Using Lie Sphere Geometry to Study Dupin Hypersurfaces in ${\bf R}^n$

A hypersurface $M$ in ${\bf R}^n$ or $S^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on $M$, i.e., the number of distinct principal curvatures is constant on $M$. The notions of Dupin and proper Dupin hypersurfaces in ${\bf R}^n$ or $S^n$ can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations. In these notes, we give a detailed introduction to this method for studying Dupin hypersurfaces in ${\bf R}^n$ or $S^n$, including proofs of several fundamental results.


Introduction
A hypersurface M in R n or S n is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M (see Pinkall [53]).
The notions of Dupin and proper Dupin hypersurfaces in R n or S n can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations (see Theorem 8.1). This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations.
In these notes, we give a detailed introduction to the method for studying Dupin hypersurfaces in R n or S n using Lie sphere geometry, including necessary and sufficient conditions for a Dupin hypersurface in S n to be equivalent to an isoparametric hypersurface in S n by a Lie sphere transformation (Theorem 9.3). As an application, we give a classification due to Pinkall [53] of the cyclides of Dupin in S n up to Lie sphere transformations (Theorem 10.1). We also give a survey of results concerning compact proper Dupin hypersurfaces and their relationship to isoparametric hypersurfaces (see Section 12).
Using linear algebra, one can show that this Lie quadric Q n+1 contains projective lines but no linear subspaces of P n+2 of higher dimension, since the metric in equation (1) has signature (n + 1, 2) (see [14, p. 21]). The oneparameter family of Lie spheres in R n ∪ {∞} corresponding to the points on a line on Q n+1 is called a parabolic pencil of spheres. It consists of all Lie spheres in oriented contact at a certain contact element (p, N) on R n ∪ {∞}, where p is a point in R n ∪{∞} and N is a unit tangent vector to R n ∪{∞} at p. That is, (p, N) is an element of the unit tangent bundle of R n ∪ {∞}. In this way, Lie also established a bijective correspondence between the manifold of contact elements on R n ∪ {∞} and the manifold Λ 2n−1 of projective lines on the Lie quadric Q n+1 .
A Lie sphere transformation is a projective transformation of P n+2 which maps the Lie quadric Q n+1 to itself. In terms of the geometry of R n , a Lie sphere transformation maps Lie spheres to Lie spheres. Furthermore, since a projective transformation maps lines to lines, a Lie sphere transformation preserves oriented contact of Lie spheres in R n .
Let R n+3 2 denote R n+3 endowed with the metric , in equation (1), and let O(n + 1, 2) denote the group of orthogonal transformations of R n+3 2 . One can show that every Lie sphere transformation is the projective transformation induced by an orthogonal transformation, and thus the group G of Lie sphere transformations is isomorphic to the quotient group O(n + 1, 2)/{±I} (see [14, pp. 26-27]). Furthermore, any Möbius (conformal) transformation of R n ∪ {∞} induces a Lie sphere transformation, and the Möbius group is precisely the subgroup of G consisting of all Lie sphere transformations that map point spheres to point spheres.
The manifold Λ 2n−1 of projective lines on the quadric Q n+1 has a contact structure, i.e., a 1-form ω such that ω ∧ (dω) n−1 does not vanish on Λ 2n−1 . The condition ω = 0 defines a codimension one distribution D on Λ 2n−1 which has integral submanifolds of dimension n − 1, but none of higher dimension. Such an integral submanifold λ : M n−1 → Λ 2n−1 of D of maximal dimension is called a Legendre submanifold. If α is a Lie sphere transformation, then α maps lines on Q n+1 to lines on Q n+1 , and the map µ = αλ is also a Legendre submanifold. The submanifolds λ and µ are said to be Lie equivalent.
Let M n−1 be an oriented hypersurface in R n . Then M n−1 naturally induces a Legendre submanifold λ : M n−1 → Λ 2n−1 , called the Legendre lift of M n−1 , as we will show in these notes. More generally, an immersed submanifold V of codimension greater than one in R n induces a Legendre lift whose domain is the unit normal bundle B n−1 of V in R n . Similarly, any submanifold of the unit sphere S n ⊂ R n+1 has a Legendre lift. We can relate properties of a submanifold of R n or S n to Lie geometric properties of its Legendre lift, and attempt to classify certain types of Legendre submanifolds up to Lie sphere transformations. This, in turn, gives classification results for the corresponding classes of Euclidean submanifolds of R n or S n . We next recall some basic ideas from Euclidean submanifold theory that are necessary for the study of Dupin hypersurfaces. For an oriented hypersurface f : M → R n with field of unit normal vectors ξ, the eigenvalues of the shape operator (second fundamental form) A of M are called principal curvatures, and their corresponding eigenspaces are called principal spaces. A submanifold S of M is called a curvature surface of M if at each point x of S, the tangent space T x S is a principal space at x. This generalizes the classical notion of a line of curvature of a surface in R 3 . If κ is a non-zero principal curvature of M at x, the point is called the focal point of M at x determined by κ. The hypersphere in R n tangent to M at f (x) and centered at the focal point f κ (x) is called the curvature sphere at x determined by κ.
It is well known that there always exists an open dense subset Ω of M on which the multiplicities of the principal curvatures are locally constant (see, for example, Singley [55]). If a principal curvature κ has constant multiplicity m in some open set U ⊂ M, then the corresponding m-dimensional distribution of principal spaces is integrable, i.e., it is an m-dimensional foliation, and the leaves of this principal foliation are curvature surfaces. Furthermore, if the multiplicity m of κ is greater than one, then by using the Codazzi equation, one can show that κ is constant along each leaf of this principal foliation (see, for example, [27, p. 24]). This is not true, in general, if the multiplicity m = 1. Analogues of these results hold for oriented hypersurfaces in the sphere S n or in real hyperbolic space H n (see, for example, [27, pp. 9-35]).
As mentioned earlier, a hypersurface M in R n or S n is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M (see Pinkall [53]).
The notions of Dupin and proper Dupin hypersurfaces in R n or S n can be generalized to a class of Legendre submanifolds in Lie sphere geometry known as Dupin submanifolds (see Section 8). In particular, the Legendre lifts of Dupin hypersurfaces in R n or S n are Dupin submanifolds in this generalized sense. The Dupin and proper Dupin properties of such submanifolds are easily seen to be invariant under Lie sphere transformations (see Theorem 8.1).
A well-known class of proper Dupin hypersurfaces consists of the cylides of Dupin in R 3 , introduced by Dupin [29] in 1822 (see Section 10). Dupin defined a cyclide to be a surface M in R 3 that is the envelope of the family of spheres tangent to three fixed spheres in R 3 . This is equivalent to requiring that M have two distinct principal curvatures at each point, and that both focal maps of M degenerate into curves (instead of surfaces). Then M is the envelope of the family of curvature spheres centered along each of the focal curves. The three fixed spheres in Dupin's definition can be chosen to be three spheres from either family of curvature spheres.
The most basic examples of cyclides of Dupin in R 3 are a torus of revolution, a circular cylinder, and a circular cone. The proper Dupin property is easily shown to be invariant under Möbius transformations of R 3 ∪ {∞}, and it turns out that all cyclides of Dupin in R 3 can be obtained from these three types of examples by inversion in a sphere in R 3 (see, for example, [26, pp. 151-166]).
Pinkall's paper [53] describing higher dimensional cyclides of Dupin in the context of Lie sphere geometry was particularly influential. Pinkall defined a cyclide of Dupin of characteristic (p, q) to be a proper Dupin submanifold λ : M n−1 → Λ 2n−1 with two distinct curvature spheres of respective multiplicities p and q at each point. In Section 10 of these notes, we present Pinkall's [53] classification of the cyclides of Dupin of arbitrary dimension in R n or S n (Theorem 10.1), which is obtained by using the methods of Lie sphere geometry.
Specifically, we show in Theorem 10.1 that any connected cyclide of Dupin of characteristic (p, q) is contained in a unique compact, connected cyclide of Dupin of characteristic (p, q). Furthermore, every compact, connected cyclide of Dupin of characteristic (p, q) is Lie equivalent to the Legendre lift of a standard product of two spheres, A standard product of two spheres in S n is an isoparametric hypersurface in S n , i.e., it has constant principal curvatures in S n . The images of isoparametric hypersurfaces in S n under stereographic projection from S n to R n form a particularly important class of proper Dupin hypersurfaces in R n .
Many results in the field deal with relationships between compact proper Dupin hypersurfaces and isoparametric hypersurfaces in spheres, including the question of which compact proper Dupin hypersurfaces are Lie equivalent to an isoparametric hypersurface in a sphere. An important result in proving such classifications is Theorem 9.3 of these notes, which gives for necessary and sufficient conditions for a Legendre submanifold to be Lie equivalent to the Legendre lift of an isoparametric hypersurface in S n .
Local examples of proper Dupin hypersurfaces in R n or S n are known to be plentiful, since Pinkall [53] introduced four constructions for obtaining a proper Dupin hypersurface W in R n+m from a proper Dupin hypersurface M in R n . These constructions involve building tubes, cylinders, cones and surfaces of revolution from M, and they are discussed in Section 11. Using these constructions, Pinkall was able to construct a proper Dupin hypersurface in Euclidean space with an arbitrary number of distinct principal curvatures with any given multiplicities (see Theorem 11.1). In general, these proper Dupin hypersurfaces obtained by using Pinkall's constructions cannot be extended to compact Dupin hypersurfaces without losing the property that the number of distinct principal curvatures is constant, as we discuss in Section 11.
Proper Dupin hypersurfaces that are locally Lie equivalent to the end product of one of Pinkall's constructions are said to be reducible. Pinkall [53] found a useful characterization of reducibility in the context of Lie sphere geometry, which we state and prove in Theorem 11.2. This theorem has been important in proving several classifications of irreducible proper Dupin hypersurfaces that are described in Section 12.
In Section 12, we give a survey of results concerning compact and irreducible proper Dupin hypersurfaces, and their relationship to isoparametric hypersurfaces. Cecil, Chi and Jensen [19] showed that every compact proper Dupin hypersurface with more than two principal curvatures is irreducible (Theorem 11.3). In fact, several classifications of compact proper Dupin hypersurfaces with g ≥ 3 principal curvatures have been obtained by assuming that the hypersurface is irreducible and working locally in the context of Lie sphere geometry using the method of moving frames. (See, for example, the papers of Pinkall [50], [52]- [53], Cecil and Chern [18], Cecil and Jensen [22]- [23], Cecil, Chi and Jensen [19], and Niebergall [48]- [49].) See also [16] for a survey of classifications of compact proper Dupin hypersurfaces.
These notes are based primarily on the author's book [14], and several passages in these notes are taken directly from that book.

