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020 Dupin Submanifolds in Lie Sphere

Geometry (updated version)

Thomas E. Cecil∗and Shiing-Shen Chern†

October 14, 2020

Abstract

A hypersurfaceMn−1 in Euclidean space En is proper Dupin if thenumber of distinct principal curvatures is constant on M

n−1, and eachprincipal curvature function is constant along each leaf of its principalfoliation. This paper was originally published in 1989 [11], and itdevelops a method for the local study of proper Dupin hypersurfaces inthe context of Lie sphere geometry using moving frames. This methodhas been effective in obtaining several classification theorems of properDupin hypersurfaces since that time. This updated version of thepaper contains the original exposition together with some remarks byT.Cecil made in 2020 (as indicated in the text) that describe progressin the field since the time of the original version, as well as someimportant remaining open problems in the field.

1 Introduction

Consider a piece of surface immersed in three-dimensional Euclidean spaceE3. Its normal lines are the common tangent lines of two surfaces, the focalsurfaces. These focal surfaces may have singularities, and a classical theoremsays that if the focal surfaces both degenerate to curves, then the curves areconics, and the surface is a cyclide of Dupin. (See, for example, [18, pp.151–166].) Equivalently, the cyclides can be characterized as those surfaces

∗Supported by NSF Grant No. DMS 87-06015†Supported by NSF Grant No. DMS 87-01609

1

in E3 whose two distinct principal curvatures are both constant along theircorresponding lines of curvature.

The cyclides have been generalized to an interesting class of hypersurfacesin En, the Dupin hypersurfaces. Initially, a hypersurface M in En was saidto be Dupin if the number of distinct principal curvatures (or focal points) isconstant onM and if each principal curvature is constant along the leaves ofits corresponding principal foliation. (See [18], [41], [22].) More recently, thishas been generalized to include cases where the number of distinct principalcurvatures is not constant. (See [37], [10].)

Remark 1.1. (added by T. Cecil in 2020) In the terminology of Pinkall [37],Dupin hypersurfaces for which the number of distinct principal curvatures isconstant on M are called proper Dupin.

The study of Dupin hypersurfaces in En is naturally situated in the con-text of Lie sphere geometry, developed by Lie [25] as part of his work oncontact transformations. The projectivized cotangent bundle PT ∗En of En

has a contact structure. In fact, if x1, . . . , xn are the coordinates in En, thecontact structure is defined by the linear differential form

dxn − p1dx1 − · · · − pn−1dx

n−1.

Lie proved that the pseudo-group of all contact transformations carrying(oriented) hyperspheres in the generalized sense (i.e., including points andoriented hyperplanes) into hyperspheres is a Lie group, called the Lie spheregroup, isomorphic to O(n + 1, 2)/ ± I, where O(n + 1, 2) is the orthogonalgroup for an indefinite inner product on Rn+3 with signature (n+1, 2). TheLie sphere group contains as a subgroup the Mobius group of conformaltransformations of En and, of course, the Euclidean group. Lie exhibiteda bijective correspondence between the set of oriented hyperspheres in En

and the points on the quadric hypersurface Qn+1 in real projective spaceP n+2 given by the equation 〈x, x〉 = 0, where 〈 , 〉 is the inner product onRn+3 mentioned above. The manifold Qn+1 contains projective lines butno linear subspaces of P n+2 of higher dimension. The 1-parameter familyof oriented spheres corresponding to the points of a projective line lying onQn+1 consists of all oriented hyperspheres which are in oriented contact at acertain contact element on En. Thus, Lie constructed a local diffeomorphismbetween PT ∗En and the manifold Λ2n−1 of projective lines which lie on Qn+1.

2

An immersed submanifold f : Mk → En naturally induces a Legendresubmanifold λ : Bn−1 → Λ2n−1, where Bn−1 is the bundle of unit normalvectors to f (take Bn−1 = Mn−1 in the case k = n − 1). This Legendremap has similarities with the familiar Gauss map, and like the Gauss map,it can be a powerful tool in the study of submanifolds of Euclidean space. Inparticular, the Dupin property for hypersurfaces in En is easily formulatedin terms of the Legendre map, and it is immediately seen to be invariantunder Lie sphere transformations.

The study of Dupin submanifolds has both local and global aspects. Thor-bergsson [41] showed that a Dupin hypersurface M with g distinct principalcurvatures at each point must be taut, i.e., every nondegenerate Euclideandistance function Lp(x) = |p − x|2, p ∈ En, must have the minimum num-ber of critical points on M . Tautness was shown to be invariant Lie spheretransformations in our earlier paper [10] (see also [8, pp. 82–95]). Usingtautness and the work of Munzner [30]–[31], Thorbergsson was able to con-clude that the number g must be 1, 2, 3, 4 or 6, as with an isoparametrichypersurface in the sphere Sn. The case g = 1 is, of course, handled by thewell-known classification of umbilic hypersurfaces. Compact Dupin hyper-surfaces with g = 2 and g = 3 were classified by Cecil and Ryan [17] andMiyaoka [26], respectively. In two recent papers, Miyaoka [27], [28], has madefurther progress on the classification of compact Dupin hypersurfaces in thecases g = 4 and g = 6. Meanwhile, Grove and Halperin [22] have determinedseveral important topological invariants of compact Dupin hypersurfaces inthe cases g = 4 and g = 6.

Remark 1.2. (added by T. Cecil in 2020) For descriptions of more recentprogress on the classification of compact proper Dupin hypersurfaces, see [8,pp. 112–123] and [19, pp. 308–322]. For g ≤ 3, every compact, connectedproper Dupin hypersurface with g principal curvatures is equivalent by a Liesphere transformation to an isoparametric hypersurface in a sphere Sn. Thisis not true for g = 4 and g = 6, however, as shown by examples constructedby Pinkall and Thorbergsson [38] for g = 4, and by different examples con-structed by Miyaoka and Ozawa [29] for g = 4 and g = 6. These examplesdo not have constant Lie curvatures (cross-ratios of the principal curvatures,see Miyaoka [27]), so they can’t be Lie equivalent to an isoparametric hy-persurface, which must have constant Lie curvatures. The classification ofcompact proper Dupin hypersurfaces in the cases g = 4 and g = 6 is still anopen problem.

3

In this paper, we study Dupin hypersurfaces in the setting of Lie spheregeometry using local techniques. In Section 2, we give a brief introductionto Lie sphere geometry. In Section 3, we introduce the basic differentialgeometric notions: the Legendre map and the Dupin property. The case ofE3 is handled in Section 4, where we handle the case of g = 2 distinct focalpoints for En. This was first done for n > 3 by Pinkall [37]. Our maincontribution lies in Section 5, where we treat the case E4 by the method ofmoving frames. This case was also studied by Pinkall [36], but our treatmentseems to be more direct and differs from his in several points. It is our hopethat this method will provide a framework and give some direction for thestudy of Dupin hypersurfaces in En for n > 4.

2 Lie Sphere Geometry

We first present a brief outline of the main ideas in Lie’s geometry of spheresin Rn. This is given in more detail in Lie’s original treatment [25], in thebook of Blaschke [2], and in our paper [10] (see also the book [8]).

The basic construction in Lie sphere geometry associates each orientedsphere, oriented plane and point sphere in Rn ∪ {∞} = Sn with a point onthe quadric Qn+1 in projective space P n+2 given in homogeneous coordinates(x1, . . . , xn+3) by the equation

〈x, x〉 = −x21 + x22 + · · ·+ x2n+2 − x2n+3 = 0. (1)

We will denote real (n+ 3)-space endowed with the metric (1) by Rn+3

2 .We can designate the orientation of a sphere in Rn by assigning a plus

or a minus sign to its radius. Positive radius corresponds to the orientationdetermined by the field of inward normals to the sphere, while a negativeradius corresponds to the orientation determined by the outward normal.(See Remark 2.1 below.) A plane in Rn is a sphere which goes through thepoint ∞. The orientation of the plane can be associated with a choice ofunit normal N . The specific correspondence between the points of Qn+1 andthe set of oriented spheres, oriented planes and points in Rn ∪ {∞} is thengiven as follows:

4

Euclidean Lie

points: u ∈ Rn[(

1+u·u2, 1−u·u

2, u, 0

)]

∞ [(1,−1, 0, 0)]

spheres: center p, signed radius r[(

1+p·p−r2

2, 1−p·p+r2

2, p, r

)]

planes: u ·N = h, unit normal N [(h,−h,N, 1)]

(2)

Here the square brackets denote the point in projective space P n+2 given bythe homogeneous coordinates in the round brackets, and u ·u is the standardEuclidean dot product in Rn.

From (2), we see that the point spheres correspond to the points in theintersection of Qn+1 with the hyperplane in P n+2 given by the equationxn+3 = 0. The manifold of point spheres is called Mobius space.

A fundamental notion in Lie sphere geometry is that of oriented contactof spheres. Two oriented spheres S1 and S2 are in oriented contact if they aretangent and their orientations agree at the point of tangency. If p1 and p2 arethe respective centers of S1 and S2, and r1 and r2 are the respective signedradii, then the condition of oriented contact can be expressed analytically by

|p1 − p2| = |r1 − r2|. (3)

If S1 and S2 are represented by [k1] and [k2] as in (2), then (3) is equivalentto the condition

〈k1, k2〉 = 0. (4)

In the case where S1 and/or S2 is a plane or a point in Rn, orientedcontact has the logical meaning. That is, a sphere S and a plane π are inoriented contact if π is tangent to S and their orientations agree at the pointof contact. Two oriented planes are in oriented contact if their unit normalsare the same. They are in oriented contact at the point ∞. A point sphereis in oriented contact with a sphere or plane S if it lies on S, and ∞ is inoriented contact with each plane. In each case, the analytic condition fororiented contact is equivalent to (4) when the two “spheres” in question are

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represented as in (2).

Remark 2.1. (added by T. Cecil in 2020) In the case of a sphere [k1] anda plane [k2] as in (2), the equation (4) is equivalent to p · N = h + r. Inorder to make this correspond to the geometric definition of oriented contactstated in the paragraph above, one must adopt the convention that the in-ward normal orientation of the sphere corresponds to a positive signed radius.

Because of the signature of the metric (1), the quadric Qn+1 containslines in P n+2 but no linear subspaces of higher dimension. A line in Qn+1

is determined by two points [x], [y] in Qn+1 satisfying 〈x, y〉 = 0. The lineson Qn+1 form a manifold of dimension 2n − 1, to be denoted by Λ2n−1. InRn, a line on Qn+1 corresponds to a 1-parameter family of oriented spheressuch that any two of the oriented spheres are in oriented contact, i.e., all theoriented spheres tangent to an oriented plane at a given point, i.e., an orientedcontact element. Of course, a contact element can also be represented by anelement of T1S

n, the bundle of unit tangent vectors to the Euclidean sphereSn in En+1 with its usual metric. This is the starting for Pinkall’s [37]considerations of Lie geometry.

A Lie sphere transformation is a projective transformation of P n+2 whichtakes Qn+1 to itself. Since a projective transformation takes lines to lines, aLie sphere transformation preserves oriented contact of spheres. The groupG of Lie sphere transformations is isomorphic to O(n + 1, 2)/{±I}, whereO(n+ 1, 2) is the group of orthogonal transformations for the inner product(1). The group of Mobius transformations is isomorphic to O(n+1, 1)/{±I}.

