compact proper Dupin hypersurfaces.15 heorem 5.1. (Thorbergsson, 1983) The number g of distinct principal cur-vatures of a compact, connected proper Dupin hypersurface M ⊂ S n must be , , , or . In proving this theorem, Thorbergsson first shows that a compact, con-nected proper Dupin hypersurface M ⊂ S n must be tautly embedded, that is,every nondegenerate spherical distance function L p ( x ) = d ( p, x ) , for p ∈ S n ,has the minimum number of critical points required by the Morse inequal-ities on M . Thorbergsson then uses the fact that M is tautly embeddedin S n to show that M divides S n into two ball bundles over the first focalsubmanifolds, M + and M − , on either side of M in S n . This gives the sametopological situation as in the isoparametric case, and the result then followsfrom M¨unzner’s proof of Theorem 4.1 above.Using Thorbergsson’s results, we can formulate the relationship betweenthe taut and proper Dupin properties as follows. Theorem 5.2.
Let M ⊂ S n be a compact, connected hypersurface on whichthe number g of distinct principal curvatures is constant. Then M is taut ifand only if M is proper Dupin.Proof. As noted above, Thorbergsson [64] proved that a compact, connectedproper Dupin hypersurface M ⊂ S n must be tautly embedded. Conversely,Pinkall [51] and Miyaoka [34] (for hypersurfaces) independently proved thata taut submanifold embedded in S n is Dupin. Thus, a taut hypersurface M is proper Dupin if g is constant on M (see also [18, pp. 65–74] for morediscussion on the relationship between taut and Dupin submanifolds).The topological situation that M divides S n into two ball bundles overthe first focal submanifolds, M + and M − , on either side of M in S n leadsto important restrictions on the multiplicities of the principal curvaturesof compact proper Dupin hypersurfaces, due to Stolz [58] for g = 4, andto Grove and Halperin [28] for g = 6. These restrictions were obtainedby using advanced topological considerations in each case, and they showthat the multiplicities of the principal curvatures of a compact proper Dupinhypersurface embedded in S n must be the same as the multiplicities of theprincipal curvatures of some isoparametric hypersurface in S n .Grove and Halperin [28] also gave a list of the integral homology of allcompact proper Dupin hypersurfaces, and Fang [26] found results on thetopology of compact proper Dupin hypersurfaces with g = 6 principal cur-vatures. 16n 1985, it was known that every compact, connected proper Dupin hy-persurface M ⊂ S n (or R n ) with g = 1 , S n . At that time, every otherknown example of a compact, connected proper Dupin hypersurface in S n was also Lie equivalent to an isoparametric hypersurface in S n . This togetherwith Thorbergsson’s Theorem 5.1 above led to the following conjecture byCecil and Ryan [17, p. 184] (which we have rephrased slightly). Conjecture 5.1. (Cecil-Ryan, 1985) Every compact, connected proper Dupinhypersurface M ⊂ S n ( or R n ) is Lie equivalent to an isoparametric hyper-surface in S n . We now discuss the current status of this conjecture for each of the values g = 1 , , , ,
6, respectively. After that, we will give more detail about theindividual cases of g . g = 1: Conjecture is true. M is totally umbilic, and so M = S n − ⊂ S n , ametric hypersphere. Thus, M is isoparametric. g = 2: Conjecture is true (Cecil-Ryan [16], 1978). M is a cyclide of Dupin.Specifically, M is M¨obius equivalent to an isoparametric hypersurface S p ( r ) × S n − p − ( s ) ⊂ S n , r + s = 1 .g = 3: Conjecture is true (Miyaoka [33], 1984). M is Lie equivalent (butnot necessarily M¨obius equivalent) to an isoparametric hypersurface W m in S m +1 , for m = 1 , , g = 4: Conjecture is false. Counterexamples due to:(a) Pinkall-Thorbergsson [52] (1989): these counterexamples are certain de-formations of FKM-type [27] isoparametric hypersurfaces.(b) Miyaoka-Ozawa [40] (1989): these counterexamples are hypersurfaces ofthe form M = h − ( W ) ⊂ S , where h : S → S is the Hopf fibration, and W ⊂ S is a compact proper Dupin hypersurface with 2 principal curvaturesthat is not isoparametric. g = 6: Conjecture is false. Counterexamples due to:17iyaoka-Ozawa [40] (1989): these counterexamples are also hypersurfaces ofthe form M = h − ( W ) ⊂ S , where h : S → S is the Hopf fibration, and W ⊂ S is a compact proper Dupin hypersurface with 3 principal curvaturesthat is not isoparametric (same construction as above).We now make some comments about the individual cases based on thevalue of g . The case g = 1 is simply the well-known case of totally umbilichypersurfaces, and thus M is a great or small hypersphere in S n .In the case g = 2, a connected proper Dupin hypersurface with twoprincipal curvatures having respective multiplicities p and q is called a cyclideof Dupin of characteristic ( p, q ). Cecil and Ryan [16] showed that a compact,connected cyclide of characteristic ( p, q ) is M¨obius equivalent to a standardproduct of spheres (which is an isoparametric hypersurface), S p ( r ) × S q ( s ) ⊂ S n (1) ⊂ R n +1 , r + s = 1 , (6)where p + q = n −