Notation and Preliminary Results
In the next few sections, we review the basic setup for the sphere geometries of Möbius and Lie, and the method for studying submanifolds of R n and S n in this context.
Let (x, y) be the indefinite scalar product on the Lorentz space R n+1 where x = (x 1 , . . . , x n+1 ) and y = (y 1 , . . . , y n+1 ). We will call this scalar product the Lorentz metric. A vector x is said to be spacelike, timelike or lightlike, respectively, depending on whether (x, x) is positive, negative or zero. We will use this terminology even when we are using an indefinite metric of different signature.
In Lorentz space, the set of all lightlike vectors, given by the equation, forms a cone of revolution, called the light cone. Timelike vectors are "inside the cone" and spacelike vectors are "outside the cone." If x is a nonzero vector in R n+1

1
, let x ⊥ denote the orthogonal complement of x with respect to the Lorentz metric. If x is timelike, then the metric restricts to a positive definite form on x ⊥ , and x ⊥ intersects the light cone only at the origin. If x is spacelike, then the metric has signature (n − 1, 1) on x ⊥ , and x ⊥ intersects the cone in a cone of one less dimension. If x is lightlike, then x ⊥ is tangent to the cone along the line through the origin determined by x. The metric has signature (n − 1, 0) on this n-dimensional plane.
Lie sphere geometry is defined in the context of real projective space P n , so we now briefly review some basic concepts from projective geometry. We define an equivalence relation on R n+1 − {0} by setting x ≃ y if x = ty for some nonzero real number t. We denote the equivalence class determined by a 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 origin in R n+1 . The rectangular coordinates (x 1 , . . . , x n+1 ) are called homogeneous coordinates of the point [x], and they are only determined up to a nonzero scalar multiple. The affine space R n can be embedded in P n as the complement of the hyperplane (x 1 = 0) at infinity by the map φ : R n → P n given by φ(u) = [(1, u)]. A scalar product on R n+1 , such as the Lorentz metric, determines a polar relationship between points and hyperplanes in P n . We will also use the notation x ⊥ to denote the polar hyperplane of [x] in P n , and we will call [x] the pole of x ⊥ . If x is a non-zero lightlike vector in R n+1 1 , then [x] can be represented by a vector of the form (1, u) for u ∈ R n . Then the equation (x, x) = 0 for the light cone becomes u · u = 1 (Euclidean dot product), i.e., the equation for the unit sphere in R n . Hence, the set of points in P n determined by lightlike vectors in R n+1 1 is naturally diffeomorphic to the sphere S n−1 .

Möbius Geometry of Unoriented Spheres
As a first step toward Lie's geometry of oriented spheres, we recall the geometry of unoriented spheres in R n known as "Möbius" or "conformal" ge-ometry. We will always assume that n ≥ 2. In this section, we will only consider spheres and planes of codimension one, and we will often omit the prefix "hyper," from the words "hypersphere" and "hyperplane." (See [14, pp. 11-14] for more detail. ) We denote the Euclidean dot product of two vectors u and v in R n by u · v. We first consider stereographic projection σ : R n → S n − {P }, where S n is the unit sphere in R n+1 given by y · y = 1, and P = (−1, 0, . . . , 0) is the south pole of S n . The well-known formula for σ(u) is Note that σ is sometimes referred to as "inverse stereographic projection," in which case its inverse map from S n − {P } to R n is called "stereographic projection." We next embed R n+1 into P n+1 by the embedding φ mentioned in the previous section. Thus, we have the map φσ : R n → P n+1 given by Let (z 1 , . . . , z n+2 ) be homogeneous coordinates on P n+1 and ( , ) the Lorentz metric on the space R n+2

1
. Then φσ(R n ) is just the set of points in P n+1 lying on the n-sphere Σ given by the equation (z, z) = 0, with the exception of the improper point [(1, −1, 0, . . . , 0)] corresponding to the south pole P . We will refer to the points in Σ other than [(1, −1, 0, . . . , 0)] as proper points, and will call Σ the Möbius sphere or Möbius space. At times, it is easier to simply begin with S n rather than R n and thus avoid the need for the map σ and the special point P . However, there are also advantages for beginning in R n .
The basic framework for the Möbius geometry of unoriented spheres is as follows. Suppose that ξ is a spacelike vector in R n+2 1 . Then the polar hyperplane ξ ⊥ to [ξ] in P n+1 intersects the sphere Σ in an (n − 1)-sphere S n−1 . The sphere S n−1 is the image under φσ of an (n − 1)-sphere in R n , unless it contains the improper point [(1, −1, 0, . . . , 0)], in which case it is the image under φσ of a hyperplane in R n . Hence, we have a bijective correspondence between the set of all spacelike points in P n+1 and the set of all hyperspheres and hyperplanes in R n .
It is often useful to have specific formulas for this correspondence. Consider the sphere in R n with center p and radius r > 0 given by the equation We wish to translate this into an equation involving the Lorentz metric and the corresponding polarity relationship on P n+1 . A direct calculation shows that equation (7) is equivalent to the equation where ξ is the spacelike vector, and φσ(u) is given by equation (6). Thus, the point u is on the sphere given by equation (7) if and only if φσ(u) lies on the polar hyperplane of [ξ]. Note that the first two coordinates of ξ satisfy ξ 1 + ξ 2 = 1, and that (ξ, ξ) = r 2 . Although ξ is only determined up to a nonzero scalar multiple, we can conclude that η 1 + η 2 is not zero for any η ≃ ξ. Conversely, given a spacelike point [z] with z 1 + z 2 nonzero, we can determine the corresponding sphere in R n as follows. Let ξ = z/(z 1 + z 2 ) so that ξ 1 + ξ 2 = 1. Then from equation (9), the center of the corresponding sphere is the point p = (ξ 3 , . . . , ξ n+2 ), and the radius is the square root of (ξ, ξ).
Thus, the improper point φ(P ) lies on the polar hyperplane of [η], and the point [η] corresponds to a hyperplane in R n . Again we can find an explicit correspondence. Consider the hyperplane in R n given by the equation A direct calculation shows that (10) is equivalent to the equation Thus, the hyperplane (10) is represented in the polarity relationship by [η].
Conversely, let z be a spacelike point with z 1 + z 2 = 0. Then (z, z) = v · v, where v = (z 3 , . . . , z n+2 ). Let η = z/|v|. Then η has the form (11) and [z] corresponds to the hyperplane (10). Thus we have explicit formulas for the bijective correspondence between the set of spacelike points in P n+1 and the set of hyperspheres and hyperplanes in R n .
Similarly, we can construct a bijective correspondence between the space of all hyperspheres in the unit sphere S n ⊂ R n+1 and the manifold of all spacelike points in P n+1 as follows. The hypersphere S in S n with center p ∈ S n and (spherical) radius ρ, 0 < ρ < π, is given by the equation for y ∈ S n . If we take [z] = φ(y) = [(1, y)], then where e 1 = (1, 0, . . . , 0). Thus equation (12) is equivalent to the equation (z, (cos ρ, p)) = 0, (13) in homogeneous coordinates in P n+1 . Therefore, y lies on the hypersphere S given by equation (12) if and only if [z] = φ(y) lies on the polar hyperplane in P n+1 of the spacelike point Remark 3.1. In these notes, we will focus on spheres in R n or S n . See [14, pp. 16-18] for a treatment of the geometry of hyperspheres in real hyperbolic space H n .
Of course, the fundamental invariant of Möbius geometry is the angle. The study of angles in this setting is quite natural, since orthogonality between spheres and planes in R n can be expressed in terms of the Lorentz metric. Let S 1 and S 2 denote the spheres in R n with respective centers p 1 and p 2 and respective radii r 1 and r 2 . By the Pythagorean Theorem, the two spheres intersect orthogonally if and only if If these spheres correspond by equation (9) to the projective points [ξ 1 ] and [ξ 2 ], respectively, then a calculation shows that equation (15) is equivalent to the condition (ξ 1 , ξ 2 ) = 0.
A hyperplane π in R n is orthogonal to a hypersphere S precisely when π passes through the center of S. If S has center p and radius r, and π is given by the equation u · N = h, then the condition for orthogonality is just p · N = h. If S corresponds to [ξ] as in (9) and π corresponds to [η] as in (11), then this equation for orthogonality is equivalent to (ξ, η) = 0. Finally, if two planes π 1 and π 2 are represented by [η 1 ] and [η 2 ] as in (11), then the orthogonality condition N 1 · N 2 = 0 is equivalent to the equation (η 1 , η 2 ) = 0. Thus, in all cases of hyperspheres or hyperplanes in R n , orthogonal intersection corresponds to a polar relationship in P n+1 given by equations (8) or (11).
We conclude this section with a discussion of Möbius transformations. Recall that a linear transformation A ∈ GL(n + 2) induces a projective transformation P (A) on P n+1 defined by P (A)[x] = [Ax]. The map P is a homomorphism of GL(n + 2) onto the group P GL(n + 1) of projective transformations of P n+1 , and its kernel is the group of nonzero multiples of the identity transformation I ∈ GL(n + 2).
The Möbius transformation α = P (B) induced by an orthogonal transformation B ∈ O(n + 1, 1) maps spacelike points to spacelike points in P n+1 , and it preserves the polarity condition (ξ, η) = 0 for any two points [ξ] and [η] in P n+1 . Therefore by the correspondence given in equations (8) and (11) above, α maps the set of hyperspheres and hyperplanes in R n to itself, and it preserves orthogonality and hence angles between hyperspheres and hyperplanes. A similar statement holds for the set of all hyperspheres in S n .
Let H denote the group of Möbius transformations and let be the restriction of the map P to O(n + 1, 1). The discussion above shows that ψ is onto, and the kernel of ψ is {±I}, the intersection of O(n + 1, 1) with the kernel of P . Therefore, H is isomorphic to the quotient group O(n + 1, 1)/{±I}. One can show that the group H is generated by Möbius transformations induced by inversions in spheres in R n . This follows from the fact that the corresponding orthogonal groups are generated by reflections in hyperplanes. In fact, every orthogonal transformation on an indefinite inner product space R n k is a product of at most n reflections, a result due to Cartan and Dieudonné. (See Cartan [10, pp. 10-12], Chapter 3 of Artin's book [2], or [14, pp. 30-34]).
Since a Möbius transformation α = P (B) for B ∈ O(n + 1, 1) maps lightlike points to lightlike points in P n+1 in a bijective way, it induces a diffeomorphism of the n-sphere Σ which is conformal by the considerations given above. It is well known that the group of conformal diffeomorphisms of the n-sphere is precisely the Möbius group.