3 Legendre Submanifolds

Here we recall the concept of a Legendre submanifold of the contact manifoldΛ2n−1(= Λ) using the notation of [10]. In this section, the ranges of theindices are as follows:

1 ≤ A,B,C ≤ n+ 3, 3 ≤ i, j, k ≤ n + 1. (5)

Instead of using an orthornormal frame for the metric 〈 , 〉 defined by (1),it is useful to consider a Lie frame, that is an ordered set of vectors YA inRn+3

2 satisfying〈YA, YB〉 = gAB, (6)

6

with

[gAB] =

J 0 00 In−1 00 0 J

, (7)

where In−1 is the (n− 1)× (n− 1) identity matrix and

J =

[

0 11 0

]

. (8)

The space of all Lie frames can be identified with the orthogonal group O(n+1, 2), of which the Lie sphere group, being isomorphic to O(n + 1, 2)/{±I},is a quotient group. In this space, we introduce the Maurer-Cartan forms

dYA =∑

ωBAYB. (9)

Through differentiation of (6), we show that the following matrix of 1-formsis skew-symmetric

[ωAB] =

ω21 ω1

1 ωi1 ωn+3

1 ωn+2

1

ω22 ω1

2 ωi2 ωn+3

2 ωn+2

2

ω2j ω1

j ωij ωn+3

j ωn+2

j

ω2n+2 ω1

n+2 ωin+2 ωn+3

n+2 ωn+2

n+2

ω2n+3 ω1

n+3 ωin+3 ωn+3

n+3 ωn+2

n+3

. (10)

Next by taking the exterior derivative of (9), we get the Maurer-Cartan

equations

dωBA =

∑

C

ωCA ∧ ωB

C . (11)

In [10], we then show that the form

ωn+2

1 = 〈dY1, Yn+3〉

gives a contact structure on the manifold Λ.Let Bn−1(= B) be an (n− 1)-dimensional smooth manifold. A Legendre

map is a smooth map λ : B → Λ which annihilates the contact form on Λ,i.e., λ∗ωn+2

1 = 0 on B. All of our calculations are local in nature. We use themethod of moving frames and consider a smooth family of Lie frames YA on

7

an open subset U of B, with the line λ(b) given by [Y1(b), Yn+3(b)] for eachb ∈ U . The Legendre map λ is called a Legendre submanifold if for a genericchoice of Y1 the forms ωi

1, 3 ≤ i ≤ n + 1, are linearly independent, i.e.,

∧ ωi1 6= 0 on U. (12)

Here and later we pull back the structure forms to Bn−1 and omit the symbolsof such pull-backs for simplicity. Note that the Legendre condition is just

ωn+2

1 = 0. (13)

We now assume that our choice of Y1 satisfies (12). By exterior differentiationof (13) and using (10), we get

∑

ωi1 ∧ ωi

n+3 = 0. (14)

Hence by Cartan’s Lemma and (12), we have

ωin+3 =

∑

hijωj1, with hij = hji. (15)

The quadratic differential form

II(Y1) =∑

i,j

hijωi1ω

j1, (16)

defined up to a non-zero factor and depending on the choice of Y1, is calledthe second fundamental form.

This form can be related to the well-known Euclidean second fundamentalform in the following way. Let en+3 be any unit timelike vector in Rn+3

2 . Foreach b ∈ U , let Y1(b) be the point of intersection of the line λ(b) with thehyperplane e⊥n+3. Y1 represents the locus of point spheres in the Mobius spaceQn+1 ∩ e⊥n+3, and we call Y1 the Mobius projection of λ determined by en+3.Let e1 and e2 be unit timelike, respectively spacelike, vectors orthogonal toen+3 and to each other, chosen so that Y1 is not the point at infinity [e1− e2]for any b ∈ U . We can represent Y1 by the vector

Y1 =1 + f · f

2e1 +

1− f · f2

e2 + f, (17)

as in (2), where f(b) lies in the space Rn of vectors orthogonal to e1, e2and en+3. We will call the map f : B → Rn the Euclidean projection of

8

λ determined by the ordered triple e1, e2, en+3. The regularity condition(12) is equivalent to the condition that f be an immersion on U into Rn.For each b ∈ U , let Yn+3(b) be the intersection of λ(b) with the orthogonalcomplement of the lightlike vector e1 − e2. Yn+3 is distinct from Y1 and thus〈Yn+3, en+3〉 6= 0. So we can represent Yn+3 by a vector of the form

Yn+3 = h(e1 − e2) + ξ + en+3, (18)

where ξ : U → Rn has unit length and h is a smooth function on U . Thus, ac-cording to (2), Yn+3(b) represents the plane in the pencil of oriented spheresin Rn corresponding to the line λ(b) on Qn+1. Note that the condition〈Y1, Yn+3〉 = 0 is equivalent to h = f · ξ, while the Legendre condition〈dY1, Yn+3〉 = 0 is the same as the Euclidean condition

ξ · df = 0. (19)

Thus, ξ is a field of unit normals to the immersion f on U . Since f is animmersion, we can choose the Lie frame vectors Y3, . . . , Yn+1 to satisfy

Yi = dY1(Xi) = (f · df(Xi))(e1 − e2) + df(Xi), 3 ≤ i ≤ n+ 1, (20)

for tangent vector fields X3, . . . , Xn+1 on U . Then, we have

ωi1(Xj) = 〈dY1(Xj), Yi〉 = 〈Yj, Yi〉 = δij . (21)

Now using (18) and (20), we compute

ωin+3(Xj) = 〈dYn+3(Xj), Yi〉 = dξ(Xj) · df(Xi) = −df(AXj) · df(Xi) = −Aij ,

(22)where A = [Aij ] is the Euclidean shape operator (second fundamental form)of the immersion f . But by (15) and (21), we have

ωin+3(Xj) =

∑

hikωk1(Xj) = hij .

Hence hij = −Aij , and [hij] is just the negative of the Euclidean shape oper-ator A of f .

Remark 3.1 The discussion above demonstrates how an immersion f :Bn−1 → Rn with field of unit normals ξ induces a Legendre submanifoldλ : Bn−1 → Λ defined by λ(b) = [Y1(b), Yn+3(b)], for Y1, Yn+3 as in (17) and

9

(18), called the Legendre lift of the immersion f with field of unit normalsξ. Further, an immersed submanifold f : Mk → Rn of codimension greaterthan one gives rise to a Legendre submanifold λ : Bn−1 → Λ, where Bn−1 isthe bundle of unit normals to f in Rn. As in the case of codimension one,λ(b) is defined to be the line on Qn+1 corresponding to the oriented contactelement determined by the unit vector b normal to f at the point x = π(b),where π is the bundle projection from Bn−1 to Mk. (In that case, λ is calledthe Legendre lift of the submanifold f .)

Remark 3.2. (added by T. Cecil in 2020) In a similar way, an immersionϕ : Bn−1 → Sn with field of unit normals η : Bn−1 → Sn in Sn inducesa Legendre submanifold µ : Bn−1 → Λ defined by µ(b) = [Z1(b), Zn+3(b)],where

Z1(b) = e1 + ϕ(b), Zn+3(b) = η(b) + en+3, (23)

and Sn is identified with the unit sphere in the space Rn+1 of vectors orthog-onal to e1 and en+3. The map µ is called the Legendre lift of the immersion ϕwith field of unit normals η. As in the Euclidean case, an immersed subman-ifold ϕ : Mk → Sn of codimension greater than one gives rise to a Legendresubmanifold µ : Bn−1 → Λ, where Bn−1 is the bundle of unit normals to ϕ inSn. As in the case of codimension one, µ(b) is defined to be the line on Qn+1

corresponding to the oriented contact element determined by the unit vectorb normal to ϕ at the point x = π(b), where π is the bundle projection fromBn−1 to Mk. In that case, µ is called the Legendre lift of the submanifold ϕ.

As one would expect, the eigenvalues of the second fundamental formhave geometric significance. Consider a curve γ(t) on B. The set of pointsin Qn+1 lying on the lines λ(γ(t)) forms a ruled surface in Qn+1. We look forthe conditions that this ruled surface be developable, i.e., consist of tangentlines to a curve in Qn+1. Let rY1+Yn+3 be the point of contact. We have by(9) and (10)

d(rY1 + Yn+3) ≡∑

i

(rωi1 + ωi

n+3)Yi,mod Y1, Yn+3. (24)

Thus, the lines λ(γ(t)) form a developable if and only if the tangent directionof γ(t) is a common solution to the equations

∑

j

(rδij + hij) ωj1 = 0, 3 ≤ i ≤ n+ 1. (25)

10

In particular, r must be a root of the equation

det(rδij + hij) = 0. (26)

By (15) the roots of (26) are all real. Denote them by r3, . . . , rn+1. Thepoints riY1 + Yn+3, 3 ≤ i ≤ n + 1, are called the focal points or curvaturespheres (Pinkall [37]) on λ(b). If Y1 and Yn+3 correspond to an immersionf : U → Rn as in (17) and (18), then these focal points on λ(b) correspondby (2) to oriented spheres in Rn tangent to f at f(b) and centered at theEuclidean focal points of f . These spheres are called the curvature spheresof f and the ri are just the principal curvatures of f , i.e., eigenvalues of theshape operator A.

If r is a root of multiplicity m, then the equations (25) define an m-dimensional subspace Tr of TbB, the tangent space to B at the point b.The space Tr is called a principal space of TbB, the latter being decomposedinto a direct sum of its principal spaces. Vectors in Tr are called principalvectors corresponding to the focal point rY1 + Yn+3. Of course, if Y1 andYn+3 correspond to an immersion f : U → Rn as in (17) and (18), thenthese principal vectors are the same as the Euclidean principal vectors for fcorresponding to the principal curvature r.

With a change of frame of the form

Y ∗

i =∑

i

cjiYj, 3 ≤ i ≤ n+ 1, (27)

where [cji ] is an (n− 1)× (n− 1) orthogonal matrix, we can diagonalize [hij ]so that in the new frame, equation (15) has the form

ωin+3 = −riωi

1, 3 ≤ i ≤ n+ 1. (28)

Note that none of the functions ri is ever infinity on U because of the as-sumption that (12) holds, i.e., Y1 is not a focal point. By associating Y1 toa Euclidean immersion f as in (17), we can apply results from Euclideangeometry to our situation. In particular, it follows from a result of Singley[39] on Euclidean shape operators that there is a dense open subset of B onwhich the number g(b) of distinct focal points on λ(b) is locally constant.We will work exclusively on open subsets U of B on which g is constant. Inthat case, each eigenvalue function r : U → R is smooth (see Nomizu [34]),and its corresponding principal distribution is a smooth m-dimensional foli-ation, where m is the multiplicity of r (see [18, p. 139]). Thus, on U we can

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find smooth vector fields X3, . . . , Xn+1 dual to smooth 1-forms ω31, . . . , ω

n+1

1 ,respectively, such that each Xi is principal for the smooth focal point mapriY1 + Yn+3 on U . If rY1 + Yn+3 is a smooth focal point map of multiplicitym on U , then we can assume that

r3 = · · · = rm+2 = r. (29)

By a different choice of the point at infinity, i.e., e1 and e2, if necessary, wecan also assume that the function r is never zero on U , i.e., Yn+3 is not afocal point on U .

We now want to consider a Lie frame Y ∗

A for which Y ∗

1 is a smooth focalpoint map of multiplicity m on U . Specifically, we make the change of frame

Y ∗

1 = rY1 + Yn+3

Y ∗

2 = (1/r)Y2

Y ∗

n+2 = Yn+2 − (1/r)Y2 (30)

Y ∗

n+3 = Yn+3

Y ∗

i = Yi, 3 ≤ i ≤ n+ 1.

We denote the Maurer-Cartan forms in this frame by θBA . Note that

dY ∗

1 = d(rY1 + Yn+3) = (dr)Y1 + rdY1 + dYn+3 =∑

θA1 Y∗

A. (31)

By examining the coefficient of Y ∗

i = Yi in (31) , we see from (28) that

θi1 = rωi1 + ωi

n+3 = (r − ri) ωi1, 3 ≤ i ≤ n + 1. (32)

From (29) and (32), we see that

θa1 = 0, 3 ≤ a ≤ m+ 2. (33)

This equation characterizes the condition that a focal point map Y ∗

1 haveconstant multiplicity m on U .