1. The proof of Cecil and Ryan [16] uses the assumptionof the compactness of M in an essential way. Later, working in the contextof Lie sphere geometry, Pinkall [50] gave a local classification of the cyclidesof Dupin by showing that every connected cyclide of characteristic ( p, q ) iscontained in a unique compact, connected cyclide of characteristic ( p, q ),and that any two compact, connected cyclides of characteristic ( p, q ) are Lieequivalent to each other (see also [6, pp. 148–159]).In the case g = 3, Miyaoka [33] proved that M is Lie equivalent (althoughnot necessarily M¨obius equivalent) to an isoparametric hypersurface. LaterCecil, Chi and Jensen [11] gave a different proof of this result, based onthe local theorem of Cecil and Jensen [14] which states that a connectedirreducible proper Dupin hypersurface in S n with three principal curvaturesmust be Lie equivalent to an isoparametric hypersurface in S n .As mentioned in Section 4, Cartan [3] had shown earlier that an isopara-metric hypersurface with g = 3 principal curvatures is a tube over a standardembedding of a projective plane FP , for F = R , C , H (quaternions) or O (Cayley numbers) in S , S , S and S , respectively (see also [18, pp. 151–155]).The case g = 4 resisted all attempts at solution for several years andfinally in papers published in 1989, counterexamples to the conjecture wereconstructed independently by Pinkall and Thorbergsson [52], and by Miyaokaand Ozawa [40]. These two constructions lead to different types of counterex-18mples, and the method of Miyaoka and Ozawa also yields counterexamplesto the conjecture with g = 6 principal curvatures.In both constructions, a fundamental Lie invariant, the Lie curvatureintroduced by Miyaoka [35], was used to show that the examples are not Lieequivalent to an isoparametric hypersurface. Specifically, if M is a properDupin hypersurface with four principal curvatures, then the Lie curvature ψ is defined to be the cross-ratio of these principal curvatures. If M hassix principal curvatures, then the Lie curvatures are the cross-ratios of theprincipal curvatures taken four at a time.Viewed in the context of projective geometry, at each point x ∈ M ,a Lie curvature is the cross-ratio of four points along a projective line on Q n +1 corresponding to four curvature spheres of M at x . Since a Lie spheretransformation maps curvature spheres to curvature spheres and preservescross-ratios, a Lie curvature is invariant under Lie sphere transformations(see [6, pp. 72–82] for more detail).From the work of M¨unzner [41]–[42], it is easy to show that in the case g = 4, the Lie curvature ψ has the constant value 1 / ψ = 1 / ψ = 1 /
2, as required for a hypersurface with g = 4 that is Lie equivalent to an isoparametric hypersurface. Using theirmethods, one can also show directly that the Lie curvature is not constanton their examples (see [18, pp. 309–314]).The construction of counterexamples to Conjecture 5.1 due to Miyaokaand Ozawa [40] (see also [6, pp. 117–123]) is based on the Hopf fibration h : S → S . Miyaoka and Ozawa first show that if W is a taut compactsubmanifold of S , then M = h − ( W ) is a taut compact submanifold of S .Using this and the fact that tautness is equivalent to proper Dupin fora compact, connected hypersurface in S n on which the number g of distinctprincipal curvatures is constant (Theorem 5.2), they show that if W is aproper Dupin hypersurface in S with g distinct principal curvatures, then19 − ( W ) is a proper Dupin hypersurface in S with 2 g principal curvatures.To complete the argument, they show that if a compact, connected hy-persurface W ⊂ S is proper Dupin but not isoparametric, then the Liecurvatures of h − ( W ) are not constant, and therefore h − ( W ) is not Lieequivalent to an isoparametric hypersurface in S . For g = 2 or 3, this givesa compact proper Dupin hypersurface M = h − ( W ) in S with g = 4 or 6,respectively, that is not Lie equivalent to an isoparametric hypersurface.As we have seen, all of