Lie Geometry of Oriented Spheres
We now turn to the construction of Lie's geometry of oriented spheres in R n . Let W n+1 be the set of vectors in R n+2 1 satisfying (ζ, ζ) = 1. This is a hyperboloid of revolution of one sheet in R n+2 1 . If α is a spacelike point in P n+1 , then there are precisely two vectors ±ζ in W n+1 with α = [ζ]. These two vectors can be taken to correspond to the two orientations of the oriented sphere or plane represented by α, as we will now describe. We first introduce one more coordinate. We embed R n+2 1 into P n+2 by the embedding z → [(z, 1)]. If ζ ∈ W n+1 , then −ζ 2 1 + ζ 2 2 + · · · + ζ 2 n+2 = 1, so the point [(ζ, 1)] in P n+2 lies on the quadric Q n+1 in P n+2 given in homogeneous coordinates by the equation The manifold Q n+1 is called the Lie quadric, and the scalar product determined by the quadratic form in (18) is called the Lie metric or Lie scalar product. We will let {e 1 , . . . , e n+3 } denote the standard orthonormal basis for the scalar product space R n+3 2 with metric , . Here e 1 and e n+3 are timelike and the rest are spacelike.
We shall now see how points on Q n+1 correspond to the set of oriented hyperspheres, oriented hyperplanes and point spheres in R n ∪ {∞}. Suppose that x is any point on the quadric with homogeneous coordinate x n+3 = 0. Then x can be represented by a vector of the form (ζ, 1), where the Lorentz scalar product (ζ, ζ) = 1. Suppose first that ζ 1 + ζ 2 = 0. Then in Möbius geometry, [ζ] represents a sphere in R n . If as in equation (9), we represent [ζ] by a vector of the form then (ξ, ξ) = r 2 . Thus ζ must be one of the vectors ±ξ/r. In P n+2 , we have We can interpret the last coordinate as a signed radius of the sphere with center p and unsigned radius r > 0. In order to interpret this geometrically, we adopt the convention that a positive signed radius corresponds to the orientation of the sphere represented by the inward field of unit normals, and a negative signed radius corresponds to the orientation given by the outward field of unit normals. Hence, the two orientations of the sphere in R n with center p and unsigned radius r > 0 are represented by the two projective points, in Q n+1 . Next if ζ 1 + ζ 2 = 0, then [ζ] represents a hyperplane in R n , as in equation (11). For ζ = (h, −h, N), with |N| = 1, we have (ζ, ζ) = 1. Then the two projective points on Q n+1 induced by ζ and −ζ are These represent the two orientations of the plane with equation u · N = h. . Then x, x = (z, z) = 0, and [z] ∈ P n+1 is simply a point of the Möbius sphere Σ. Thus we have the following bijective correspondence between objects in Euclidean space and points on the Lie quadric:

Euclidean
Lie spheres: center p, signed radius r In Lie sphere geometry, points are considered to be spheres of radius zero, or "point spheres." Point spheres do not have an orientation. From now on, we will use the term Lie sphere or simply "sphere" to denote an oriented sphere, oriented plane or a point sphere in R n ∪ {∞}. We will refer to the coordinates on the right side of equation (21) as the Lie coordinates of the corresponding point, sphere or plane. In the case of R 2 and R 3 , respectively, these coordinates were classically called pentaspherical and hexaspherical coordinates (see Blaschke [5]).
At times it is useful to have formulas to convert Lie coordinates back into Cartesian equations for the corresponding Euclidean object. Suppose first that [x] is a point on the Lie quadric with x 1 + x 2 = 0. Then x = ρy, for some ρ = 0, where y is one of the standard forms on the right side of the table above. From the table, we see that y 1 + y 2 = 1, for all proper points and all spheres. Hence if we divide x by x 1 + x 2 , the new vector will be in standard form, and we can read off the corresponding Euclidean object from the table. In particular, if x n+3 = 0, then [x] represents the point sphere u = (u 3 , . . . , u n+2 ) where If x n+3 = 0, then [x] represents the sphere with center p = (p 3 , . . . , p n+2 ) and signed radius r given by Finally, suppose that x by x n+3 to make the last coordinate 1. Then if we set N = (N 3 , . . . , N n+2 ) and h according to the conditions x, x = 0 and x 1 + x 2 = 0 force N to have unit length. Thus [x] corresponds to the hyperplane u · N = h, with unit normal N and h as in equation (24). If we wish to consider oriented hyperspheres and point spheres in the unit sphere S n in R n+1 , then the table (21) above can be simplified. First, we have shown that in Möbius geometry, the unoriented hypersphere S in S n with center p ∈ S n and spherical radius ρ, 0 < ρ < π, corresponds to the point [ξ] = [(cos ρ, p)] in P n+1 . To correspond the two orientations of this sphere to points on the Lie quadric, we first note that (ξ, ξ) = − cos 2 ρ + 1 = sin 2 ρ.
Since sin ρ > 0 for 0 < ρ < π, we can divide ξ by sin ρ and consider the two vectors ζ = ±ξ/ sin ρ that satisfy (ζ, ζ) = 1. We then map these two points into the Lie quadric to get the points in Q n+1 . We can incorporate the sign of the last coordinate into the radius and thereby arrange that the oriented sphere S with signed radius ρ = 0, where −π < ρ < π, and center p corresponds to the point in Q n+1 . This formula still makes sense if the radius ρ = 0, in which case it yields the point sphere [(1, p, 0)]. We adopt the convention that the positive radius ρ in (25) corresponds to the orientation of the sphere given by the field of unit normals which are tangent vectors to geodesics in S n from −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 signed radius ρ, −π < ρ < π, or with center −p and the appropriate signed radius ρ ± π.
For a given point [x] in the quadric Q n+1 , we can determine the corresponding oriented hypersphere or point sphere in S n as follows. Multiplying by −1, if necessary, we can arrange that that the first coordinate x 1 of x is nonnegative. If x 1 is positive, then it follows from equation (25) that the center p and signed radius ρ, −π/2 < ρ < π/2, are given by If x 1 = 0, then x n+3 is nonzero, and we can divide by x n+3 to obtain a point with coordinates (0, p, 1). This corresponds to the oriented hypersphere in S n with center p and signed radius π/2, which is a great sphere in S n .

Oriented Contact of Spheres
As we saw in Section 3, the angle between two spheres is the fundamental geometric quantity in Möbius geometry, and it is the quantity that is preserved by Möbius transformations. In Lie's geometry of oriented spheres, the corresponding fundamental notion is that of oriented contact of spheres.
(See [14, pp. 19-23] for more detail.) By definition, two oriented spheres S 1 and S 2 in R n are in oriented contact if they are tangent to each other, and they have the same orientation at the point of contact There are two geometric possibilities depending on whether the signed radii of S 1 and S 2 have the same sign or opposite signs. In either case, if p 1 and p 2 are the respective centers of S 1 and S 2 , and r 1 and r 2 are their respective signed radii, then the analytic condition for oriented contact is Similarly, we say that an oriented hypersphere sphere S with center p and signed radius r and an oriented hyperplane π with unit normal N and equation u·N = h are in oriented contact if π is tangent to S and their orientations agree at the point of contact. This condition is given by the equation Next we say that two oriented planes π 1 and π 2 are in oriented contact if their unit normals N 1 and N 2 are the same. These planes can be considered to be two oriented spheres in oriented contact at the improper point. Finally, a proper point u in R n is in oriented contact with a sphere or a plane if it lies on the sphere or plane, and the improper point is in oriented contact with each plane, since it lies on each plane. An important fact in Lie sphere geometry is that if S 1 and S 2 are two Lie spheres which are represented as in equation (21) by [k 1 ] and [k 2 ], then the analytic condition for oriented contact is equivalent to the equation This can be checked easily by a direct calculation. By standard linear algebra in indefinite inner product spaces (see, for example, [14, p. 21]), the fact that the signature of R n+3 2 is (n + 1, 2) implies that the Lie quadric contains projective lines in P n+2 , but no linear subspaces of P n+2 of higher dimension. These projective lines on Q n+1 play a crucial role in the theory of submanifolds in the context of Lie sphere geometry.
One can show further (see [14, pp. 21-23]), that if [k 1 ] and [k 2 ] are two points of Q n+1 , then the line [k 1 , k 2 ] in P n+2 lies on Q n+1 if and only if the spheres corresponding to [k 1 ] and [k 2 ] are in oriented contact, i.e., k 1 , k 2 = 0. Moreover, if the line [k 1 , k 2 ] lies on Q n+1 , then the set of spheres in R n corresponding to points on the line [k 1 , k 2 ] is precisely the set of all spheres in oriented contact with both of these spheres. Such a 1-parameter family of spheres is called a parabolic pencil of spheres in R n ∪ {∞}.
Each parabolic pencil contains exactly one point sphere, and if that point sphere is a proper point, then the parabolic pencil contains exactly one hyperplane π in R n , and the pencil consists of all spheres in oriented contact with the oriented plane π at p. Thus, we can associate the parabolic pencil with the point (p, N) in the unit tangent bundle of R n ∪ {∞}, where N is the unit normal to the oriented plane π.
If the point sphere in the pencil is the improper point, then the parabolic pencil is a family of parallel hyperplanes in oriented contact at the improper point. If N is the common unit normal to all of these planes, then we can associate the pencil with the point (∞, N) in the unit tangent bundle of Similarly, we can establish a correspondence between parabolic pencils and elements of the unit tangent bundle T 1 S n that is expressed in terms of the spherical metric on S n . If ℓ is a line on the quadric, then ℓ intersects both e ⊥ 1 and e ⊥ n+3 at exactly one point, where e 1 = (1, 0, . . . , 0) and e n+3 = (0, . . . , 0, 1). So the parabolic pencil corresponding to ℓ contains exactly one point sphere (orthogonal to e n+3 ) and one great sphere (orthogonal to e 1 ), given respectively by the points, Since ℓ lies on the quadric, we know that k 1 , k 2 = 0, and this condition is equivalent to the condition p · ξ = 0, i.e., ξ is tangent to S n at p. Thus, the parabolic pencil of spheres corresponding to the line ℓ can be associated with the point (p, ξ) in T 1 S n . More specifically, the line ℓ can be parametrized as From equation (25) above, we see that [K t ] corresponds to the oriented sphere in S n with center p t = cos t p + sin t ξ, and signed radius t. The pencil consists of all oriented spheres in S n in oriented contact with the great sphere corresponding to [k 2 ] at the point (p, ξ) in T 1 S n . Their centers p t lie along the geodesic in S n with initial point p and initial velocity vector ξ. Detailed proofs of all these facts are given in [14, pp. 21-23]. We conclude this section with a discussion of Lie sphere transformations. By definition, a Lie sphere transformation is a projective transformation of P n+2 which maps 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, and since it is a projective transformation, it maps lines on Q n+1 to lines on Q n+1 . Thus, it preserves oriented contact of spheres in R n or S n . Conversely, Pinkall [53] (see also [14, pp. 28-30]) proved the so-called "Fundamental Theorem of Lie sphere geometry," which states that any line preserving diffeomorphism of Q n+1 is the restriction to Q n+1 of a projective transformation, that is, a transformation of the space of oriented spheres which preserves oriented contact is a Lie sphere transformation.
By the same type of reasoning as given in Section 3 for Möbius transformations, one can show that the group G of Lie sphere transformations is isomorphic to the group O(n + 1, 2)/{±I}, where O(n + 1, 2) is the group of orthogonal transformations of R n+3 2 . As with the Möbius group, it follows from the theorem of Cartan and Dieudonné (see [14, pp. 30-34]) that the Lie sphere group G is generated by Lie inversions, that is, projective transformations that are induced by reflections in O(n + 1, 2).
The Möbius group H can be considered to be a subgroup of G in the following manner. Each Möbius transformation on the space of unoriented spheres, naturally induces two Lie sphere transformations on the space Q n+1 of oriented spheres as follows. If A is in O(n + 1, 1), then we can extend A to a transformation B in O(n + 1, 2) by setting B = A on R n+2 1 and B(e n+3 ) = e n+3 . In terms the standard orthonormal basis in R n+3 2 , the transformation B has the matrix representation, Although A and −A induce the same Möbius transformation in H, the Lie transformation P (B) is not the same as the Lie transformation P (C) induced by the matrix where ≃ denotes equivalence as projective transformations. Note that P (B) = ΓP (C), where Γ is the Lie transformation represented in matrix form by From equation (21), we see that Γ has the effect of changing the orientation of every oriented sphere or plane. The transformation Γ is called the change of orientation transformation or "Richtungswechsel" in German. Hence, the two Lie sphere transformations induced by the Möbius transformation P (A) differ by this change of orientation factor. Thus, the group of Lie sphere transformations induced from Möbius transformations is isomorphic to O(n + 1, 1). This group consists of those Lie transformations that map [e n+3 ] to itself, and it is a double covering of the Möbius group H. Since these transformations are induced from orthogonal transformations of R n+3 2 , they also map e ⊥ n+3 to itself, and thereby map point spheres to point spheres. When working in the context of Lie sphere geometry, we will refer to these transformations as "Möbius transformations."