We now introduce the concept of a Dupin submanifold and then see whatfurther restrictions it allows us to place on the structure forms.

A connected submanifold N ⊂ B is called a curvature submanifold if itstangent space is everywhere a principal space. The Legendre submanifold iscalled Dupin if for every curvature submanifold N ⊂ B, the lines λ(b), b ∈ N ,all pass through a fixed point, i.e., each focal point map is constant along its

12

curvature submanifolds. This definition of “Dupin” is the same as that ofPinkall [37]. It is weaker than the definition of Dupin for Euclidean hyper-surfaces used by Thorbergsson [41], Miyaoka [26], Grove-Halperin [22] andCecil-Ryan [18], all of whom assumed that the number g of distinct curvaturespheres is constant on B. As we noted above, g is locally constant on a denseopen subset of any Legendre submanifold, but g is not necessarily constanton the whole of a Dupin submanifold, as the example of the Legendre sub-manifold induced from a tube B3 over a torus T 2 ⊂ R3 ⊂ R4 demonstrates(see Pinkall [36], Cecil-Ryan [18, p. 188]).

It is easy to see that the Dupin property is invariant under Lie spheretransformations as follows. Suppose that λ : B → Λ2n−1 is a Legendresubmanifold and that α is a Lie transformation. The map αλ : B → Λ2n−1 isalso a Legendre submanifold with αλ(b) = [αY1(b), αYn+3(b)]. Furthermore,if k = rY1 + Yn+3 is a curvature sphere of λ at a point b ∈ B, then since αis a linear transformation, αk is a curvature sphere of αλ at b with the sameprincipal space as k. Thus λ and αλ have the same curvature submanifoldson B, and the Dupin property clearly holds for λ if and only if it holds forαλ.

We now return to the calculation that led to equation (33). We have thatY ∗

1 = rY1 + Yn+3 is a smooth focal point map of multiplicity m on the openset U and its corresponding principal space is spanned by the vector fieldsX3, . . . , Xm+2. The Dupin condition that Y ∗

1 be constant along the leaves ofTr is simply

dY ∗

1 (Xa) ≡ 0, mod Y ∗

1 , 3 ≤ a ≤ m+ 2. (34)

From (21), (32) and (33), we have that

dY ∗

1 (Xa) = θ11(Xa)Y1 + θn+3

1 (Xa)Yn+3, 3 ≤ a ≤ m+ 2. (35)

Comparing (34) and (35), we see that

θn+3

1 (Xa) = 0, 3 ≤ a ≤ m+ 2. (36)

We now show that we can make one more change of frame and make θn+3

1 = 0.We can write the form θn+3

1 in terms of the basis ω31, . . . , ω

n+1

1 as

θn+3

1 =∑

aiωi1.

From (36), we see that we actually have

θn+3

1 =n+1∑

b=m+3

abωb1. (37)

13

Using (21), (32) and (37), we compute for all m+ 3 ≤ b ≤ n+ 1,

dY ∗

1 (Xb) = θ11(Xb)Y∗

1 + θb1(Xb)Yb + θn+3

1 (Xb)Yn+3

= θ11(Xb)Y∗

1 + (r − rb)Yb + abYn+3 (38)

= θ11(Xb)Y∗

1 + (r − rb)(Yb + (ab/(r − rb))Yn+3).

We now make the change of Lie frame,

Y 1 = Y ∗

1 , Y 2 = Y ∗

2 ,

Y a = Ya, 3 ≤ a ≤ m+ 2,

Y b = Yb + (ab/(r − rb))Yn+3, m+ 3 ≤ b ≤ n+ 1, (39)

Y n+2 = −n+1∑

b=m+3

(ab/(r − rb))Yb + Yn+2 −1

2

n+1∑

b=m+3

(ab/(r − rb))2Yn+3,

Y n+3 = Yn+3.

Let αBA be the Maurer–Cartan forms for this new frame. We still have

αa1 = 〈dY 1, Y a〉 = 〈dY ∗

1 , Ya〉 = θa1 = 0, 3 ≤ a ≤ m+ 2. (40)

Furthermore, since Y 1 = Y ∗

1 , the Dupin condition (34) still yields

αn+3

1 (Xa) = 0, 3 ≤ a ≤ m+ 2. (41)

Finally, for m+ 3 ≤ b ≤ n+ 1, we have from (38) and (39)

αn+3

1 (Xb) = 〈dY 1(Xb), Y n+2〉 = 〈θ11(Xb)Y 1 + (r − rb)Y b, Y n+2〉 = 0. (42)

From (41) and (42), we conclude that

αn+3

1 = 0. (43)

Thus, our main result of this section is that the assumption that the focalpoint map Y 1 = rY1+Yn+3 has constant multiplicity m and is constant alongthe leaves of its principal foliation Tr allows us to produce a Lie frame Y A

whose structure forms satisfy

αa1 = 0, 3 ≤ a ≤ m+ 2, (44)

αn+3

1 = 0.

14

4 Cyclides of Dupin

Dupin [20, p. 200] initiated the study of this subject in 1822 when he defineda cyclide to be a surface M2 in E3 which is the envelope of the family ofspheres tangent to three fixed spheres. This was shown to be equivalent re-quiring that both sheets of the focal set of M2 in E3 degenerate into curves.ThenM2 is the envelope of each of the two families of curvature spheres. Thekey step in the classical Euclidean proof (see, for example, Eisenhart [21, pp.312–314] or Cecil-Ryan [18, pp. 151–166]) is to show that the two focal curvesare a pair of so-called “focal conics” in E3, i.e., an ellipse and hyperbola inmutually orthogonal planes such that the vertices of the ellipse are the fociof the hyperbola and vice-versa, or a pair of parabolas in orthogonal planessuch that the vertex of each is the focus of the other. This classical proof islocal, i.e., one needs only a small piece of the surface to determine the focalconics and reconstruct the whole cyclide. Of course, envelopes of familiesof spheres can have singularities and some of the cyclides have one or twosingular points in E3. It turns out, however, that all of the different forms ofcyclides in Euclidean space induce Legendre submanifolds which are locallyLie equivalent. In other words, they are all various Euclidean projectionsof one Legendre submanifold. Pinkall [37] generalized this result to higherdimensional Dupin submanifolds. He defined a cyclide of characteristic (p, q)to be a Dupin submanifold with the property that at each point it has ex-actly two distinct focal points with respective multiplicities p and q. He thenproved the following.

Theorem 4.1. (Pinkall [37]): (a) Every connected cyclide of Dupin is con-tained in a unique compact and connected cyclide of Dupin.(b) Any two cyclides of the same characteristic are locally Lie equivalent,each being Lie equivalent to an open subset of a standard product of spheresin Sn.

In this section, we give a proof of Pinkall’s result using the method ofmoving frames. Let λ : B → Λ be the Dupin cyclide. The main step inthe proof of Theorem 4.1 is to show that the two focal point maps k1 andk2 from B to Qn+1 are such that the image k1(B) lies in the intersectionof Qn+1 with a (p + 1)-dimensional subspace E of P n+2 while k2(B) lies inthe intersection of the (q + 1)-dimensional subspace E⊥ with Qn+1. Thisgeneralizes the key step in the classical Euclidean proof that the two focal

15

curves are focal conics. Once this fact has been established for k1 and k2, itis relatively easy to complete the proof of the Theorem.

We begin the proof by taking advantage of the results of the previoussection. As we showed in (44), on any neighborhood U in B, we can find alocal Lie frame, which we now denote by YA, whose Maurer-Cartan forms,now denoted ωB

A , satisfy

ωa1 = 0, 3 ≤ a ≤ p+ 2, (45)

ωn+3

1 = 0.

In this frame, Y1 is a focal point map of multiplicity p from U to Qn+1. By thehypotheses of Theorem 4.1, there is one other focal point of multiplicity q =n−1−p at each point of B. By repeating the procedure used in constructingthe frame YA, we can construct a new frame Y A which has as Y n+3 the otherfocal point map sY1 + Yn+3, where s is a root of (26) of multiplicity q. Theprincipal space Ts is the span of the vectors Xp+3, . . . , Xn+1 in the notationof the previous section. The fact that Y n+3 is a focal point yields

ωbn+3 = 0, p+ 3 ≤ b ≤ n+ 1, (46)

in analogy to (33). The Dupin condition analogous to (34) is

dY n+3(Xb) ≡ 0, mod Y n+3, p+ 3 ≤ b ≤ n+ 1. (47)

This eventually leads toω1

n+3 = 0. (48)

One can check that this change of frame does not affect condition (45). Wenow drop the bars and call this last frame YA with Maurer-Cartan forms ωB

A

satisfying,

ωa1 = 0, 3 ≤ a ≤ p+ 2,

ωbn+3 = 0, p + 3 ≤ b ≤ n+ 1, (49)

ωn+3

1 = 0, ω1

n+3 = 0.

Furthermore, the following forms are easily shown to be a basis for the cotan-gent space at each point of U ,

{ω3

n+3, . . . , ωp+2

n+3, ωp+3

1 , . . . , ωn+1

1 }. (50)

16

We begin by taking the exterior derivative of the equations ωa1 = 0 and

ωbn+3 = 0 in (49). Using (10), (11) and (49), we obtain

0 = ωp+3

1 ∧ ωap+3 + · · ·+ ωn+1

1 ∧ ωan+1, 3 ≤ a ≤ p+ 2, (51)

0 = ω3

n+3 ∧ ωb3 + · · ·+ ωp+2

n+3 ∧ ωbp+2, p+ 3 ≤ b ≤ n+ 1. (52)

We now show that (51) and (52) imply that

ωab = 0, 3 ≤ a ≤ p+ 2, p+ 3 ≤ b ≤ n+ 1. (53)

To see this, note that since ωab = −ωb

a, each of the terms ωab occurs in exactly

one of the equations (51) and in exactly one of the equations (52). Equation(51) involves the basis forms ωp+3

1 , . . . , ωn+1

1 , while equation (52) involves thebasis forms ω3

n+3, . . . , ωp+2

n+3. We now show how to handle the form ω3p+3; the

others are treated in a similar fashion. The equations from (51) and (52),respectively, involving ω3

p+3 = −ωp+3

3 are

0 = ωp+3

1 ∧ ω3

p+3 + ωp+4

1 ∧ ω3

p+4 + · · ·+ ωn+1

1 ∧ ω3

n+1, (54)

0 = ω3

n+3 ∧ ωp+3

3 + ω4

n+3 ∧ ωp+3

4 + · · ·+ ωp+2

n+3 ∧ ωp+3

p+2. (55)

We take the wedge product of (54) with ωp+4

1 ∧ · · · ∧ ωn+1

1 and get

0 = ω3

p+3 ∧ (ωp+3

1 ∧ · · · ∧ ωn+1

1 ),

which implies that ωp+3 is in the span of ωp+3

1 , . . . , ωn+1

1 . On the other hand,taking the wedge product of (55) with ω4

n+3 ∧ · · · ∧ ωp+2

n+3 yields

0 = ω3

p+3 ∧ (ω3

n+3 ∧ · · · ∧ ωp+2

n+3),

and thus that ω3p+3 is in the span of ω3

n+3, . . . , ωp+2

n+3. We conclude that ω3p+3 =

0, as desired.We next differentiate ωn+3

1 = 0 and use (10), (13) and (49) to obtain

0 = dωn+3

1 = ωp+3

1 ∧ ωn+3

p+3 + · · ·+ ωn+1

1 ∧ ωn+3

n+1. (56)

This implies that

ωn+3

b ∈ Span {ωp+3

1 , . . . , ωn+1

1 }, p+ 3 ≤ b ≤ n + 1. (57)

17

Similarly, differentiation of the equation ω1n+3 = 0 yields

0 = ω3

n+3 ∧ ω1

3 + · · ·+ ωp+2

n+3 ∧ ω1

p+2, (58)

which implies that

ωa1 ∈ Span {ω3

n+3, . . . , ωp+2

n+3}, 3 ≤ a ≤ p + 2. (59)

We next differentiate equation (53). Using the skew-symmetry relations(10) and equations (49) and (53), we see that all terms drop out except thefollowing,

0 = dωab = ω2

b ∧ ωa2 + ωn+3

b ∧ ωan+3

= −(ω1

a ∧ ωb1) + ωn+3

b ∧ ωan+3.