the hypersurfaces described above are shown to becounterexamples to Conjecture 5.1 by proving that they do not have constantLie curvatures. This led to a revision of Conjecture 5.1 by Cecil, Chi andJensen [12, p. 52] in 2007 that contains the additional assumption of constantLie curvatures. This revised conjecture is still an open problem, although ithas been shown to be true in some cases, which we will describe after statingthe conjecture. Conjecture 5.2. (Cecil-Chi-Jensen, 2007) Every compact, connected properDupin hypersurface in S n with four or six principal curvatures and constantLie curvatures is Lie equivalent to an isoparametric hypersurface. In 1989, Miyaoka [35]–[36] showed that if some additional assumptionsare made regarding the intersections of the leaves of the various principalfoliations, then this revised conjecture is true in both cases g = 4 and 6. Thusfar, however, it has not been proven that Miyaoka’s additional assumptionsare satisfied in general.Cecil, Chi and Jensen [11] made progress on the revised conjecture in thecase g = 4 by using the fact that compactness implies irreducibility for aproper Dupin hypersurface with g ≥ g = 4.Case g = 4: There is only one Lie curvature, ψ = ( µ − µ )( µ − µ )( µ − µ )( µ − µ ) , (7)when we fix the order of the principal curvatures of M to be, µ < µ < µ < µ . (8)20or an isoparametric hypersurface with four principal curvatures ordered asin equation (8), M¨unzner’s results [41]–[42] imply that the Lie curvature ψ = 1 /
2, and the multiplicities satisfy m = m , m = m . Furthermore,if M ⊂ S n is a compact, connected proper Dupin hypersurface with g = 4,then the multiplicities of the principal curvatures must be the same as thoseof an isoparametric hypersurface by the work of Stolz [58], so they satisfy m = m , m = m .Cecil-Chi-Jensen [11] proved the following local classification of irreducibleproper Dupin hypersurfaces with four principal curvatures and constant Liecurvature ψ = 1 /
2. In the case where all the multiplicities equal one, thistheorem was first proven by Cecil and Jensen [15].
Theorem 5.3. (Cecil-Chi-Jensen, 2007) Let M ⊂ S n be a connected irre-ducible proper Dupin hypersurface with four principal curvatures ordered asin equation (8) having multiplicities, m = m ≥ , m = m = 1 , (9) and constant Lie curvature ψ = 1 / . Then M is Lie equivalent to an isopara-metric hypersurface in S n . By Theorem 3.2 above, we know that compactness implies irreducibilityfor proper Dupin hypersurfaces with more than two principal curvatures.Furthermore, Miyaoka [35] proved that if ψ is constant on a compact properDupin hypersurface M ⊂ S n with g = 4, then ψ = 1 / M , when theprincipal curvatures are ordered as in equation (8). As a consequence, weget the following corollary of Theorem 5.3. Corollary 5.1.
Let M ⊂ S n be a compact, connected proper Dupin hyper-surface with four principal curvatures having multiplicities m = m ≥ , m = m = 1 , and constant Lie curvature ψ . Then M is Lie equivalent to an isoparametrichypersurface in S n . Question:
What about the general case where m is also allowed to begreater than one, i.e., m = m ≥ , m = m ≥ m i is greater than one, but no correspondingsums if all m i equal one. These sums make the calculations significantlymore difficult. So far this method has not led to a proof in the general case(10), although Cecil, Chi and Jensen were able to handle the case (9), where m is allowed to be greater than one, but m is restricted to equal one. Keyelements in the proof are the Lie geometric criteria for reducibility due toPinkall [50] (see also [6, pp. 141–148]) and for Lie equivalence to an isopara-metric hypersurface [6, p. 77].Case g = 6: Grove and Halperin (1987) proved that if M ⊂ S n is a compactproper Dupin hypersurface with g = 6 principal curvatures, then all the prin-cipal curvatures must have the same multiplicity m , and m = 1 or 2 as is thecase for an isoparametric hypersurface, as shown by Abresch [1]. They alsoproved other topological results about compact proper Dupin hypersurfacesthat support Conjecture 5.2 in the case g = 6.As mentioned above, Miyaoka [36] showed that if some additional as-sumptions are made regarding the intersections of the leaves of the variousprincipal foliations, then Conjecture 5.2 is true in the case g = 6. However,it has not been proven that Miyaoka’s additional assumptions are satisfied ingeneral, and so Conjecture 5.2 remains as an open problem in the case g = 6. References [1] U. Abresch,
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