Legendre Submanifolds
The goal of this section is to define a contact structure on the unit tangent bundle T 1 S n and on the (2n − 1)-dimensional manifold Λ 2n−1 of projective lines on the Lie quadric Q n+1 , and to describe its associated Legendre submanifolds. This will enable us to study submanifolds of R n or S n within the context of Lie sphere geometry in a natural way. This theory was first developed extensively in a modern setting by Pinkall [53] (see also Cecil-Chern [17]- [18] or the books [14, pp. 51-60], [27, pp. 202-212]).
We consider T 1 S n to be the (2n − 1)-dimensional submanifold of given by As shown in the previous section, the points on a line ℓ lying on Q n+1 correspond to the spheres in a parabolic pencil of spheres in S n . In particular, as in equation (30) . Thus we get a bijective correspondence between the points (x, ξ) of T 1 S n and the space Λ 2n−1 of lines on Q n+1 given by the map: where We use this correspondence to place a natural differentiable structure on Λ 2n−1 in such a way as to make the map in equation (35) a diffeomorphism. We now show how to define a contact structure on the manifold T 1 S n . By the diffeomorphism in equation (35), this also determines a contact structure on Λ 2n−1 . Recall that a (2n − 1)-dimensional manifold V 2n−1 is said to be a contact manifold if it carries a globally defined 1-form ω such that at all points of V 2n−1 . Such a form ω is called a contact form. A contact form ω determines a codimension one distribution (the contact distribution) D on V 2n−1 defined by for p ∈ V 2n−1 . This distribution is as far from being integrable as possible, in that there exist integral submanifolds of D of dimension n − 1, but none of higher dimension (see, for example, [14, p. 57]). The distribution D determines the corresponding contact form ω up to multiplication by a nonvanishing smooth function. A tangent vector to T 1 S n at a point (x, ξ) can be written in the form Differentiation of the condition x · ξ = 0 implies that (X, Z) also satisfies Using the method of moving frames, one can show that the form ω defined by is a contact form on T 1 S n (see, for example, Cecil-Chern [17] or the book [14, pp. 52-56]), and we will omit the proof here. At a point (x, ξ), the distribution D is the (2n − 2)-dimensional space of vectors (X, Z) satisfying X · ξ = 0, as well as the equations (39) and (40). The equation X · ξ = 0 together with equation (40) implies that for vectors (X, Z) in D.
Returning to the general theory of contact structures, we let V 2n−1 be a contact manifold with contact form ω and corresponding contact distribution D, as in equation (38). An immersion φ : [4, p. 36].) It is well known (see, for example, [14, p. 57]) that the contact distribution D has integral submanifolds of dimension n−1, but none of higher dimension. These integral submanifolds of maximal dimension are called Legendre submanifolds of the contact structure.
In our specific case, we now formulate conditions for a smooth map µ : M n−1 → T 1 S n to be a Legendre submanifold. We consider T 1 S n as a submanifold of S n × S n as in equation (34), and so we can write µ = (f, ξ), where f and ξ are both smooth maps from M n−1 to S n . We have the following theorem (see [14, p. 58]) giving necessary and sufficient conditions for µ to be a Legendre submanifold. (1) Scalar product conditions: (2) Immersion condition: there is no nonzero tangent vector X at any point x ∈ M n−1 such that df (X) and dξ(X) are both equal to zero.
Note that by equation (34), the scalar product conditions are precisely the conditions necessary for the image of the map µ = (f, ξ) to be contained in T 1 S n . Next, since dµ(X) = (df (X), dξ(X)), Condition (2) is necessary and sufficient for µ to be an immersion. Finally, from equation (41), we see that ω(dµ(X)) = df (X) · ξ(x), for each X ∈ T x M n−1 . Hence Condition (3) is equivalent to the requirement that µ * ω = 0 on M n−1 .
We now want to translate these conditions into the projective setting, and find necessary and sufficient conditions for a smooth map λ : M n−1 → Λ 2n−1 to be a Legendre submanifold. We again make use of the diffeomorphism defined in equation (35) between T 1 S n and Λ 2n−1 .
For each x ∈ M n−1 , we know that λ( The map [Y 1 ] from M n−1 to Q n+1 is called the Möbius projection or point sphere map of λ, and the map [Y n+3 ] from M n−1 to Q n+1 is called the great sphere map. The maps f and ξ are called the spherical projection of λ, and the spherical field of unit normals of λ, respectively. In this way, λ determines a map µ = (f, ξ) from M n−1 to T 1 S n , and because of the diffeomorphism (35), λ is a Legendre submanifold if and only if µ satisfies the conditions of Theorem 6.1.
It is often useful to have conditions for when λ determines a Legendre submanifold that do not depend on the special parametrization of λ in terms of the point sphere and great sphere maps, [Y 1 ] and [Y n+3 ]. In fact, in many applications of Lie sphere geometry to submanifolds of S n or R n , it is better to consider λ = [Z 1 , Z n+3 ], where Z 1 and Z n+3 are not the point sphere and great sphere maps.
Pinkall [53] gave the following projective formulation of the conditions needed for a Legendre submanifold. In his paper, Pinkall referred to a Legendre submanifold as a "Lie geometric hypersurface." The proof that the three conditions of the theorem below are equivalent to the three conditions of Theorem 6.1 can be found in [14, pp. 59-60].
where Z 1 and Z n+3 are smooth maps from M n−1 into R n+3 2 . Then λ determines a Legendre submanifold if and only if Z 1 and Z n+3 satisfy the following conditions.
(1) Scalar product conditions: for each x ∈ M n−1 , the vectors Z 1 (x) and Z n+3 (x) are linearly independent and (2) Immersion condition: there is no nonzero tangent vector X at any point x ∈ M n−1 such that dZ 1 (X) and dZ n+3 (X) are both in (3) Contact condition: dZ 1 , Z n+3 = 0.
Every oriented hypersurface in S n or R n naturally induces a Legendre submanifold of Λ 2n−1 , as does every submanifold of codimension m > 1 in these spaces. Conversely, a Legendre submanifold naturally induces a smooth map into S n or R n , which may have singularities. We now study the details of these maps.
Let f : M n−1 → S n be an immersed oriented hypersurface with field of unit normals ξ : M n−1 → S n . The induced Legendre submanifold is given by the map λ : The map λ is called the Legendre lift of the immersion f with field of unit normals ξ.
To show that λ is a Legendre submanifold, we check the conditions of Theorem 6.2. Condition (1) is satisfied since both f and ξ are maps into S n , and ξ(x) is tangent to S n at f (x) for each x in M n−1 . Since f is an immersion, dY 1 (X) = (0, df (X), 0) is not in Span {Y 1 (x), Y n+3 (x)}, for any nonzero vector X ∈ T x M n−1 , and so Condition (2) is satisfied. Finally, Condition (3) is satisfied since because ξ is a field of unit normals to f .
In the case of a submanifold φ : V → S n of codimension m + 1 greater than one, the domain of the Legendre lift is be the unit normal bundle B n−1 of the submanifold φ(V ). We consider B n−1 to be the submanifold of V × S n given by where Geometrically, λ(x, ξ) is the line on the quadric Q n+1 corresponding to the parabolic parabolic pencil of spheres in S n in oriented contact at the contact element (φ(x), ξ) ∈ T 1 S n . In [14, pp. 61-62], we show that λ satisfies the conditions of Theorem 6.2, and we omit the proof here. Similarly, suppose that F : M n−1 → R n is an oriented hypersurface with field of unit normals η : M n−1 → R n , where we identify R n with the subspace of R n+3 2 spanned by {e 3 , . . . , e n+2 }. The Legendre lift of (F, η) is the map By equation (21), [Y 1 (x)] corresponds to the point sphere and [Y n+3 (x)] corresponds to the hyperplane in the parabolic pencil determined by the line λ(x) for each x ∈ M n−1 . One can easily verify that Conditions (1)-(3) of Theorem 6.2 are satisfied in a manner similar to the spherical case. In the case of a submanifold ψ : V → R n of codimension greater than one, the Legendre lift of ψ is the map λ from the unit normal bundle B n−1 to Λ 2n−1 defined by λ( The verification that the pair {Y 1 , Y n+3 } satisfies conditions (1)-(3) of Theorem 6.2 is similar to that for submanifolds of S n of codimension greater than one, and we omit that proof here also.
Conversely, suppose that λ : M n−1 → Λ 2n−1 is an arbitrary Legendre submanifold. We have seen above that we can parametrize λ as for the spherical projection f and spherical field of unit normals ξ. Both f and ξ are smooth maps, but neither need be an immersion or even have constant rank (see [14, Here [Z 1 (x)] corresponds to the unique point sphere in the parabolic pencil determined by λ(x), and [Z n+3 (x)] corresponds to the unique plane in this pencil. As in the spherical case, the smooth maps F and η need not have constant rank.