Thus,

ω1

a ∧ ωb1 = ωn+3

b ∧ ωan+3, 3 ≤ a ≤ p+ 2, p+ 3 ≤ b ≤ n + 1. (60)

We now show that equation (60) implies that there is some function α on Usuch that

ω1

a = α ωan+3, 3 ≤ a ≤ p+ 2, (61)

ωn+3

b = −α ωb1, p+ 3 ≤ b ≤ n+ 1.

To see this, note that for any a, 3 ≤ a ≤ p+ 2, (59) gives

ω1

a = c3ω3

n+3 + · · ·+ cp+2ωp+2

n+3 for some c3, . . . , cp+2, (62)

while for any b, p+ 3 ≤ b ≤ n+ 1, (57) gives

ωn+3

b = dp+3ωp+3

1 + · · ·+ dn+1ωn+1

1 for some dp+3, . . . , dn+1. (63)

Thus,

ω1

a ∧ ωb1 = c3ω

3

n+3 ∧ ωb1 + · · ·+ caω

an+3 ∧ ωb

1 + · · ·+ cp+2ωp+2

n+3 ∧ ωb1, (64)

ωn+3

b ∧ωan+3 = dp+3ω

p+3

1 ∧ωan+3+· · ·+dbωb

1∧ωan+3+· · ·+dn+1ω

n+1

1 ∧ωan+3. (65)

From (60), we know that the right-hand sides of (64) and (65) are equal. Butthese expressions contain no common terms from the basis of 2-forms except

18

those involving ωan+3∧ωb

1. Thence, all of the coefficients except ca and db arezero, and we have

caωan+3 ∧ ωb

1 = dbωb1 ∧ ωa

n+3 = (−db)ωan+3 ∧ ωb

1.

Thus ca = −db. If we set αa = ca and µb = db, we have shown that (62) and(63) reduce to

ω1

a = αaωan+3, ωn+3

b = µbωb1, with µb = −αa.

This procedure works for any choice of a and b in the appropriate ranges. Byholding a fixed and varying b, we see that all of the quantities µb are equalto each other and to −αa. Similarly, all the quantities αa are the same, and(61) holds. We now consider the expression (9) for dYa, 3 ≤ a ≤ p + 2. Weomit the terms that vanish because of (10), (49) or (53),

dYa = ω1

aY1 + ω3

aY3 + · · ·+ ωp+2

a Yp+2 + ωn+2

a Yn+2 + ωn+3

a Yn+3.

Using (61) and the fact from (10) that ωn+2a = −ωa

n+3, this becomes

dYa = ωan+3(αY1 − Yn+2) + ω3

aY3 + · · ·+ ωp+2

a Yp+2 + ωn+3

a Yn+3. (66)

Similarly, for p+ 3 ≤ b ≤ n + 1, we find

dYb = ω1

bY1 + ω2

b (Y2 + αYn+3) + ωp+3

b Yp+3 + · · ·+ ωn+1

b Yn+1. (67)

We make the change of frame,

Y ∗

2 = Y2 + αYn+3, Y ∗

n+2 = Yn+2 − αY1, (68)

Y ∗

β = Yβ, β 6= 2, n+ 2.

We now drop the asterisks but use the new frame. From (66) and (67), wesee that in this new frame, we have

dYa = ωan+3(−Yn+2) + ω3

aY3 + · · ·+ ωp+2

a Yp+2 + ωn+3

a Yn+3, (69)

dYb = ω1

bY1 + ω2

bY2 + ωp+3

b Yp+3 + · · ·+ ωn+1

b Yn+1. (70)

That is, in the new frame, we have

ω1

a = 0, 3 ≤ a ≤ p+ 2, (71)

19

ωn+3

b = 0, p + 3 ≤ b ≤ n + 1. (72)

Our goal now is to show that the two spaces

E = Span {Y1, Y2, Yp+3, . . . , Yn+1} (73)

and its orthogonal complement,

E⊥ = Span {Y3, . . . , Yp+2, Yn+2, Yn+3} (74)

are invariant under d.Concerning E, we have that dYb ∈ E for p + 3 ≤ b ≤ n + 1 by (70).

Furthermore, (10), (13) and (49) imply that

dY1 = ω1

1Y1 + ωp+3

1 Yp+3 + · · ·+ ωn+1

1 Yn+1,

which is in E. Thus, it only remains to show that dY2 is in E. To do this, wedifferentiate (71). As before, we omit terms which are zero because of (10),(49), (53) and (71). We see that formula (11) for dω1

a reduces to

0 = dω1

a = ωn+2

a ∧ω1

n+2 = −ωan+3∧ω1

n+2 = ωan+3∧ωn+3

2 , 3 ≤ a ≤ p+2. (75)

Similarly, by differentiating equation (72), we find that

0 = dωn+3

b = ω2

b ∧ ωn+3

2 = −ωb1 ∧ ωn+3

2 , p+ 3 ≤ b ≤ n+ 1.

From this and (75), we see that the wedge product of ωn+3

2 with every formin the basis (50) is zero, and hence ωn+3

2 = 0. Using this and the fact thatωn+2

2 = −ω1n+3 = 0, and that by (10) and (71),

ωa2 = −ω1

a = 0, 3 ≤ a ≤ p+ 2,

we havedY2 = ω2

2Y2 + ωp+3

2 Yp+3 + · · ·+ ωn+1

2 Yn+1,

which is in E. So E is invariant under d and is thus a fixed subspace of P n+2,independent of the choice of point of U . Obviously then, the space E⊥ in(74) is also a fixed subspace of P n+2.

Note that E has signature (q+1, 1) as a vector subspace of Rn+3

2 , and E⊥

has signature (p + 1, 1). Take an orthornormal basis {e1, . . . , en+3} of Rn+3

2

with e1 and en+3 timelike and

E = Span {e1, . . . , eq+2}, E⊥ = Span {eq+3, . . . , en+3}.

20

Then E ∩ Qn+1 is given in homogeneous coordinates (x1, . . . , xn+3) with re-spect to this basis by

x21 = x22 + · · ·+ x2q+2, xq+3 = · · · = xn+3 = 0. (76)

This quadric is diffeomorphic to the unit sphere Sq in the span Eq+1 of thespacelike vectors e2, . . . , eq+2 with the diffeomorphism φ : Sq → E ∩ Qn+1

being given byφ(u) = [e1 + u], u ∈ Sq. (77)

Similarly, E⊥ ∩Qn+1 is the quadric given in homogeneous coordinates by

x2n+3 = x2q+3 + · · ·+ x2n+2, x1 = · · · = xq+2 = 0. (78)

E⊥ ∩ Qn+1 is diffeomorphic to the unit sphere Sp in the span Ep+1 ofeq+3, . . . , en+2 with the diffeomorphism ψ : Sp → E⊥ ∩ Qn+1 being givenby

ψ(v) = [v + en+3], v ∈ Sp. (79)

The focal point map Y1 of our Dupin submanifold is constant on the leavesof the principal foliation Tr, and so Y1 factors through an immersion of theq-dimensional space of leaves U/Tr into the q-sphere given by the quadric(76). Hence, the image of Y1 is an open subset of this quadric. Similarly,Yn+3 factors through an immersion of the p-dimensional space of leaves U/Tsonto an open subset of the p-sphere given by the quadric (78). From this itis clear that the unique compact cyclide containing λ : U → Λ as an opensubmanifold is given by the Dupin submanifold λ : Sq × Sp → Λ with

λ(u, v) = [k1(u, v), k2(u, v)], (u, v) ∈ Sq × Sp,

wherek1(u, v) = φ(u), k2(u, v) = ψ(v).

Geometrically, the image of λ consists of all lines joining a point on thequadric in equation (76) to a point on the quadric (78).

Thus any choice of (q + 1)-plane E in P n+2 with signature (q + 1, 1)and corresponding orthogonal complement E⊥ determines a unique compactcyclide of characteristic (p, q) and vice-versa. The local Lie equivalence ofany two cyclides of the same characteristic is then clear.

From the standpoint of Euclidean geometry, if we consider the pointspheres to be given by Qn+1 ∩ e⊥n+3, as in Section 2, then the Legendre sub-

manifold λ above is induced in the usual way from the unit normal bundle

21

Bn−1 = Sq×Sp of the standard embedding of Sq as a great q-sphere Eq+1∩Sn,where Eq+1 is the span of e2, . . . , eq+2 and S

n is the unit sphere in En+1, thespan of e2, . . . , en+2. The spheres Sq and Sp = Sn ∩Ep+1, where Ep+1 is thespan of eq+3, . . . , en+2, are the two focal submanifolds in Sn of a standardproduct of spheres Sp×Sq in Sn (see [18, pp. 295–296] or [19, pp. 110–111]).

5 Dupin Submanifolds for n = 4.

The classification of Dupin submanifolds induced from surfaces in R3 followsfrom the results of the last section. In his doctoral dissertation [35] (laterpublished as [36]), U. Pinkall obtained a local classification up to Lie equiv-alence of all Dupin submanifolds induced from hypersurfaces in R4. As weshall see, this is a far more complicated calculation than that of the previoussection, and as yet, no one has obtained a similar classification of Dupin hy-persurfaces in Rn for n ≥ 5. In this section, we will prove Pinkall’s theoremusing the method of moving frames.

Remark 5.1. (added by T. Cecil in 2020) See [19, pp. 301–308] for a surveyof local classifications of higher dimensional proper Dupin hypersurfaces thathave been found since the time that the original version of this paper waswritten.

We follow the notation used in Sections 3 and 4. We consider a Dupinsubmanifold

λ : B → Λ (80)

wheredimB = 3, dimΛ = 7, (81)

and the image λ(b), b ∈ B, is the line [Y1, Y7] of the Lie frame Y1, . . . , Y7. Weassume that there are three distinct focal points on each line λ(b).

By (44), we can choose the frame so that

ω31 = ω7

1 = 0, (82)

ω47 = ω1

7 = 0.

22

By making a change of frame of the form

Y ∗

1 = αY1, Y ∗

2 = (1/α)Y2, (83)

Y ∗

7 = βY7, Y ∗

6 = (1/β)Y6,

for suitable smooth functions α and β on B, we can arrange that Y1 + Y7represents the third focal point for each point of B. Then by using the factthat B is Dupin, we can use the method employed at the end of Section 3to make a change of frame leading to the following equations similar to (44)(and to (82)),

ω51 + ω5

7 = 0, (84)

ω11 − ω7

7 = 0.

This completely fixes the Yi, i = 3, 4, 5, and Y1, Y7 are determined up to atransformation of the form

Y ∗

1 = τY1, Y ∗

7 = τY7. (85)

Each of the three focal point maps Y1, Y7, Y1 + Y7 is constant along theleaves of its corresponding principal foliation. Thus, each focal point mapfactors through an immersion of the corresponding 2-dimensional space ofleaves of its principal foliation into Q5. (See Section 4 of Chapter 2 of thebook [18] or [19, pp. 18–32] for more detail on this point.) In terms of movingframes, this implies that the forms ω4

1, ω51, ω

37 are linearly independent on B,

i.e.,ω4

1 ∧ ω5

1 ∧ ω3

7 6= 0. (86)

This can also be seen by expressing the forms above in terms of a Lieframe {Y 1, . . . , Y n+3}, where Y 1 satisfies the regularity condition (12), andusing the fact that each focal point has multiplicity one. For simplicity, wewill also use the notation,

θ1 = ω4

1, θ2 = ω5

1, θ3 = ω3

7. (87)

Analytically, the Dupin conditions are three partial differential equations,and we are treating an over-determined system. The method of movingframes reduces the handling of its integrability conditions to a straightfor-ward algebraic problem, viz., that of repeated exterior differentiations.

23

We begin by computing the exterior derivatives of the three equationsω31 = 0, ω4

7 = 0, ω51 + ω5

7 = 0. Using the skew-symmetry relations (10), aswell as (82) and (84), the exterior derivatives of these three equations yieldthe system

0 = ω4

1 ∧ ω4

3 + ω5

1 ∧ ω5

3,

0 = ω5

1 ∧ ω5

4 + ω3

7 ∧ ω4

3,

0 = ω4

1 ∧ ω5

4 + ω3

7 ∧ ω5

3.

If we take the wedge product of the first of these with ω41, we conclude that

ω53 is in the span of ω4

1 and ω51. On the other hand, taking the wedge product

of of the third equation with ω41 yields that ω5

3 is in the span of ω41 and ω3

7.Consequently, ω5

3 = ρω41, for some smooth function ρ on B. Similarly, one

can show that there exist smooth functions σ and τ such that ω43 = σω5

1 andω54 = τω3

7 . Then, if we substitute these into the three equations above, weget that ρ = σ = τ , and hence we have,

ω5

3 = ρω4

1, ω4

3 = ρω5

1, ω5

4 = ρω3

7. (88)

Next we differentiate the equations ω71 = 0, ω1

7 = 0, ω11 − ω7

7 = 0. As above,use of the skew-symmetry relations (10) and the equations (82) and (84)yields the existence of smooth functions a, b, c, p, q, r, s, t, u on B such thatthe following relations hold:

ω7

4 = −ω4

6 = aω4

1 + bω5

1, (89)

ω7

5 = −ω5

6 = bω4

1 + cω5

1;

ω1

3 = −ω3

2 = pω3

7 − qω5

1, (90)

ω1

5 = −ω5

2 = qω3

7 − rω5

1;

ω1

4 = −ω4

2 = bω5

1 + sω4

1 + tω3

7, (91)

ω3

6 = −ω7

3 = qω5

1 + tω4

1 + uω3

7.

We next see what can be deduced from taking the exterior derivatives ofthe equations (88)–(91). First, we take the exterior derivatives of the threebasis forms ω4

1, ω51, ω

37. For example, using the relations that we have derived

so far, we have from the Maurer-Cartan equation (11),

dω4

1 = ω1

1 ∧ ω4

1 + ω5

1 ∧ ω4

5 = ω1

1 ∧ ω4

1 − ρω5

1 ∧ ω3

7.

24

We obtain similar equations for dω51 and dω3

7. When we use the forms θ1, θ2,θ3 defined in (87) for ω4

1, ω51, ω

37, we have

dθ1 = ω1

1 ∧ θ1 − ρ θ2 ∧ θ3,dθ2 = ω1

1 ∧ θ2 − ρ θ3 ∧ θ1, (92)

dθ3 = ω1

1 ∧ θ3 − ρ θ1 ∧ θ2.

We next differentiate (88). We have ω43 = ρω5

1. On the one hand,

dω4

3 = ρ dω5

1 + dρ ∧ ω5

1.

Using the second equation in (92) with ω51 = θ2, this becomes

dω4

3 = ρω1

1 ∧ ω5

1 − ρ2ω3

7 ∧ ω4

1 + dρ ∧ ω5

1.

On the other hand, we can compute dω43 from the Maurer-Cartan equation

(11) and use the relationships that we have derived to find

dω4

3 = (−p− ρ2 − a)(ω4

1 ∧ ω3

7)− qω5

1 ∧ ω4

1 + bω3

7 ∧ ω5

1.

If we equate these two expressions for dω43, we get

(−p− a− 2ρ2) ω4

1 ∧ ω3

7 = (dρ+ ρω1

1 − qω4

1 − bω3

7) ∧ ω5

1. (93)

Because of the independence of ω41, ω

51 and ω3

7, both sides of the equationabove must vanish. Thus, we conclude that

2ρ2 = −a− p, (94)

and that dρ+ ρω11 − qω4

1 − bω37 is a multiple of ω5

1. Similarly, differentiationof the equation ω5

4 = ρω37 yields the following analogue of (93),

(s− a− r + 2ρ2) ω4

1 ∧ ω5

1 = (dρ+ ρω1

1 + tω5

1 − qω4

1) ∧ ω3

7, (95)

and differentiation of ω53 = ρω4

1 yields

(c+ p + u− 2ρ2) ω5

1 ∧ ω3

7 = (−dρ− ρω1

1 − tω5

1 + bω3

7) ∧ ω4

1. (96)

In each of the equations (93), (95), (96), both sides of the equation mustvanish. From the vanishing of the left sides of the equations, we get thefundamental relationship,

2ρ2 = −a− p = a + r − s = c + p+ u. (97)

25

Furthermore, from the vanishing of the right-hand sides of the three equations(93), (95) and (96), we can determine after some algebra that

dρ+ ρω1

1 = qω4

1 − tω5

1 + bω3

7. (98)

The last equation shows the importance of ρ. Following the notation intro-duced in equation (87), we write equation (98) as,

dρ+ ρω1

1 = ρ1θ1 + ρ2θ2 + ρ3θ3, (99)

whereρ1 = q, ρ2 = −t, ρ3 = b, (100)

are the “covariant derivatives” of ρ.Using the Maurer-Cartan equations, we can compute

dω1

1 = ω4

1 ∧ ω1

4 + ω5

1 ∧ ω1

5

= ω4

1 ∧ (bω5

1 + tω3

7) + ω5

1 ∧ (qω3

7 − rω5

1)

= bω4

1 ∧ ω5

1 + qω5

1 ∧ ω3

7 − tω3

7 ∧ ω4

1.

Using (87) and (100), this can be rewritten as

dω1

1 = ρ3 θ1 ∧ θ2 + ρ1 θ2 ∧ θ3 + ρ2 θ3 ∧ θ1. (101)

The trick now is to express everything in terms of ρ and its successive co-variant derivatives.

We first derive a general form for these covariant derivatives. Supposethat σ is a smooth function which satisfies a relation of the form

dσ +mσω1

1 = σ1 θ1 + σ2 θ2 + σ3 θ3, (102)

for some integer m. (Note that (98) is such a relationship for the function ρwith m = 1.) By taking the exterior derivative of (102) and using (92) and(101) to express both sides in terms of the standard basis of 2-forms θ1 ∧ θ2,θ2 ∧ θ3 and θ3 ∧ θ1, one finds that the functions σ1, σ2, σ3 satisfy equations ofthe form

dσα + (m+ 1)σαω1

1 = σα1 θ1 + σα2 θ2 + σα3 θ3, α = 1, 2, 3, (103)

where the coefficient functions σαβ satisfy the commutation relations,

σ12 − σ21 = −mσρ3 − ρσ3,

σ23 − σ32 = −mσρ1 − ρσ1, (104)

σ31 − σ13 = −mσρ2 − ρσ2.

26

In particular, from equation (99), we have the following commutation rela-tions on ρ1, ρ2, ρ3:

ρ12 − ρ21 = −2ρρ3

ρ23 − ρ32 = −2ρρ1 (105)

ρ31 − ρ13 = −2ρρ2.

We next take the exterior derivative of equations (89)–(91). We first differ-entiate the equation

ω7

4 = aω4

1 + bω5

1. (106)

On the one hand, from the Maurer-Cartan equation (11) for dω74, we have

(by not writing those terms which have already been shown to vanish),

dω7

4 = ω2

4 ∧ ω7

2 + ω3

4 ∧ ω7

3 + ω5

4 ∧ ω7

5 + ω7

4 ∧ ω7

7 (107)

= −ω4

1 ∧ ω7

2 + (−ρω5

1) ∧ (−qω5

1 − tω4

1 − uω3

7)

+ ρω3

7 ∧ (bω4

1 + cω5

1) + (aω4

1 + bω5

1) ∧ ω1

1.

On the other hand, differentiation of the right-hand side of equation (106)yields

dω7

4 = da ∧ ω4

1 + a dω4

1 + db ∧ ω5

1 + b dω5

1 (108)

= da ∧ ω4

1 + a(ω1

1 ∧ ω4

1 − ρω5

1 ∧ ω3

7)

+ db ∧ ω5

1 + b(ω1

1 ∧ ω5

1 − ρω4

1 ∧ ω3

7).

Equating (107) and (108) yields

(da + 2aω1

1 − 2bρω3

7 − ω7

2) ∧ ω4

1 (109)

+ (db+ 2bω1

1 + (a+ u− c)ρω3

7) ∧ ω5

1 + ρtω4

1 ∧ ω5

1 = 0.

Since b = ρ3, it follows from (98) and (103) that

db+ 2bω1

1 = dρ3 + 2ρ3ω1

1 = ρ31 θ1 + ρ32 θ2 + ρ33 θ3. (110)

By examining the coefficient of ω51 ∧ω3

7 = θ2 ∧ θ3 in equation (109) and using(110), we get that

ρ33 = ρ(c− a− u). (111)

Furthermore, the remaining terms in (109) are

(da+ 2aω1

1 − ω7

2 − 2ρbω3

7 − (ρt+ ρ31)ω5

1) ∧ ω4

1 (112)

+ terms involving ω5

1 and ω3

7 only.

27

Thus, the coefficient in parentheses must be a multiple of ω41, call it aω

41. We

can write this using (87) and (100) as

da+ 2aω1

1 = ω7

2 + aθ1 + (ρ31 − ρρ2)θ2 + 2ρρ3θ3. (113)

In a similar manner, if we differentiate

ω7

5 = bω4

1 + cω5

1,

we obtain,

dc+ 2cω1

1 = ω7

2 + (ρ32 + ρρ1)θ1 + cθ2 − 2ρρ3θ3. (114)

Thus, from the two equations in (89), we have obtained equations (111),(113) and (114). In completely analogous fashion, we can differentiate thetwo equations in (90) to obtain

ρ11 = ρ(s + r − p), (115)

dp+ 2pω1

1 = −ω7

2 + 2ρρ1θ1 + (−ρ13 − ρρ2)θ2 + pθ3, (116)

dr + 2rω1

1 = −ω7

2 − 2ρρ1θ1 + rθ2 + (−ρ12 + ρρ3)θ3, (117)

while differentiation of (91) yields

ρ22 + ρ33 = ρ(p− r − s), (118)

ds+ 2sω1

1 = sθ1 + (ρ31 + ρρ2)θ2 + (−ρ21 + ρρ3)θ3, (119)

du+ 2uω1

1 = (−ρ23 − ρρ1)θ1 + (ρ13 − ρρ2)θ2 + uθ3. (120)

In these equations, the coefficients a, c, p, r, s, u remain undetermined.However, by differentiating equation (97) and using the appropriate equationsamong those involving these quantities above, one can show that

a = −6ρρ1, c = 6ρρ2,

p = −6ρρ3, r = 6ρρ2, (121)

s = −12ρρ1, u = 12ρρ3.

From equations (111), (115), (118) and (97), we easily compute that

ρ11 + ρ22 + ρ33 = 0. (122)

28

Using (121), equations (119) and (120) can be rewritten as

ds+ 2sω1

1 = −12ρρ1θ1 + (ρ31 + ρρ2)θ2 + (−ρ21 + ρρ3)θ3, (123)

du+ 2uω1

1 = (−ρ23 − ρρ1)θ1 + (ρ13 − ρρ2)θ2 + 12ρρ3θ3. (124)

By taking the exterior derivatives of these two equations and making use ofequation (122) and of the commutation relations (104) for ρ and its variousderivatives, one can ultimately show after a lengthy calculation that thefollowing fundamental equations hold:

ρρ12 + ρ1ρ2 + ρ2ρ3 = 0,

ρρ21 + ρ1ρ2 − ρ2ρ3 = 0,

ρρ23 + ρ2ρ3 + ρ2ρ1 = 0, (125)

ρρ32 + ρ2ρ3 − ρ2ρ1 = 0,

ρρ31 + ρ3ρ1 + ρ2ρ2 = 0,

ρρ13 + ρ3ρ1 − ρ2ρ2 = 0.