Curvature Spheres
To motivate the definition of a curvature sphere we consider the case of an oriented hypersurface f : M n−1 → S n with field of unit normals ξ : M n−1 → S n . (We could consider an oriented hypersurface in R n , but the calculations are simpler in the spherical case.) The shape operator of f at a point x ∈ M n−1 is the symmetric linear transformation A : T x M n−1 → T x M n−1 defined on the tangent space T x M n−1 by the equation The eigenvalues of A are called the principal curvatures, and the corresponding eigenvectors are called the principal vectors. We next recall the notion of a focal point of an immersion. For each real number t, define a map For each x ∈ M n−1 , the point f t (x) lies an oriented distance t along the normal geodesic to Thus, df t (X) equals zero for X = 0 if and only if cot t is a principal curvature of f at x, and X is a corresponding principal vector. Hence, p = f t (x) is a focal point of f at x of multiplicity m if and only if cot t is a principal curvature of multiplicity m at x. Note that each principal curvature κ = cot t, 0 < t < π, produces two distinct antipodal focal points on the normal geodesic with parameter values t and t + π. The oriented hypersphere centered at a focal point p and in oriented contact with f (M n−1 ) at f (x) is called a curvature sphere of f at x. The two antipodal focal points determined by κ are the two centers of the corresponding curvature sphere. Thus, the correspondence between principal curvatures and curvature spheres is bijective. The multiplicity of the curvature sphere is by definition equal to the multiplicity of the corresponding principal curvature. We now formulate the notion of curvature sphere in the context of Lie sphere geometry. As in equation (43), the Legendre lift λ : For each x ∈ M n−1 , the points on the line λ(x) can be parametrized as where f t is given in equation (51) above. By equation (25), the point [K t (x)] in Q n+1 corresponds to the oriented sphere in S n with center f t (x) and signed radius t. This sphere is in oriented contact with the oriented hypersurface f (M n−1 ) at f (x). Given a tangent vector X ∈ T x M n−1 , we have Thus, dK t (X) = (0, 0, 0) for a nonzero vector X ∈ T x M n−1 if and only if df t (X) = 0, i.e., p = f t (x) is a focal point of f at x corresponding to the principal curvature cot t. The vector X is a principal vector corresponding to the principal curvature cot t, and it is also called a principal vector corresponding to the curvature sphere [K t ]. This characterization of curvature spheres depends on the parametrization of λ = [Y 1 , Y n+3 ] given by the point sphere and great sphere maps [Y 1 ] and [Y n+3 ], respectively, and it has only been defined in the case where the spherical projection f is an immersion. We now give a projective formulation of the definition of a curvature sphere that is independent of the parametrization of λ and is valid for an arbitrary Legendre submanifold.
Let λ : M n−1 → Λ 2n−1 be a Legendre submanifold parametrized by the pair {Z 1 , Z n+3 }, as in Theorem 6.2. Let x ∈ M n−1 and r, s ∈ R with at least one of r and s not equal to zero. The sphere, is called a curvature sphere of λ at x if there exists a nonzero vector X in T x M n−1 such that The vector X is called a principal vector corresponding to the curvature sphere [K]. This definition is invariant under a change of parametrization of the form considered in the statement of Theorem 6.2. Furthermore, if we take the special parametrization Z 1 = Y 1 , Z n+3 = Y n+3 given in equation (53), then condition (56) holds if and only if r dY 1 (X) + s dY n+3 (X) actually equals (0, 0, 0). From equation (56), it is clear that the set of principal vectors corresponding to a given curvature sphere [K] at x is a subspace of T x M n−1 . This set is called the principal space corresponding to the curvature sphere [K]. Its dimension is the multiplicity of [K]. The reader is referred to Cecil-Chern [17]- [18] for a development of the notion of a curvature sphere in the context of Lie sphere geometry, without beginning with submanifolds of S n or R n .
We next show that a Lie sphere transformation maps curvature spheres to curvature spheres. We first need to discuss the notion of Lie equivalent Legendre submanifolds. Let λ : M n−1 → Λ 2n−1 be a Legendre submanifold parametrized by λ = [Z 1 , Z n+3 ]. Suppose β = P (B) is the Lie sphere transformation induced by an orthogonal transformation B in the group O(n + 1, 2). Since B is orthogonal, the maps, W 1 = BZ 1 , W n+3 = BZ n+3 , satisfy the Conditions (1)-(3) of Theorem 6.2, and thus γ = [W 1 , W n+3 ] is a Legendre submanifold which we denote by βλ : M n−1 → Λ 2n−1 . We say that the Legendre submanifolds λ and βλ are Lie equivalent. In terms of submanifolds of real space forms, we say that two immersed submanifolds of R n or S n are Lie equivalent if their Legendre lifts are Lie equivalent. Proof. Let λ = [Z 1 , Z n+3 ] and βλ = [W 1 , W n+3 ] as above. For a tangent vector X ∈ T x M n−1 and real numbers r and s, at least one of which is not zero, we have r dW 1 (X) + s dW n+3 (X) = r d(BZ 1 )(X) + s d(BZ n+3 )(X) (57) = B(r dZ 1 (X) + s dZ n+3 (X)), since B is a constant linear transformation. Thus, we see that We next consider the case when the Lie sphere transformation β is a spherical parallel transformation P t defined in terms of the standard basis of R n+3 2 by P t e 1 = cos t e 1 + sin t e n+3 , P t e n+3 = − sin t e 1 + cos t e n+3 , The transformation P t has the effect of adding t to the signed radius of each oriented sphere in S n while keeping the center fixed (see, for example, [14, pp. 48-49]).
From this, we see that P t λ has spherical projection and spherical unit normal field given, respectively, by The minus sign occurs because P t takes a sphere with center f −t (x) and radius −t to the point sphere f −t (x). We call P t λ a parallel submanifold of λ. Formula (61) shows the close correspondence between these parallel submanifolds and the parallel hypersurfaces f t to f , in the case where f is an immersed hypersurface.
In the case where the spherical projection f is an immersion at a point x ∈ M n−1 , we know that the number of values of t in the interval [0, π) for which f t is not an immersion is at most n − 1, the maximum number of distinct principal curvatures of f at x. Pinkall [53, p. 428] proved that this statement is also true for an arbitrary Legendre submanifold, even if the spherical projection f is not an immersion at x by proving the following theorem (see also [14, pp. 68-72] for a proof). As a consequence of Pinkall's theorem, one can pass to a parallel submanifold, if necessary, to obtain the following important corollary by using well known results concerning immersed hypersurfaces in S n . Note that parts

Dupin Submanifolds
We now recall some basic concepts from the theory of Dupin hypersurfaces in S n (see, for example, [27, pp. 9-35] for more detail), and then generalize the notion of Dupin to Legendre submanifolds in Lie sphere geometry. Let f : M → S n be an immersed hypersurface, and let ξ be a locally defined field of unit normals to f (M). A curvature surface of M is a smooth submanifold S ⊂ M such that for each point x ∈ S, the tangent space T x S is equal to a principal space (i.e., an eigenspace) of the shape operator A of M at x. This generalizes the classical notion of a line of curvature for a principal curvature of multiplicity one. The hypersurface M is said to be Dupin if: (a) along each curvature surface, the corresponding principal curvature is constant.
Furthermore, a Dupin hypersurface M is called proper Dupin if, in addition to 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, and so are those hypersurfaces in R n obtained from isoparametric hypersurfaces in S n via stereographic projection (see, for example, [27, pp. 28-30]). In particular, the well-known ring cyclides of Dupin in R 3 are obtained in this way from a standard product torus S 1 (r) × S 1 (s) in S 3 , where r 2 + s 2 = 1.
Using the Codazzi equation, one can show that if a curvature surface S has dimension greater than one, then the corresponding principal curvature is constant on S. This is not necessarily true on a curvature surface of dimension equal to one (i.e., a line of curvature).
Second, Condition (b) is equivalent to requiring that each continuous principal curvature function has constant multiplicity on M. Further, for any hypersurface M in S n , there exists a dense open subset of M on which the number of distinct principal curvatures is locally constant (see, for example, Singley [55]).
It also follows from the Codazzi equation that if a continuous principal curvature function µ has constant multiplicity m on a connected open subset U ⊂ M, then µ is a smooth function on U, and the distribution T µ of principal spaces corresponding to µ is a smooth foliation whose leaves are the curvature surfaces corresponding to µ on U. This principal curvature function µ is constant along each of its curvature surfaces in U if and only if these curvature surfaces are open subsets of m-dimensional great or small spheres in S n (see [27, pp. 24-32]).
We can generalize the notion of a curvature surface for hypersurfaces in real space forms to Legendre submanifolds. Specifically, let λ : M n−1 → Λ 2n−1 be a Legendre submanifold. A connected submanifold S of M n−1 is called a curvature surface if at each x ∈ S, the tangent space T x S is equal to some principal space T i , as in Corollary 7.1. For example, if dim T i is constant on an open subset U of M n−1 , then each leaf of the principal foliation T i is a curvature surface on U.
There exist many examples of Dupin hypersurfaces in S n or R n that are not proper Dupin, because the number of distinct principal curvatures is not constant on the hypersurface. This also results in curvature surfaces that are not leaves of a principal foliation. An example due to Pinkall [53] is a tube M 3 in R 4 of constant radius over a torus of revolution T 2 ⊂ R 3 ⊂ R 4 (see also [14, p. 69] for a description of Pinkall's example).
One consequence of the results mentioned above is that proper Dupin hypersurfaces in S n or R n are algebraic, as is the case with isoparametric hypersurfaces, as shown by Münzner [46]- [47]. This result is most easily formulated for hypersurfaces in R n . It states that a connected proper Dupin hypersurface f : M → R n must be contained in a connected component of an irreducible algebraic subset of R n of dimension n − 1. Pinkall [51] sent the author a letter in 1984 that contained a sketch of a proof of this result, but he did not publish a proof. In 2008, Cecil, Chi and Jensen [21] used methods of real algebraic geometry to give a proof of this result based on Pinkall's sketch. The proof makes use of the various principal foliations whose leaves are open subsets of spheres to construct an analytic algebraic parametrization of a neighborhood of f (x) for each point x ∈ M. In contrast to the situation for isoparametric hypersurfaces, however, a connected proper Dupin hypersurface does not necessarily lie in a compact connected proper Dupin hypersurface, as Pinkall's example mentioned above of a tube M 3 in R 4 of constant radius over a torus of revolution T 2 ⊂ R 3 ⊂ R 4 shows.
Next we generalize the definition of a Dupin hypersurface in a real space form to the setting of Legendre submanifolds in Lie sphere geometry. We say that a Legendre submanifold λ : M n−1 → Λ 2n−1 is a Dupin submanifold if:

Lifts of Isoparametric Hypersurfaces
In this section, we give a Lie sphere geometric characterization of the Legendre lifts of isoparametric hypersurfaces in the sphere S n (Theorem 9.3). This result has been used in several papers to prove that under certain conditions a proper Dupin submanifold is Lie equivalent to the Legendre lift of an isoparametric hypersurface.
Let λ : M n−1 → Λ 2n−1 be an arbitrary Legendre submanifold. As before, where f and ξ are the spherical projection and spherical field of unit normals, respectively. For x ∈ M n−1 , the points on the line λ(x) can be written in the form, that is, we take µ as an inhomogeneous coordinate along the projective line λ(x). Then the point sphere [Y 1 ] corresponds to µ = ∞. The next two theorems give the relationship between the coordinates of the curvature spheres of λ and the principal curvatures of f , in the case where f has constant rank.
In the first theorem, we assume that the spherical projection f is an immersion on M n−1 . By Theorem 7.2, we know that this can always be achieved locally by passing to a parallel submanifold.
Theorem 9.1. Let λ : M n−1 → Λ 2n−1 be a Legendre submanifold whose spherical projection f : M n−1 → S n is an immersion. Let Y 1 and Y n+3 be the point sphere and great sphere maps of λ as in equation (62). Then the curvature spheres of λ at a point x ∈ M n−1 are Recall that the point sphere and great sphere maps for λ are given as in equation (45) by (64) where κ 1 , . . . , κ g−1 are the distinct principal curvatures of the shape operator A ξ , and κ g = ∞. For 1 ≤ i ≤ g − 1, the multiplicity of the curvature sphere [K i ] equals the multiplicity of the principal curvature κ i , while the multiplicity The proof of this theorem is similar to that of Theorem 9.1, but one must introduce local coordinates on the unit normal bundle to get a complete proof (see [14, p. 74]).
We close this section with a local Lie geometric characterization of Legendre submanifolds that are Lie equivalent to the Legendre lift of an isoparametric hypersurface in S n (see [12] or [14, p. 77]). Here a line in P n+2 is called timelike if it contains only timelike points. This means that an orthonormal basis for the 2-plane in R n+3 2 determined by the timelike line consists of two timelike vectors. An example is the line [e 1 , e n+3 ]. This theorem has been useful in obtaining several classification results for proper Dupin hypersurfaces.
Proof. If λ is the Legendre lift of an isoparametric hypersurface in S n , then all the spheres in a family [K i ] have the same radius ρ i , where 0 < ρ i < π. By formula (25), this is equivalent to the condition K i , P i = 0, where are g points on the timelike line [e 1 , e n+3 ] (see [14, pp. 17-18] (25), this means that the spheres corresponding to β[K i ] have constant radius on M n−1 . By applying a parallel transformation P t , if necessary, we can arrange that none of these curvature spheres has radius zero. Then P t βλ is the Legendre lift of an isoparametric hypersurface in S n .

Cyclides of Dupin
The classical cyclides of Dupin in R 3 were studied intensively by many leading mathematicians in the nineteenth century, including Liouville [40], Cayley [11], and Maxwell [41], whose paper contains stereoscopic figures of the various types of cyclides. A good account of the history of the cyclides in the nineteenth century is given by Lilienthal [39] (see also Klein [36, [3], and Cecil and Ryan [26, pp. 151-166]).
We now turn our attention to Pinkall's classification of the cyclides of Dupin of arbitrary dimension, which is obtained by using the methods of Lie sphere geometry. Our presentation here is based on the accounts of this subject given in [14, pp. 148-159] and [27, pp. 263-283]. A proper Dupin submanifold λ : M n−1 → Λ 2n−1 with two distinct curvature spheres of respective multiplicities p and q at each point is called a cyclide of Dupin of characteristic (p, q).
We will prove that any connected cyclide of Dupin of characteristic (p, q) is contained in a unique compact, connected cyclide of Dupin of characteristic (p, q). Furthermore, every compact, connected cyclide of Dupin of characteristic (p, q) is Lie equivalent to the Legendre lift of a standard product of two spheres, where p and q are positive integers such that p + q = n − 1. Thus any two compact, connected cyclides of Dupin of the same characteristic are Lie equivalent.
It is well known that the product S q (1/ √ 2)×S p (1/ √ 2) is an isoparametric hypersurface in S n with two distinct principal curvatures having multiplicities m 1 = p and m 2 = q (see, for example, [27, pp. 110-111]). Furthermore, every compact isoparametric hypersurface in S n with two principal curvatures of multiplicities p and q is Lie equivalent is a good model for the cyclides, it is often easier to work with the two focal submanifolds S q (1)×{0} and {0}×S p (1) in proving classification results. The Legendre lifts of these two focal submanifolds are Lie equivalent to the Legendre lift of S q (1/ √ 2) × S p (1/ √ 2), since they are parallel submanifolds of the Legendre lift of is a tube of spherical radius π/4 in S n over either of its two focal submanifolds.
We now describe our standard model of a cyclide of characteristic (p, q) in the context of Lie sphere geometry, as in Pinkall's paper [53] (see also [14, p. 149]). Let {e 1 , . . . , e n+3 } be the standard orthonormal basis for R n+3 2 , with e 1 and e n+3 unit timelike vectors, and {e 2 , . . . , e n+2 } unit spacelike vectors. Then S n is the unit sphere in the Euclidean space R n+1 spanned by These spaces have signatures (q + 1, 1) and (p + 1, 1), respectively. The intersection Ω ∩ Q n+1 is the quadric given in homogeneous coordinates by This set is diffeomorphic to the unit sphere S q in Similarly, the quadric Ω ⊥ ∩ Q n+1 is diffeomorphic to the unit sphere S p in R p+1 = Span {e q+3 , . . . , e n+2 } by the diffeomorphism ψ : The model that we will use for the cyclides in Lie sphere geometry is the Legendre submanifold λ : It is easy to check that the Legendre Conditions (1)-(3) of Theorem 6.2 are satisfied by the pair {k 1 , k 2 }. To find the curvature spheres of λ, we decompose the tangent space to S p × S q at a point (u, v) as Then dk 1 (X, 0) = 0 for all X ∈ T u S p , and dk 2 (Y ) = 0 for all Y in T v S q . Thus, [k 1 ] and [k 2 ] are curvature spheres of λ with respective multiplicities p and q. Furthermore, the image of [k 1 ] lies in the quadric Ω ∩ Q n+1 , and the image of [k 2 ] is contained in the quadric Ω ⊥ ∩ Q n+1 . The point sphere map of λ is [k 1 ], and thus λ is the Legendre lift of the focal submanifold S q × {0} ⊂ S n , considered as a submanifold of codimension p + 1 in S n . As noted above, this Legendre lift λ of the focal submanifold is Lie equivalent to the Legendre lift of the standard product of spheres by means of a parallel transformation.
We now prove Pinkall's [53] classification of proper Dupin submanifolds with two distinct curvature spheres at each point. Pinkall's proof depends on establishing the existence of a local principal coordinate system. This can always be done in the case of g = 2 curvature spheres, because the the sum of the dimensions of the two principal spaces is n − 1, the dimension of M n−1 (see, for example, [27, p. 249]). Such a coordinate system might not exist in the case g > 2. In fact, if M is an isoparametric hypersurface in S n with more than two distinct principal curvatures, then there cannot exist a local principal coordinate system on M (see, for example, [26, pp. 180-184] or [27, pp. 248-252]).
For a different proof of Pinkall's theorem (Theorem 10.1 below) using the method of moving frames, see the paper of Cecil-Chern [18] or [27, pp. 266-273]. That approach generalizes to the study of proper Dupin hypersurfaces with g > 2 curvature spheres (see, for example, Cecil and Jensen [22], [23]).
Note that before Pinkall's paper, Cecil and Ryan [24]- [25] (see also [26, pp. 166-179]) proved a classification of complete cyclides in R n using techniques of Euclidean submanifold theory. However, the proof used the assumption of completeness in an essential way, and that theorem did not contain part (a) of Pinkall's Theorem 10.1 below.
where p + q = n − 1. Thus, any two compact, connected cyclides of Dupin of characteristic (p, q) are Lie equivalent.
Proof. Suppose that λ : M n−1 → Λ 2n−1 is a connected cyclide of Dupin of characteristic (p, q) with p + q = n − 1. We may take λ = [k 1 , k 2 ], where [k 1 ] and [k 2 ] are the curvature spheres with respective multiplicities p and q. Each curvature sphere map factors through an immersion of the space of leaves of its principal foliation T i for i = 1, 2. Since the sum of the dimensions of T 1 and T 2 equals the dimension of M n−1 , locally we can take a principal coordinate system (u, v) (see, for example, [27, p. 249 Now let (u, v) and (ū,v) be any two points in W . From (i), we have the following key equation,

]) defined on an open set
The dimensions of E and E ⊥ as subspaces of P n+2 satisfy dim E + dim E ⊥ = n + 1 = p + q + 2.
We claim that dim E = q + 1 and dim E ⊥ = p + 1. To see this, suppose first that dim E > q + 1. Then dim E ⊥ ≤ p, and E ⊥ ∩ Q n+1 cannot contain the p-dimensional submanifold k 2 (W ), contradicting equation (72). Similarly, assuming that dim E ⊥ > p + 1 leads to a contradiction, since then dim E ≤ q, and E ∩ Q n+1 cannot contain the q-dimensional submanifold k 1 (W ).
Then E ∩ Q n+1 is given in homogeneous coordinates (x 1 , . . . , x n+3 ) with respect to this basis by This quadric is diffeomorphic to the unit sphere S q in the span R q+1 of the spacelike vectors w 2 , . . . , w q+2 with the diffeomorphism γ : S q → E ∩ Q n+1 given by Similarly E ⊥ ∩ Q n+1 is the quadric given in homogeneous coordinates by This space E ⊥ ∩Q n+1 is diffeomorphic to the unit sphere S p in the span R p+1 of the spacelike vectors w q+3 , . . . , w n+2 with the diffeomorphism given by The image of the curvature sphere map k 1 of multiplicity p is contained in the q-dimensional quadric E ∩ Q n+1 given by equation (75), which is diffeomorphic to S q . The map k 1 is constant on each leaf of its principal foliation T 1 , and so k 1 factors through an immersion of the q-dimensional space of leaves W/T 1 into the q-dimensional quadric E ∩ Q n+1 . Hence, the image of k 1 is an open subset of this quadric, and each leaf of T 1 corresponds to a point γ(v) of the quadric.
Similarly, the curvature sphere map k 2 of multiplicity q factors through an immersion of its p-dimensional space of leaves W/T 2 onto an open subset of the p-dimensional quadric E ⊥ ∩ Q n+1 given by equation (77), and each leaf of T 2 corresponds to a point of δ(u) of that quadric.
From this it is clear that the restriction of the Legendre map λ to the neighborhood W ⊂ M is contained in the compact, connected cyclide where for the maps γ and δ defined above. By a standard connectedness argument, the Legendre map λ : M → Λ 2n−1 is also the restriction of the compact, connected cyclide ν to an open subset of S p × S q . This proves part (a) of the theorem.
In projective space P n+2 , the image of ν consists of all lines joining a point on the quadric E ∩Q n+1 in equation (75) to a point on the quadric E ⊥ ∩Q n+1 in equation (77). Thus any choice of a (q + 1)-plane E in P n+2 with signature (q + 1, 1) and corresponding orthogonal complement E ⊥ with signature (p + 1, 1) determines a unique compact, connected cyclide of characteristic (p, q) and vice-versa.
The Lie equivalence of any two compact, connected cyclides of the same characteristic stated in part (b) of the theorem is then clear, since given any two choices E and F of (q + 1)-planes in P n+2 with signature (q + 1, 1), there is a Lie sphere transformation that maps E to F and E ⊥ to F ⊥ .
In particular, if we take F to be the space Ω in equation (67), then the corresponding cyclide is our standard model. So our given compact, connected cyclide ν in equation (79) is Lie equivalent to our standard model. As noted before the statement of Theorem 10.1, our standard model is Lie equivalent to the Legendre lift of the standard product of two spheres, where p + q = n − 1, via parallel transformation. Thus, any compact, connected cyclide of Dupin of characteristic (p, q) is Lie equivalent to the Legendre lift of a standard product of two spheres given in equation (81).
Remark 10.1. We can also see that the submanifold λ in Theorem 10.1 is Lie equivalent to the Legendre lift of an isoparametric hypersurface in S n with two principal curvatures by invoking Theorem 9.3, because the two curvature sphere maps [k 1 ] and [k 2 ] are orthogonal to the two points w n+3 and w 1 , respectively, on the timelike line [w 1 , w n+3 ] in P n+2 .
Theorem 10.1 is a classification of the cyclides of Dupin in the context of Lie sphere geometry. It is also useful to have a Möbius geometric classification of the cyclides of Dupin M n−1 ⊂ R n . This is analogous to the classical characterizations of the cyclides of Dupin in R 3 obtained in the nineteenth century (see, for example, [26, pp. 151-166]). K. Voss [59] announced the classification in Theorem 10.2 below for the higher-dimensional cyclides, but he did not publish a proof. The theorem follows quite directly from Theorem 10.1 and known results on surfaces of revolution.
The theorem is phrased in terms embedded hypersurfaces in R n . Thus we are excluding the standard model given in equation (69), where the spherical projection (and thus the Euclidean projection) is not an immersion. Of course, the spherical projections of all parallel submanifolds of the standard model in the sphere are embedded isoparametric hypersurfaces in the sphere S n , except for the Legendre lift of the other focal submanifold. The proof of the following theorem using techniques of Lie and Möbius geometry was given in [13], and later versions of the proof together with computer graphic illustrations of the cyclides are given in [14, pp. 151-159] and [27, pp. 273-281]. These proofs use same notation that we have been using in this section. We omit the proof here and refer the reader to these two references.
Theorem 10.2. (a) Every connected cyclide of Dupin M n−1 ⊂ R n of characteristic (p, q) is Möbius equivalent to an open subset of a hypersurface of revolution obtained by revolving a q-sphere S q ⊂ R q+1 ⊂ R n about an axis R q ⊂ R q+1 or a p-sphere S p ⊂ R p+1 ⊂ R n about an axis R p ⊂ R p+1 .
(b) Two hypersurfaces obtained by revolving a q-sphere S q ⊂ R q+1 ⊂ R n about an axis of revolution R q ⊂ R q+1 are Möbius equivalent if and only if they have the same value of ρ = |r|/a, where r is the signed radius of the profile sphere S q and a > 0 is the distance from the center of S q to the axis of revolution.
Remark 10.2. Note that in this theorem, the profile sphere S q ⊂ R q+1 ⊂ R n is allowed to intersect the axis of revolution R q ⊂ R q+1 , in which case the hypersurface of revolution has singularities in R n . Under Möbius transformation, this leads to cyclides which have Euclidean singularities, such as the classical horn cyclides and spindle cyclides (see, for example, [14, pp. 151-159] for more detail). In such cases, however, the corresponding Legendre map λ : S p × S q → Λ 2n−1 is still an immersion.