We now briefly outline the details of this calculation. By (123), we have

s1 = −12ρρ1, s2 = ρ31 + ρρ2, s3 = ρρ3 − ρ21. (126)

The commutation relation (104) for s with m = 2 gives

s12 − s21 = −2sρ3 − ρs3 = −2sρ3 − ρ(ρρ3 − ρ21). (127)

On the other hand, we can directly compute by taking covariant derivativesof (126) that

s12 − s21 = −12ρρ12 − 12ρ2ρ1 − (ρ311 + ρ1ρ2 + ρρ21). (128)

The main problem now is to get ρ311 into a usable form. By taking thecovariant derivative of the third equation in (105), we find

ρ311 − ρ131 = −2ρ1ρ2 − 2ρρ21. (129)

Then using the commutation relation,

ρ131 = ρ113 − 2ρ1ρ2 − ρρ12,

29

we get from (129) that

ρ311 = ρ113 − 4ρ1ρ2 − ρρ12 − 2ρρ21. (130)

Taking the covariant derivative of ρ11 = ρ(s + r − p) and substituting theexpression obtained for ρ113 into (130), we get

ρ311 = ρ3(s+ r − p)− 3ρρ21 − 2ρρ12 + 8ρ2ρ3 − 4ρ1ρ2. (131)

If we substitute (131) for ρ311 in (128) and then equate the right-hand sidesof (127) and (128), we obtain the first equation in (125). The cyclic permu-tations are obtained in a similar way from s23 − s32, etc.

Our frame attached to the line [Y1, Y7] is still not completely determined,viz., the following change is allowable:

Y ∗

2 = α−1Y2 + µY7, Y ∗

6 = α−1Y6 − µY1. (132)

The Yi’s, i = 3, 4, 5 being completely determined , we have under this change,

ω4∗

1 = αω4

1, ω5∗

1 = αω5

1, ω3∗

7 = αω3

7,

ω7∗

4 = α−1ω7

4 + µω4

1,

ω1∗

3 = α−1ω1

3 − µω3

7,

which implies that

a∗ = α−2a+ α−1µ, p∗ = α−2p− α−1µ.

We choose µ to make a∗ = p∗. After dropping the asterisks, we have from(97) that

a = p = −ρ2, r = 3ρ2 + s, c = 3ρ2 − u. (133)

Now using the fact that a = p, we can subtract (116) from (113) and getthat

ω7

2 = 4ρρ1θ1 − ((ρ31 + ρ13)/2)θ2 − 4ρρ3θ3. (134)

We are finally in position to proceed toward the main results. Ultimately,we show that the frame can be chosen so that the function ρ is constant, andthe classification naturally splits into two cases ρ = 0 and ρ 6= 0.

The case ρ 6= 0.

30

We now assume that the function ρ is never zero on B. The following lemmais the key in this case. This is Pinkall’s Lemma [36, p. 108], where hisfunction c is the negative of our function ρ. Since ρ 6= 0, the fundamentalequations (125) allow one to express all of the second covariant derivativesραβ in terms of ρ and its first derivatives ρα. This enables us to give a some-what simpler proof than Pinkall gave for the lemma.

Lemma 5.1. Suppose that ρ never vanishes on B. Then ρ1 = ρ2 = ρ3 = 0at every point of B.

Proof: First note that if the function ρ3 vanishes identically, then (125) andthe assumption that ρ 6= 0 imply that ρ1 and ρ2 also vanish identically. Wenow complete the proof by showing that ρ3 must vanish everywhere. This isaccomplished by considering the expression s12 − s21. By the commutationrelations (104), we have

s12 − s21 = −2sρ3 − ρs3.

By (125) and (126), we see that

ρs3 = ρ2ρ3 − ρρ21 = ρ1ρ2,

and sos12 − s21 = −2sρ3 − ρ1ρ2. (135)

On the other hand, we can compute s12 directly from the equation s1 =−12ρρ1. Then using the expression for ρ12 obtained from (125), we get

s12 = −12ρ2ρ1 − 12ρρ12 = −12(ρ2ρ1 + ρρ12) (136)

= −12(ρ2ρ1 + (−ρ2ρ1 − ρ2ρ3)) = 12ρ2ρ3.

Next we have from (126),s2 = ρ31 + ρρ2.

Using (125), we can write

ρ31 = −ρ3ρ1ρ−1 − ρρ2,

and thus,s2 = −ρ3ρ1/ρ. (137)

31

Then, we compute

s21 = −(ρ(ρ3ρ11 + ρ31ρ1)− ρ3ρ2

1)/ρ2.

Using (115) for ρ11 and (125) to get ρ31, this becomes

s21 = −ρ3(s+ r − p) + 2ρ3ρ2

1ρ−2 + ρ1ρ2. (138)

Now equate the expression in (135) for s12 − s21 with that obtained by sub-tracting equation (138) from equation (136) to get

−2sρ3 − ρ1ρ2 = 12ρ2ρ3 + ρ3(s+ r − p)− 2ρ3ρ2

1ρ−2 − ρ1ρ2.

This can be rewritten as

0 = ρ3(12ρ2 + 3s+ r − p− 2ρ21ρ

−2). (139)

Using the expressions in (133) for r and p, we see that

3s+ r − p = 4s+ 4ρ2,

and so (139) can be written as

0 = ρ3(16ρ2 + 4s− 2ρ21ρ

−2). (140)

Suppose that ρ3 6= 0 at some point b ∈ B. Then ρ3 does not vanish on someneighborhood U of b. By (140), we have

16ρ2 + 4s− 2ρ21ρ−2 = 0 (141)

on U . We now take the θ2-covariant derivative of (141) and obtain

32ρρ2 + 4s2 − 4ρ1ρ12ρ−2 + 4ρ21ρ2ρ

−3 = 0. (142)

We now substitute the expression (137) for s2 and the formula

ρ12 = −ρ1ρ2ρ−1 − ρρ3

obtained from (125) into (142). After some algebra, (142) reduces to

ρ2(32ρ4 + 8ρ21) = 0.

32

Since ρ 6= 0, this implies that ρ2 = 0 on U . But then the left side of theequation (125)

ρρ21 + ρ1ρ2 = ρ2ρ3,

must vanish on U . Since ρ 6= 0, we conclude that ρ3 = 0 on U , a contradictionto our assumption. Hence, ρ3 vanishes identically on B, and the lemma isproven.

We now continue with the case ρ 6= 0. According to Lemma 5.1, allthe covariant derivatives of ρ are zero, and our formulas simplify greatly.Equations (111) and (115) give

c− a− u = 0, s+ r − p = 0.

These combined with (133) give

c = r = ρ2, u = −s = 2ρ2. (143)

By (134) we have ω72 = 0. So the differentials of the frame vectors can now

be written

dY1 − ω1

1Y1 = ω4

1Y4 + ω5

1Y5,

dY7 − ω1

1Y7 = ω3

7Y3 − ω5

1Y5,

dY2 + ω1

1Y2 = ρ2(ω3

7Y3 + 2ω4

1Y4 + ω5

1Y5),

dY6 + ω1

1Y6 = ρ2(2ω3

7Y3 + ω4

1Y4 − ω5

1Y5), (144)

dY3 = ω3

7Z3 + ρ(ω5

1Y4 + ω4

1Y5),

dY4 = −ω4

1Z4 + ρ(−ω5

1Y3 + ω3

7Y5),

dY5 = ω5

1Z5 + ρ(−ω4

1Y3 − ω3

7Y4),

where

Z3 = −Y6 + ρ2(−Y1 − 2Y7),

Z4 = Y2 + ρ2(2Y1 + Y7), (145)

Z5 = −Y2 + Y6 + ρ2(−Y1 + Y7).

From this, we notice that

Z3 + Z4 + Z5 = 0, (146)

so that the points Z3, Z4 and Z5 lie on a line.

33

From (99) and (101) and the lemma, we see that

dρ+ ρω1

1 = 0, dω1

1 = 0. (147)

We now make a change of frame of the form

Y ∗

1 = ρY1, Y ∗

2 = (1/ρ)Y2,

Y ∗

7 = ρY7, Y ∗

6 = (1/ρ)Y6, (148)

Y ∗

i = Yi, 3 ≤ i ≤ 5.

Then set

Z∗

i = (1/ρ)Zi, 3 ≤ i ≤ 5, (149)

ω4∗

1 = ρω4

1, ω5∗

1 = ρω5

1, ω3∗

7 = ρω3

7.

The effect of this change is to make ρ∗ = 1 and ω1∗1 = 0, for we can compute

the following:

dY ∗

1 = ω4∗

1 Y4 + ω5∗

1 Y5,

dY ∗

7 = ω3∗

7 Y3 − ω5∗

1 Y5,

dY ∗

2 = ω3∗

7 Y3 + 2ω4∗

1 Y4 + ω5∗

1 Y5,

dY ∗

6 = 2ω3∗

7 Y3 + ω4∗

1 Y4 − ω5∗

1 Y5, (150)

dY3 = ω3∗

7 Z∗

3 + ω5∗

1 Y4 + ω4∗

1 Y5,

dY4 = −ω4∗

1 Z∗

4 − ω5∗

1 Y3 + ω3∗

7 Y5,

dY5 = ω5∗

1 Z∗

5 − ω4∗

1 Y3 − ω3∗

7 Y4,

with

dZ∗

3 = 2(−2ω3∗

7 Y3 − ω4∗

1 Y4 + ω5∗

1 Y5),

dZ∗

4 = 2(ω3∗

7 Y3 + 2ω4∗

1 Y4 + ω5∗

1 Y5), (151)

dZ∗

5 = 2(ω3∗

7 Y3 − ω4∗

1 Y4 − 2ω5∗

1 Y5),

and

dω4∗

1 = −ω5∗

1 ∧ ω3∗

7 , i.e., dθ∗1 = −θ∗2 ∧ θ∗3,dω5∗

1 = −ω3∗

7 ∧ ω4∗

1 , i.e., dθ∗2 = −θ∗3 ∧ θ∗1,dω3∗

7 = −ω4∗

1 ∧ ω5∗

1 , i.e., dθ∗3 = −θ∗1 ∧ θ∗2.

34

Comparing the last equation with (92), we see that ω1∗1 = 0 and ρ∗ = 1.

This is the final frame which we will need in this case ρ 6= 0. So, we againdrop the asterisks.

We are now ready to prove Pinkall’s classification result for the case ρ 6= 0[36, p. 117]. As with the cyclides, there is only one compact model, up toLie equivalence. This is Cartan’s isoparametric hypersurface M3 in S4 (see[3]–[6]). It is a tube of constant radius over each of its two focal submanifolds,which are standard Veronese surfaces in S4. (See [18, pp. 296–299], [19, pp.151–155 ], [42] for more detail.) We will describe the Veronese surface afterstating the theorem.

Theorem 5.2: (Pinkall [36]) (a) Every connected Dupin submanifold withρ 6= 0 is contained in a unique compact connected Dupin submanifold withρ 6= 0.(b) Any two Dupin submanifolds with ρ 6= 0 are locally Lie equivalent, eachbeing Lie equivalent to an open subset of Cartan’s isoparametric hypersurfacein S4.

Our method of proof differs from that of Pinkall in that we will provedirectly that each of the focal submanifolds can naturally be considered tobe an open subset of a Veronese surface in a hyperplane P 5 ⊂ P 6. The Dupinsubmanifold can then be constructed from these focal submanifolds.

We now recall the definition of a Veronese surface. First consider the mapfrom the unit sphere y21 + y22 + y23 = 1 in R3 into R5 given by

(x1, . . . , x5) = (2y2y3, 2y3y1, 2y1y2, y2

1, y2

2). (152)

This map takes the same value on antipodal points of the 2-sphere, so it in-duces a map φ : P 2 → R5. One can show by an elementary direct calculationthat φ is an embedding of P 2 and that φ is substantial in R5, i.e., does notlie in any hyperplane. Any embedding of P 2 into P 5 which is projectivelyequivalent to φ is called a Veronese surface. (See Lane [23, pp. 424–430] formore detail.)

Let k1 = Y1, k2 = Y7, k3 = Y1 + Y7 be the focal point maps of the Dupinsubmanifold λ : B → Λ with ρ 6= 0. Each ki is constant along the leaves ofits principal foliation Ti, so each ki factors through an immersion φi of the2-dimensional space of leaves B/Ti into P

6. We will show that each of theseφi is an open subset of a Veronese surface in some P 5 ⊂ P 6.

35

We wish to integrate the differential system (150), which is completelyintegrable. For this purpose we drop the asterisks and write the system asfollows:

dY1 = θ1Y4 + θ2Y5,

dY7 = θ3Y3 − θ2Y5,

dY2 = θ3Y3 + 2θ1Y4 + θ2Y5,

dY6 = 2θ3Y3 + θ1Y4 − θ2Y5, (153)

dY3 = θ3Z3 + θ2Y4 + θ1Y5,

dY4 = −θ1Z4 − θ2Y3 + θ3Y5,

dY5 = θ2Z5 − θ1Y3 − θ3Y4,

with

dZ3 = 2(−2θ3Y3 − θ1Y4 + θ2Y5),

dZ4 = 2(θ3Y3 + 2θ1Y4 + θ2Y5), (154)

dZ5 = 2(θ3Y3 − θ1Y4 − 2θ2Y5),

where

dθ1 = −θ2 ∧ θ3,dθ2 = −θ3 ∧ θ1, (155)

dθ3 = −θ1 ∧ θ2.

and

Z3 = −Y1 − Y6 − 2Y7,

Z4 = 2Y1 + Y2 + Y7, (156)

Z5 = −Y1 − Y2 + Y6 + Y7,

so thatZ3 + Z4 + Z5 = 0. (157)

PutW1 = −Y1 + Y6 − 2Y7, W2 = −2Y1 + Y2 − Y7. (158)

We find from (153) thatdW1 = dW2 = 0, (159)

36

so that the points W1 and W2 are fixed. Their inner products are

〈W1,W1〉 = 〈W2,W2〉 = −4, 〈W1,W2〉 = −2, (160)

and the line [W1,W2] consists entirely of timelike points. Its orthogonalcomplement in R7

2 is spanned by Y3, Y4, Y5, Z4, Z5. It consists entirely ofspacelike points and has no point in common with Q5. We will denote it asR5.

It suffices to solve the system (153) in R5 for Y3, Y4, Y5, Z4, Z5. For wehave

d(Z4 − Z5 − 6Y1) = 0, d(Z4 + 2Z5 − 6Y7) = 0, (161)

so that there exist constant vectors C1 and C2 such that

Z4 − Z5 − 6Y1 = C1, Z4 + 2Z5 − 6Y7 = C2. (162)

Thus, Y1 and Y7 are determined by these equations, and then Y2 and Y6 aredetermined by (158). Note that C1 and C2 are timelike points, and the line[C1, C2] consists entirely of timelike points.

Equations (155) are the structure equations of SO(3). It is thus naturalto take the latter as the parameter space, whose points are the 3×3 matrices

A = [aik], 1 ≤ i, j, k ≤ 3,

satisfyingAtA = AAt = I, detA = 1. (163)

The first equations above, when expanded, are

∑

aijaik =∑

ajiaki = δjk. (164)

The Maurer–Cartan forms of SO(3) are

αik =∑

akjdaij = −αki. (165)

They satisfy the Maurer–Cartan equations,

dαik =∑

αij ∧ αjk. (166)

If we setθ1 = α23, θ2 = α31, θ3 = α12, (167)

37

these equations reduce to (155). With the θi given by (167), we shall writedown an explicit solution of (153).

Let EA, 1 ≤ A ≤ 5, be a fixed linear frame in R5. Let

Fi = 2ai2ai3E1 + 2ai3ai1E2 + 2ai1ai2E3 + a2i1E4 + a2i2E5, (168)

1 ≤ i, j, k ≤ 3. Sincea2i1 + a2i2 + a2i3 = 1,

we see from (152), with yj = aij , that Fi is a Veronese surface for 1 ≤ i ≤ 3.Using equation (164), we compute that

F1 + F2 + F3 = E4 + E5 = constant. (169)

Since the coefficients in Fi are quadratic, the partial derivatives,

∂2Fi

∂aij∂aik,

are independent of i. Moreover, the quantities Gik defined below satisfy

Gik =∑

j

aij∂Fk

∂akj= Gki.

We use these facts in the following computation:

dGik =∑ ∂Fk

∂akjdaij +

∑

aij∂2Fk

∂akj∂akldakl (170)

=∑ ∂Fk

∂akjdaij +

∑

aij∂2Fi

∂aij∂aildakl

=∑ ∂Fk

∂akjdaij +

∑ ∂Fi

∂aijdakj,

where the last step follows from the linear homogeneity of ∂Fi/∂ail. In termsof αij, we have

dGik =∑ ∂Fk

∂akjaljαil +

∑ ∂Fi

∂aijaljαkl, (171)

which gives, when expanded,

dG23 = 2(F3 − F2)θ1 +G12θ2 −G13θ3, (172)

38

and its cyclic permutations.On the other hand, by the same manipulation, we have

dFi =∑ ∂Fi

∂aijdaij =

∑ ∂Fi

∂aijakjαik, (173)

givingdF1 = −G31θ2 +G12θ3,

and its cyclic permutations.One can now immediately verify that a solution of (153) is given by

Y3 = G12, Y4 = −G23, Y5 = G31, (174)

Z3 = 2(F2 − F1), Z4 = 2(F3 − F2), Z5 = 2(F1 − F3).

This is also the most general solution of (153), for the solution is determinedup to a linear transformation, and our choice of frame EA is arbitrary.

By (174), the functions Z1, Z2, Z3 are expressible in terms of F1, F2, F3,and then by (162), so also are Y1, Y7, Y1 + Y7. Specifically, by (162), (169)and (174), we have

6Y1 = Z4 − Z5 − C1 = 2(−F1 − F2 + 2F3)− C1

= 6F3 − 2(E4 + E5)− C1,

so that the focal map Y1, up to an additive constant, is the Veronese surfaceF3. Similarly, the focal maps Y7 and Y1 + Y7 are the Veronese surfaces F1

and −F2, respectively, up to additive constants.We see from (158) that

〈Y1,W1〉 = 0, 〈Y7,W2〉 = 0, 〈Y1 + Y7,W1 −W2〉 = 0.

Thus Y1 is contained in the Mobius space Σ4 = Q5∩W⊥

1 . Let e7 =W1/2 ande1 = (2W2−W1)/

√12. Then e1 is the unique unit vector on the timelike line

[W1,W2] which is orthogonal to W1. In a manner similar to that of Section3, we can write

Y1 = e1 + f,

where f maps B into the unit sphere S4 in the Euclidean space R5 = [e1, e7]⊥

in R72. We call f the spherical projection of the Legendre map λ determined

by the ordered pair {e7, e1} (see [10] for more detail). We know that f isconstant along the leaves of the principal foliation T1 corresponding to Y1,

39

and f induces a map f : B/T1 → S4. By what we have shown above, f mustbe an open subset of a spherical Veronese surface.

Note that the unit timelike vector W2/2 satisfies

W2/2 = (√3/2)e1 + (1/2)e7 = sin(π/3)e1 + cos(π/3)e7.

If we consider the points in the Mobius space Σ to represent point spheresin S4, then as we show in [10], points in Q5 ∩W⊥

2 represent oriented spheresin S4 with oriented radius −π/3. In a way similar to that above, the secondfocal submanifold Y7 ⊂ Q5∩W⊥

2 induces a spherical Veronese surface. Whenconsidered from the point of view of the Mobius space Σ, the points Y7represent oriented spheres of radius −π/3 centered at points of this Veronesesurface. These spheres must be in oriented contact with the point spheres ofthe first Veronese surface determined by Y1 in Q5 ∩P5. Thus, the points inthe second Veronese surface must lie at a distance π/3 along normal geodesicsin S4 to the first Veronese surface f . In fact (see, for example [18, pp. 296–299]), the set of all points in S4 at a distance π/3 from a spherical Veronesesurface is another spherical Veronese surface.

Thus, with this choice of coordinates, the Dupin submanifold in questionis simply an open subset of the Legendre submanifold induced in the stan-dard way by considering B to be the unit normal bundle to the sphericalVeronese embedding f induced by Y1. For values of t = kπ/3, k ∈ Z, theparallel hypersurface at oriented distance t to f in S4 is a Veronese surface.For other values of t, the parallel hypersurface is an isoparametric hypersur-face in S4 with three distinct principal curvatures (Cartan’s isoparametrichypersurface). All of these parallel hypersurfaces are Lie equivalent to eachother and to the Legendre submanifolds induced by the Veronese surfaces.

Remark 5.2. (added by T. Cecil in 2020) There is a somewhat shorter wayto complete the proof of Theorem 5.2 by using the following characterizationof isoparametric hypersurfaces in Lie sphere geometry (see [7], [8, p. 77] or[19, p. 221]). Let λ : Mn−1 → Λ2n−1 be a Legendre submanifold with g dis-tinct curvature spheres K1, . . . , Kg at each point. Then λ is Lie equivalent tothe Legendre lift of an isoparametric hypersurface with g principal curvaturesin Sn if and only if there exist g points P1, . . . , Pg, on a timelike line in P n+2

such that 〈Ki, Pi〉 = 0, 1 ≤ i ≤ g. In the proof of Theorem 5.2 above, we seethat the three curvature sphere maps Y1, Y7 and Y1 + Y7 are orthogonal tothe three points W1, W2, and W1−W2 on the timelike line in [W1,W2] in P

6

40

by equations (158)–(160). Thus, λ is Lie equivalent to the Legendre lift of anisoparametric hypersurface with three principal curvatures in S4. The proofof Theorem 5.2 is then completed by invoking Cartan’s [4] classification ofsuch isoparametric hypersurfaces.

The case ρ = 0.

We now consider the case where ρ is identically zero on B. It turns outthat no new examples occur here, in that these Dupin submanifolds can allbe constructed from Dupin cyclides by certain standard constructions. Tomake this precise, we recall’s Pinkall’s [37, p. 437] notion of reducibility. OurDupin submanifold can be considered, as in Section 3, to have been inducedfrom a Dupin hypersurface M3 ⊂ E4. The Dupin submanifold is reducibleif M3 is obtained from a Dupin surface S ⊂ E3 ⊂ E4 by one of the fourfollowing standard constructions.

i. M is a cylinder S ×R in E4.

ii. M is the hypersurface of revolution obtained by revolving

S about a plane π disjoint from S in E3. (175)

iii. Project S stereographically onto a surface N ⊂ S3 ⊂ E4.

M is the cone R ·N over N.

iv. M is a tube of constant radius around S in E4.

Remark 5.3. (added by T. Cecil in 2020) More generally, the Dupin sub-manifold λ : B → Λ7 is said to be reducible if it is locally Lie equivalent tothe Legendre lift of a proper Dupin hypersurface in E4 obtained by one ofPinkall’s standard constructions given in equation (175). See [37, p. 437] or[8, pp. 125–148] for more detail on Pinkall’s standard constructions in thecontext of Lie sphere geometry.

Pinkall proved [37, p. 438] that the Dupin submanifold λ : B → Λ7 isreducible if and only if some focal point map is contained in a 4-dimensionalsubspace P 4 ⊂ P 6.