Local Constructions
Pinkall [53] introduced four constructions for obtaining a Dupin hypersurface W in R n+m from a Dupin hypersurface M in R n . We first describe these constructions in the case m = 1 as follows.
Begin with a Dupin hypersurface M n−1 in R n and then consider R n as the linear subspace R n × {0} in R n+1 . The following constructions yield a Dupin hypersurface W n in R n+1 .
(1) Let W n be the cylinder M n−1 × R in R n+1 .
(2) Let W n be the hypersurface in R n+1 obtained by rotating M n−1 around an axis (a linear subspace) R n−1 ⊂ R n .
(3) Let W n be a tube of constant radius in R n+1 around M n−1 .
(4) Project M n−1 stereographically onto a hypersurface V n−1 ⊂ S n ⊂ R n+1 . Let W n be the cone over V n−1 in R n+1 .
In general, these constructions introduce a new principal curvature of multiplicity one which is constant along its lines of curvature. The other principal curvatures are determined by the principal curvatures of M n−1 , and the Dupin property is preserved for these principal curvatures. These constructions can be generalized to produce a new principal curvature of multiplicity m by considering R n as a subset of R n ×R m rather than R n ×R. (See [14, pp. 125-148] for a detailed description of these constructions in full generality in the context of Lie sphere geometry.) Although Pinkall gave these four constructions, his Theorem 4 [53, p. 438] showed that the cone construction is redundant, since it is Lie equivalent to a tube. (See the proof of Theorem 11.2 and Remark 11.1 below.) For this reason, we will only study three standard constructions: tubes, cylinders and surfaces of revolution.
A Dupin submanifold obtained from a lower-dimensional Dupin submanifold via one of these standard constructions is said to be reducible. More generally, a Dupin submanifold which is locally Lie equivalent to such a Dupin submanifold is called reducible.
Using these constructions, Pinkall was able to produce a proper Dupin hypersurface in Euclidean space with an arbitrary number of distinct principal curvatures, each with any given multiplicity (see Theorem 11.1 below). In general, these proper Dupin hypersurfaces cannot be extended to compact Dupin hypersurfaces without losing the property that the number of distinct principal curvatures is constant, as we will discuss after the proof of the theorem. To get a proper Dupin hypersurface in R 5 with three principal curvatures having respective multiplicities m 1 = m 2 = 1, m 3 = 2, one simply takes for the set U above. To obtain a proper Dupin hypersurface M 4 in R 5 with four principal curvatures, first invert the hypersurface M 3 above in a 3-sphere in R 4 , chosen so that the image of M 3 contains an open subset W 3 on which no principal curvature vanishes. The hypersurface W 3 is proper Dupin, since the proper Dupin property is preserved by Möbius transformations. Now take M 4 to be the cylinder W × R in R 4 × R = R 5 .
In general, there are problems in trying to produce compact proper Dupin hypersurfaces by using these constructions. We now examine some of the problems involved with the the cylinder, surface of revolution, and tube constructions individually (see [14, pp. 127-141] for more details).
For the cylinder construction, the new principal curvature of W n is identically zero, while the other principal curvatures of W n are equal to those of M n−1 . Thus, if one of the principal curvatures µ of M n−1 is zero at some points but not identically zero, then the number of distinct principal curvatures is not constant on W n , and so W n is Dupin but not proper Dupin.
For the surface of revolution construction, if the focal point corresponding to a principal curvature µ at a point x of the profile submanifold M n−1 lies on the axis of revolution R n−1 , then the principal curvature of W n at x determined by µ is equal to the new principal curvature of W n resulting from the surface of revolution construction. Thus, if the focal point of M n−1 corresponding to µ lies on the axis of revolution for some but not all points of M n−1 , then W n is not proper Dupin.
If W n is a tube in R n+1 of radius ε over M n−1 , then there are exactly two distinct principal curvatures at the points in the set M n−1 × {±ε} in W n , regardless of the number of distinct principal curvatures on M n−1 . Thus, W n is not a proper Dupin hypersurface unless the original hypersurface M n−1 is totally umbilic, i.e., it has only one distinct principal curvature at each point.
Another problem with these constructions is that they may not yield an immersed hypersurface in R n+1 . In the tube construction, if the radius of the tube is the reciprocal of one of the principal curvatures of M n−1 at some point, then the constructed object has a singularity. For the surface of revolution construction, a singularity occurs if the profile submanifold M n−1 intersects the axis of revolution.
Many of the issues mentioned in the preceding paragraphs can be resolved by working in the context of Lie sphere geometry and considering Legendre lifts of hypersurfaces in Euclidean space (see [14, pp. 127-148]). In that context, a proper Dupin submanifold λ : M n−1 → Λ 2n−1 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.
Pinkall [53] found the following useful characterization of reducibility in the context of Lie sphere geometry. For simplicity, we deal with the constructions as they are written at the beginning of this section, i.e., we take the case where the multiplicity of the new principal curvature is m = 1. Here we give Pinkall's proof [53, p. 438] (see also [14, pp. 143-144]). Proof. We first note that the following manifolds of spheres are hyperplane sections of the Lie quadric Q n+1 : a) the hyperplanes in R n , b) the spheres with a fixed signed radius r, c) the spheres that are orthogonal to a fixed sphere.
To see this, we use the Lie coordinates given in equation (21). In Case a), the hyperplanes are characterized by the equation x 1 + x 2 = 0, which clearly determines a hyperplane section of Q n+1 . In Case b), the spheres with signed radius r are determined by the linear equation In Case c), it can be assumed that the fixed sphere is a hyperplane H through the origin in R n . A sphere is orthogonal to H if and only if its center lies in H. This clearly imposes a linear condition on the vector in equation (21) representing the sphere.
The sets a), b), c) are each of the form with w, w = 0, −1, 1 in Cases a), b), c), respectively.
We can now see that every reducible Dupin hypersurface has a family of curvature spheres that is contained in two hyperplane sections of the Lie quadric as follows.
For the cylinder construction, the tangent hyperplanes of the cylinder are curvature spheres that are orthogonal to a fixed hyperplane in R n . Thus, that family of curvature spheres is contained in a n-dimensional linear subspace E of P n+2 such that the signature of , on the polar subspace E ⊥ of E is (0, +).
For the surface of revolution construction, the new family of curvature spheres all have their centers in the axis of revolution, which is a linear subspace of codimension 2 in R n . Thus, that family of curvature spheres is contained in a n-dimensional linear subspace E of P n+2 such that the signature of , on the polar subspace E ⊥ of E is (+, +).
For the tube construction, the new family of curvature spheres all have the same radius, and their centers all lie in the hyperplane of R n containing the manifold over which the tube is constructed. Thus, that family of curvature spheres is contained in a n-dimensional linear subspace E of P n+2 such that the signature of , on the polar subspace E ⊥ of E is (−, +).
Conversely, suppose that K : W n−1 → P n+2 is a family of curvature spheres that is contained in an n-dimensional linear subspace E of P n+2 . Then , must have signature (+, +), (0, +) or (−, +) on the polar subspace E ⊥ , because otherwise E ∩ Q n+1 would be empty or would consist of a single point.
If the signature of E ⊥ is (+, +), then there exists a Lie sphere transformation A which takes E to a space F = A(E) such that F ∩ Q n+1 consists of all spheres that have their centers in a fixed (n − 2)-dimensional linear subspace R n−2 of R n . Since one family of curvature spheres of this Dupin submanifold Aλ lies in F ∩ Q n+1 , and the Dupin submanifold Aλ is the envelope of these spheres, Aλ must be a surface of revolution with the axis R n−2 (see [14, pp. 142-143] for more detail on envelopes of families of spheres in this situation), and so λ is reducible.
If the signature of E ⊥ is (0, +), then there exists a Lie sphere transformation A which takes E to a space F = A(E) such that F ∩ Q n+1 consists of hyperplanes orthogonal to a fixed hyperplane in R n . Since one family of curvature spheres of this Dupin submanifold Aλ lies in F ∩ Q n+1 , and the Dupin submanifold Aλ is the envelope of these spheres, Aλ is obtained as a result of the cylinder construction, and so λ is reducible.
If the signature of E ⊥ is (−, +), then there exists a Lie sphere transformation A which takes E to a space F = A(E) such that F ∩ Q n+1 consists of spheres that all have the same radius and whose centers lie in a hyperplane R n−1 of R n . Since one family of curvature spheres of this Dupin submanifold Aλ lies in F ∩ Q n+1 , and the Dupin submanifold Aλ is the envelope of these spheres, Aλ is obtained as a result of the tube construction, and so λ is reducible.
Remark 11.1. Note that for the cone construction (4) at the beginning of this section, the new family [K] of curvature spheres consists of hyperplanes through the origin that are tangent to the cone along the rulings. In the Lie coordinates (21), the origin corresponds to the point [e 1 + e 2 ], while the hyperplanes are orthogonal to the improper point [e 1 − e 2 ]. Thus, the hyperplanes through the origin correspond by equation (21) to points in the linear subspace E, whose orthogonal complement E ⊥ is spanned {e 1 + e 2 , e 1 − e 2 }. This space E ⊥ is also spanned by {e 1 , e 2 }, and so the signature of E ⊥ is (−, +), the same as for the tube construction. Therefore, the cone construction and the tube construction are Lie equivalent. (See Remark 5.13 of [14, p. 144] for more detail.) Finally, there is one more geometric interpretation of the tube construction. Note that a family [K] of curvature spheres that lies in a linear subspace whose orthogonal complement has signature (−, +) can also be considered to consist of spheres in S n of constant radius in the spherical metric whose centers lie in a hyperplane. The corresponding proper Dupin submanifold can thus be considered to be a tube in the spherical met-ric over a lower-dimensional submanifold that lies in a hyperplane section of S n .
As we noted after the proof of Theorem 11.1, there are difficulties in constructing compact proper Dupin hypersurfaces by using Pinkall's constructions. We can construct a reducible compact proper Dupin hypersurface with two principal curvatures by revolving a circle C in R 3 about an axis R 1 ⊂ R 3 that is disjoint from C to obtain a torus of revolution. Of course, this can be generalized to higher dimensions, as in Theorem 10.2, by revolving a q-sphere S q ⊂ R q+1 ⊂ R n about an axis R q ⊂ R q+1 to obtain a compact cyclide of Dupin of characteristic (p, q), where p + q = n − 1. Such a cyclide has two principal curvatures at each point having respective multiplicities p and q.
However, Cecil, Chi and Jensen [19] (see also [14, pp. 146-147]) showed that every compact proper Dupin hypersurface with more than two principal curvatures is irreducible, as stated in the following theorem. The proof uses known facts about the topology of a compact proper Dupin hypersurface and the topology of a compact hypersurface obtained by one of Pinkall's constructions (see [19] or [14, pp. 146-148] for a complete proof).