If ρ is identically zero on B, then by (99), all of the covariant derivativesof ρ are also equal to zero. From (100) and (133), we see that the functionsin equations (89)–(91) satisfy

q = t = b = 0, a = p = 0, r = s, c = −u.

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Then from (134), we have ω72 = 0. From these and the other relations among

the Maurer-Cartan forms which we have derived, we see that the differentialsof the frame vectors can be written

dY1 − ω1

1Y1 = ω4

1Y4 + ω5

1Y5,

dY7 − ω1

1Y7 = ω3

7Y3 − ω5

1Y5,

dY2 + ω1

1Y2 = s(−ω4

1Y4 + ω5

1Y5),

dY6 + ω1

1Y6 = u(ω3

7Y3 + ω5

1Y5), (176)

dY3 = ω3

7(−Y6 + uY7),

dY4 = ω4

1(sY1 − Y2),

dY5 = ω5

1(−sY1 − Y2 + Y6 − uY7).

Note that from (123) and (124), we have

ds+ 2sω1

1 = 0, du+ 2uω1

1 = 0, (177)

and from (92) thatdθi = ω1

1 ∧ θi, i = 1, 2, 3. (178)

From (101), we have that dω11 = 0. Hence on any local disk neighborhood U

in B, we have thatω1

1 = dσ, (179)

for some smooth function σ on U . We next consider a change of frame of theform,

Y ∗

1 = e−σY1, Y ∗

7 = e−σY7,

Y ∗

2 = eσY2, Y ∗

6 = eσY6, (180)

Y ∗

i = Yi, i = 3, 4, 5.

The effect of this change is to make ω1∗1 = 0 while keeping ρ∗ = 0. If we set

ω4∗

1 = e−σω4

1, ω5∗

1 = e−σω5

1, ω3∗

7 = e−σω3

7,

42

then we can compute that from (176) that

dY ∗

1 = ω4∗

1 Y4 + ω5∗

1 Y5,

dY ∗

7 = ω3∗

7 Y3 − ω5∗

1 Y5,

dY ∗

2 = s∗(−ω4∗

1 Y4 + ω5∗

1 Y5)

dY ∗

6 = u∗(ω3∗

7 Y3 + ω5∗

1 Y5), (181)

dY3 = ω3∗

7 Z∗

3 , where Z∗

3 = −Y ∗

6 − u∗Y ∗

7 ,

dY4 = ω4∗

1 Z∗

4 , where Z∗

4 = s∗Y ∗

1 − Y ∗

2 ,

dY5 = ω5∗

1 Z∗

5 , where Z∗

5 = −s∗Y ∗

1 − Y ∗

2 + Y ∗

6 − u∗Y ∗

7 ,

wheres∗ = se2σ, u∗ = ue2σ. (182)

Using (177) and (182), we can then compute that

ds∗ = 0, du∗ = 0, (183)

i.e., s∗ and u∗ are constant functions on the local neighborhood U .The frame in equation (180) is our final frame, and we drop the asterisks

in further references to equations (180)–(183). Since the functions s and uare now constant, we can compute from equation (181) that

dZ3 = −2uω3

7Y3,

dZ4 = 2sω4

1Y4, (184)

dZ5 = 2(u− s)ω5

1Y5.

From this we see that the following 4-dimensional subspaces,

Span {Y1, Y4, Y5, Z4, Z5},Span {Y7, Y3, Y5, Z3, Z5}, (185)

Span {Y1 + Y7, Y3, Y4, Z3, Z4},

are invariant under exterior differentiation and are thus constant. Thus,each of the three focal point maps Y1, Y7 and Y1 + Y7 is contained in a 4-dimensional subspace of P 6, and our Dupin submanifold is reducible in threedifferent ways. Each of the three focal point maps is thus an immersion ofthe space of leaves of its principal foliation onto an open subset of a cyclideof Dupin in a space Σ3 = P 4 ∩Q5. See [8, p. 188] for more detail.

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We state this result to due Pinkall [36] as follows:

Theorem 5.3: Every Dupin submanifold with ρ = 0 is reducible. Thus, it isobtained from a cyclide inR3 by one of the four standard constructions (175).

Pinkall [36, p. 111] then proceeds to classify Dupin submanifolds withρ = 0 up to Lie equivalence. We will not prove his result here. The readercan follow his proof using the fact that his constants α and β are our con-stants s and −u, respectively.

Remark 5.4. (added by T. Cecil in 2020) We now briefly describe some localresults for higher dimensional Dupin hypersurfaces that have been obtainedusing an approach similar to that used in proving Theorems 5.2 and 5.3above. We also describe some remaining open problems in the field. (See[19, pp. 301–308], [9], for more complete surveys.)

These classifications primarily involve finding proper Dupin submanifoldsthat are irreducible in the sense of Pinkall [37, p. 438] as in Theorem 5.2. Ingeneral, a proper Dupin submanifold λ :Mn−1 → Λ2n−1 is said to be reducibleif it is locally Lie equivalent to the Legendre lift of a proper Dupin hyper-surface in Rn obtained by one of Pinkall’s standard constructions (suitablygeneralized to Rn) given in equation (175).

Pinkall [37] proved the following characterization of reducibility in termsof Lie sphere geometry. A connected proper Dupin submanifold λ :Mn−1 →Λ2n−1 is reducible if and only if there exists a curvature sphere map [K] of λthat lies in a linear subspace of P n+2 of codimension at least two. (See also[8, pp. 127–148] and [19, pp. 233–256] for more detail.)

After Pinkall’s results, Niebergall [32] next proved that every connectedproper Dupin hypersurface in R5 with three principal curvatures is reducible.Following that, Cecil and Jensen [15] found the following local classificationof irreducible Dupin hypersurfaces with three principal curvatures of arbi-trary dimension. Suppose Mn−1 is a connected irreducible proper Dupinhypersurface in Sn with three distinct principal curvatures of multiplicitiesm1, m2, m3. Then m1 = m2 = m3 = m, and Mn−1 is Lie equivalent to theLegendre lift of an isoparametric hypersurface in Sn.

It then follows from Cartan’s [4] classification of isoparametric hypersur-faces with g = 3 that m = 1, 2, 4 or 8, and the isoparametric hypersurfacemust be a tube of constant radius over a standard embedding of a projec-tive plane FP 2 into S3m+1 (see, for example, [18, pp. 296–299]), [19, pp.

44

151–155]), [42]), where F is the division algebra R, C, H (quaternions), O(Cayley numbers), for m = 1, 2, 4, 8, respectively. Thus, up to congruence,there is only one such family for each value of m.

Note that in the original paper [15], the theorem of Cecil and Jensenwas proven under the assumption that Mn−1 is locally irreducible, i.e., thatMn−1 does not contain any reducible open subset. However, Cecil, Chi andJensen [14] later proved that proper Dupin hypersurfaces are analytic, andas a consequence local irreducibility is equivalent to irreducibility (see, [13]or [8, pp. 145–146]).

An open problem is the classification up to Lie equivalence of reducibleDupin hypersurfaces with three principal curvatures in Rn, n ≥ 5. As notedabove, Pinkall [36] found such a classification in the case of M3 ⊂ R4. Itmay be possible to generalize Pinkall’s results to higher dimensions using theapproach of [15].

We next discuss the problem of finding local classifications of irreducibleproper Dupin submanifolds λ with four curvature spheres. As mentioned inRemark 1.2, Pinkall and Thorbergsson [38] (for g = 4), and Miyaoka andOzawa [29] (for g = 4 and g = 6) gave constructions of compact properDupin hypersurfaces in spheres that do not have constant Lie curvatures(cross-ratios of the principal curvatures), and so they are not Lie equivalentto the Legendre lift of an isoparametric hypersurface. These examples areirreducible (see [19, pp. 255–256]). So irreducibility itself without additionalassumptions does not lead to the conclusion that λ is Lie equivalent to anisoparametric hypersurface, as in the case g = 3.

This leads to the problem of finding necessary and sufficient conditionsfor an irreducible proper Dupin hypersurface with g = 4 curvature spheresto be Lie equivalent to the Legendre lift of an isoparametric hypersurface.Considerable progress has been made on this problem, as we now describe.

Let λ : Mn−1 → Λ2n−1 be a connected proper Dupin submanifold withfour curvature spheres at each point. Then one can find a Lie frame in whichthe four curvature spheres are Y1, Yn+3, Y1 + Yn+3 and Y1 +ΨYn+3, where Ψis the Lie curvature of λ, having respective multiplicities m1, m2, m3 and m4.Corresponding to the one function ρ in the case Theorems 5.2 and 5.3, thereare four sets of functions, F α

pa, Fµpa, F

µαa, F

µαp, where 1 ≤ a ≤ m1, m1+1 ≤ p ≤

m1 +m2, m1 +m2 +1 ≤ α ≤ m1 +m2 +m3, m1 +m2+m3 +1 ≤ µ ≤ n− 1.Munzner [30]–[31] proved that the multiplicities of the principal curva-

tures of an isoparametric hypersurface with four principal curvatures in Sn

must satisfy m1 = m2, m3 = m4, when the principal curvatures are appropri-

45

ately ordered (see also [1], [40], [12]). Thus, in the papers of Cecil and Jensen[16], and Cecil, Chi and Jensen [13], those restrictions on the multiplicities areassumed. Furthermore, the assumption that that the Lie curvature Ψ = −1is made in [13], since that is also is true for an isoparametric hypersurfacewith four principal curvatures (with the appropriate ordering of the principalcurvatures).

In [16, pp. 3–4], Cecil and Jensen conjectured that an irreducible con-nected proper Dupin hypersurface in Sn with four principal curvatures hav-ing multiplicities satisfying m1 = m2, m3 = m4 and constant Lie curvature Ψmust be Lie equivalent to the Legendre lift of an open subset of an isopara-metric hypersurface in Sn.

In that paper [16], Cecil and Jensen proved that the conjecture is true ifall the multiplicities are equal to one (see also Niebergall [33], who obtainedthe same conclusion under additional assumptions). In the second paper [13]mentioned above, Cecil, Chi and Jensen proved that the conjecture is trueif m1 = m2 ≥ 1, m3 = m4 = 1, and the Lie curvature is assumed to satisfyΨ = −1. They conjectured that their theorem is still true if m3 = m4 is alsoallowed to be greater than one, but the calculations involved in that case areformidable, and it remains an open problem.

Note that the assumption of irreducibility is needed in the papers [13]and [16] mentioned in the preceding paragraph, since Cecil [7] constructedan example of a reducible noncompact proper Dupin submanifold with g = 4curvature spheres having all multiplicities equal and constant Lie curvatureΨ = −1 that is not Lie equivalent to an open subset of an isoparametrichypersurface with four principal curvatures. (See also [8, pp. 80–82] or [19,pp.304–306].)

An important step in in the theorems of Cecil-Jensen [16] and Cecil-Chi-Jensen [13] mentioned above is proving that under the assumptions m1 =m2 ≥ 1, m3 = m4 = 1, and Ψ = −1, the Dupin submanifold λ is reducible ifthere exists some fixed index, say a, such that

F αpa = F µ

pa = F µαa = 0, for all p, α, µ. (186)

Thus, if λ is irreducible, no such index a can exist, and it can be shownafter a lengthy argument that λ is Lie equivalent to the Legendre lift of anopen subset of an isoparametric hypersurface, as in Remark 5.2. This is ageneralization of the case ρ 6= 0 for g = 3 handled in Theorem 5.2 above.

46

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Thomas E. CecilDepartment of Mathematics and Computer ScienceCollege of the Holy CrossWorcester, MA 01610email: tcecil@holycross.edu

Shiing-Shen ChernDepartment of MathematicsUniversity of CaliforniaBerkeley, CA 94720andMathematical Sciences Research Institute1000 Centennial DriveBerkeley, CA 94720

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