Classifications of Dupin Hypersurfaces
In this section, we discuss classification results concerning proper Dupin hypersurfaces in R n or S n that have been obtained using the techniques of Lie sphere geometry. These primarily concern two important classes: compact proper Dupin hypersurfaces, and irreducible proper Dupin hypersurfaces. Of course, Theorem 11.3 shows there is a strong connection between these two classes of hypersurfaces, and many classifications of compact proper Dupin hypersurfaces with g ≥ 3 principal curvatures have been obtained by assuming that the hypersurface is irreducible and working locally in the context of Lie sphere geometry using the method of moving frames. (See, for example, the papers of Pinkall [50], [52]- [53], Cecil and Chern [18], Cecil and Jensen [22]- [23], and Cecil, Chi and Jensen [19].) See also [16] for a survey of classifications of compact proper Dupin hypersurfaces.
Two key tools in many of these classifications are: 1) the Lie sphere geometric characterization of Legendre lifts of isoparametric hypersurfaces given in Theorem 9.3, 2) Pinkall's characterization of reducible proper Dupin hypersurfaces given in Theorem 11.2.
We now summarize these classifications and give references to their proofs.
We begin by recalling some important facts about compact proper Dupin hypersurfaces embedded in S n . Following Münzner's work [46]- [47] on isoparametric hypersurfaces, Thorbergsson [57] proved the following theorem which shows that Münzner's restriction on the number g of distinct principal curvatures of an isoparametric hypersurface also holds for compact proper Dupin hypersurfaces embedded in S n . This is in stark contrast to Pinkall's Theorem 11.1 which states that there are no restrictions on the number of distinct principal curvatures or their multiplicities for non-compact proper Dupin hypersurfaces.
In proving this theorem, Thorbergsson first shows that a compact, connected proper Dupin hypersurface M ⊂ S n must be tautly embedded, that is, every nondegenerate spherical distance function L p (x) = d(p, x) 2 , for p ∈ S n , has the minimum number of critical points required by the Morse inequalities on M. Thorbergsson then uses the fact that M is tautly embedded in S n to show that M divides S n into two ball bundles over the first focal submanifolds, M + and M − , on either side of M in S n . This gives the same topological situation as in the isoparametric case, and the theorem then follows from Münzner's [47] proof of the restriction on g for isoparametric hypersurfaces.
The topological situation that M divides S n into two ball bundles over the first focal submanifolds, M + and M − , on either side of M in S n also leads to important restrictions on the multiplicities of the principal curvatures of compact proper Dupin hypersurfaces, due to Stolz [56] for g = 4, and to Grove and Halperin [34] for g = 6. These restrictions were obtained by using advanced topological considerations in each case, and they show that the multiplicities of the principal curvatures of a compact proper Dupin hypersurface embedded in S n must be the same as the multiplicities of the principal curvatures of some isoparametric hypersurface in S n .
Grove and Halperin [34] also gave a list of the integral homology of all compact proper Dupin hypersurfaces, and Fang [31] found results on the topology of compact proper Dupin hypersurfaces with g = 6 principal curvatures.
In 1985, it was known that every compact, connected proper Dupin hypersurface M ⊂ S n (or R n ) with g = 1, 2 or 3 principal curvatures is Lie equivalent to an isoparametric hypersurface in S n . At that time, every other known example of a compact, connected proper Dupin hypersurface in S n was also Lie equivalent to an isoparametric hypersurface in S n . This together with Thorbergsson's Theorem 12.1 above led to the following conjecture by Cecil and Ryan [26, p. 184] (which we have rephrased slightly). We now discuss the state of the conjecture for each of the values of g. The case g = 1 is simply the case of totally umbilic hypersurfaces, and M is a great or small hypersphere in S n . In the case g = 2, Cecil and Ryan [24] showed that M is a cyclide of Dupin (see Section 10), and thus it is Möbius equivalent to a standard product of spheres S p (r) × S n−1−p (s) ⊂ S n (1) ⊂ R n+1 , r 2 + s 2 = 1, which is an isoparametric hypersurface.
In the case g = 3, Miyaoka [42] proved that M is Lie equivalent to an isoparametric hypersurface (see also Cecil-Chi-Jensen [19] for a different proof using the fact that compactness implies irreducibility, i.e., Theorem 11.3). Earlier, Cartan [7] had shown that an isoparametric hypersurface with g = 3 principal curvatures is a tube over a standard embedding of a projective plane FP 2 , for F = R, C, H (quaternions) or O (Cayley numbers) in S 4 , S 7 , S 13 and S 25 , respectively. For F = R, a standard embedding is a spherical Veronese surface (see also [27, pp. 151-155]).
All attempts to verify Conjecture 12.1 in the cases g = 4 and 6 were unsuccessful, however. Finally, in 1988, Pinkall and Thorbergsson [54] and Miyaoka and Ozawa [45] gave two different methods for producing counterexamples to Conjecture 12.1 with g = 4 principal curvatures. The method of Miyaoka and Ozawa also yields counterexamples to the conjecture in the case g = 6.
These examples were shown to be counterexamples to the conjecture by a consideration of their Lie curvatures, which were introduced by Miyaoka [43]. Lie curvatures are cross-ratios of the principal curvatures taken four at a time, and they are equal to the cross-ratios of the corresponding curvature spheres along a projective line by Theorems 9.1 and 9.2. Since Lie sphere transformations map curvature spheres to curvature spheres by Theorem 7.1, and they preserve cross-ratios of four points along a projective line (since they are projective transformations), Lie curvatures are invariant under Lie sphere transformations. Obviously, the Lie curvatures must be constant for a Legendre submanifold that is Lie equivalent to the Legendre lift of an isoparametric hypersurface in a sphere.
The examples of Pinkall and Thorbergsson are obtained by taking certain deformations of the isoparametric hypersurfaces of FKM-type constructed by Ferus, Karcher and Münzner [32] using representations of Clifford algebras. Pinkall and Thorbergsson proved that their examples are not Lie equivalent to an isoparametric hypersurface by showing that the Lie curvature does not have the constant value ψ = 1/2, as required for a hypersurface with g = 4 that is Lie equivalent to an isoparametric hypersurface (if the principal curvatures are appropriately ordered). Using their methods, one can also show directly that the Lie curvature is not constant for their examples (see [27, pp. 309-314]).
The construction of counterexamples to Conjecture 12.1 due to Miyaoka and Ozawa [45] (see also [14, pp. 117-123]) is based on the Hopf fibration h : S 7 → S 4 . Miyaoka and Ozawa show that if W 3 is a proper Dupin hypersurface in S 4 with g distinct principal curvatures, then M = h −1 (W 3 ) is a proper Dupin hypersurface in S 7 with 2g principal curvatures. Next they show that if a compact, connected hypersurface W 3 ⊂ S 4 is proper Dupin but not isoparametric, then the Lie curvatures of h −1 (W 3 ) are not constant, and therefore h −1 (W 3 ) is not Lie equivalent to an isoparametric hypersurface in S 7 . For g = 2 or 3, this gives a compact proper Dupin hypersurface h −1 (W 3 ) in S 7 with g = 4 or 6, respectively, that is not Lie equivalent to an isoparametric hypersurface.
As noted above, all of these hypersurfaces are shown to be counterexamples to Conjecture 12.1 by proving that they do not have constant Lie cur-vatures. This led to a revision of Conjecture 12.1 by Cecil, Chi and Jensen [20, p. 52] in 2007 that contains the additional assumption of constant Lie curvatures. This revised conjecture is still an open problem, although it has been shown to be true in some cases, which we will describe after stating the conjecture. We first note that in 1989, Miyaoka [43]- [44] showed that if some additional assumptions are made regarding the intersections of the leaves of the various principal foliations, then this revised conjecture is true in both cases g = 4 and 6. Thus far, however, it has not been proven that Miyaoka's additional assumptions are satisfied in general.
Cecil, Chi and Jensen [19] made progress on the revised conjecture in the case g = 4 by using the fact that compactness implies irreducibility for a proper Dupin hypersurface with g ≥ 3 (see Theorem 11.3), and then working locally with irreducible proper hypersurfaces in the context of Lie sphere geometry.
If we fix the order of the principal curvatures of M to be, then there is only one Lie curvature, For an isoparametric hypersurface with four principal curvatures ordered as in equation (82), Münzner's results [46]- [47] imply that the Lie curvature ψ = 1/2, and the multiplicities satisfy m 1 = m 3 , m 2 = m 4 . 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 those of an isoparametric hypersurface by the work of Stolz [56], so they satisfy m 1 = m 3 , m 2 = m 4 .
Cecil-Chi-Jensen [19] proved the following local classification of irreducible proper Dupin hypersurfaces with four principal curvatures and constant Lie curvature ψ = 1/2. In the case where all the multiplicities equal one, this theorem was first proven by Cecil and Jensen [23].
and constant Lie curvature ψ = 1/2. Then M is Lie equivalent to an isoparametric hypersurface in S n .
Key elements in the proof of Theorem 12.2 are the Lie geometric criteria for reducibility (Theorem 11.2) due to Pinkall [53], and the criterion for Lie equivalence to an isoparametric hypersurface (Theorem 9.3).
By Theorem 11.3 above, we know that compactness implies irreducibility for proper Dupin hypersurfaces with more than two principal curvatures. Furthermore, Miyaoka [43] proved that if ψ is constant on a compact proper Dupin hypersurface M ⊂ S n with g = 4, then ψ = 1/2 on M, when the principal curvatures are ordered as in equation (82). As a consequence, we get the following corollary of Theorem 12.2. The remaining open question is what happens if m 2 is also allowed to be greater than one, i.e., and constant Lie curvature ψ.
Regarding this question, we note that the local proof of Theorem 12.2 of Cecil, Chi and Jensen [19] uses the method of moving frames, and it involves a large system of equations that contains certain sums if some m i is greater than one, but no corresponding sums if all m i equal one. These sums make the calculations significantly more difficult, and so far this method has not led to a proof in the general case (85). Even so, this approach to proving Conjecture 12.2 in the case g = 4 could possibly be successful with some additional insight regarding the structure of the calculations involved.
Finally, Grove and Halperin [34] proved in 1987 that if M ⊂ S n is a compact proper Dupin hypersurface with g = 6 principal curvatures, then all the principal curvatures must have the same multiplicity m, and m = 1 or 2. This was shown earlier for isoparametric hypersurfaces with g = 6 by Abresch [1]. Grove and Halperin also proved other topological results about compact proper Dupin hypersurfaces that support Conjecture 12.2 in the case g = 6.
As mentioned above, Miyaoka [44] showed that if some additional assumptions are made regarding the intersections of the leaves of the various principal foliations, then Conjecture 12.2 is true in the case g = 6. However, it has not been proven that Miyaoka's additional assumptions are satisfied in general, and so Conjecture 12.2 remains as an open problem in the case g = 6.