aa r X i v : . [ m a t h . DG ] J u l THE ISOPARAMETRIC STORY, A HERITAGE OF´ELIE CARTAN
QUO-SHIN CHI
Abstract.
In this article, we survey along the historical route theclassification of isoparametric hypersurfaces in the sphere, payingattention to the employed techniques in the case of four principalcurvatures. Prologue
When writing this survey article, I kept in mind that there had beenan extensive body of research papers on the subject of isoparametricsubmanifolds and beyond. With the addition of the comprehensivebooks [2, 8] that wrapped up much of what had been known up to thetime of their publications, and the survey articles [60, 61] that wentin-depth beyond the isoparametric category, I would therefore devotethis article primarily to the classification part of the hypersurface case.As in [16], I once more followed the historical development of theextensive studies to let the flow of presentation as motivated and seam-less as possible. The difference from [16] is that I included much moredetailed expositions and proofs in the present article. Since the sub-ject is so all-encompassing that I was compelled to assume backgroundknowledge in the first place, a comfortable understanding of symmet-ric spaces is preferred, for which I would refer to the two volumes byLoos [42], Volume 2 of Kobayashi and Nomizu [38], and Helgason [32].Meanwhile, in [14], I wrote a fairly detailed exposition on the com-prehensive commutative algebra engaged in the isoparametric story, towhich I would thus refer without dwelling more on it than is necessary.Also, I would only report on the methods entailed in the classification,in the case of four principal curvatures, done by my collaborators andmyself [7], and subsequently by myself [11, 13, 15], leaving the classifi-cation in the case of six principal curvatures [22, 44, 45] to the expertiseof the authors themselves.
Mathematics Subject Classification.
Primary 53C40.
Key words and phrases.
Isoparametric hypersurfaces.
The codimension 2 estimates prevalent in the classification derivedand developed from the powerful criterion of Serre on prime ideals isstressed in this article, whose unifying capacity in conjunction withthe underlying isoparametric geometry lifts us from the jungle of in-tertwined components of tensors to the canopy of ideals of polynomialrings, to enable us to see the entire landscape of classification.I would like to thank the referees for many valuable comments tobetter the exposition, and Zhenxiao Xie for his careful reading throughthe manuscript during his visit at Washington University.2.
The dawn, 1918-1940
When we stroll along a beach, the last arriving wavefront gentlybrushes our feet to a halt, where we often see cusps forming of thewavefront due to the different speeds at front points. Waves form sin-gularity. The same phenomenon, if applied to the lenses of our eyesshone upon by lights refracted through different media with differentresulting speeds, most of us will feel disoriented after some exposuretime, because the formation of wave singularity plays the trick. Itwould then be interesting to understand the wavefronts that travel ata constant speed each moment. This was investigated by Laura [40]in 1918. He concluded that such wavefronts were either planar, cylin-drical, or spherical, as our daily experiences would almost certainlyconvince us that this is the case, by seeing laser beams of planar wave-fronts, fluorescent tube light of cylindrical wavefronts, and candle lightof spherical wavefronts.To see the equations that govern the wavefronts traveling at a con-stant speed each moment, let us start with the wave equation,∆ φ = ∂ φ∂t . Wavefronts are level surfaces of φ , at each moment t , which propagatealong the normal directions of the level surfaces. That the wavefrontspeed remains constant on each level surface means ds/dt = b ( φ ) and(2.1) |∇ φ | = change per unit length of φ along the normal = a ( φ ) , for some smooth functions a and b , where s is the distance by which awavefront travels. Therefore, we derive ∂φ∂t = ∂φ∂s dsdt = a ( φ ) b ( φ ) := c ( φ ) , and, as a result,(2.2) ∆ φ = ∂ φ∂t = c ′ ( φ ) c ( φ ) . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 3
Definition 2.1.
A smooth function f over a Riemannian manifold istransnormal if |∇ f | = A ( f ) for some smooth function A . A transnormal function is isoparametricif, for some smooth function B , ∆ f = B ( f ) . Let c be a regular value of an isoparametric function f . The levelsurface f − ( c ) is called an isoparametric hypersurface. Somigliana’s paper (1918-1919) brought mean curvature to the fore-ground.
Theorem 2.1. (Somigliana, [55])
Over R , a transnormal function f isisoparametric if and only if each regular level surface of f has constantmean curvature.Proof. For each regular level surface M := f − ( c ) of a transnormalfunction f , n = ∇ f / |∇ f | = ∇ f /A ( f )is a unit normal field to M . The shape operator S of the surface M is S ( X ) := − d n ( X ) for a tangent vector X of M . However, d ( ∇ f )( X ) = d ( A ( f ) n )( X ) = A ′ ( f ) df ( X ) n + A ( f ) d n ( X )= A ( f ) d n ( X ) = − A ( f ) S ( X ) . On the other hand, as a linear operator, d ( ∇ f ) : X Hessian( f ) X. Taking trace, we obtain∆ f = trace ( d ( ∇ f )) = − A ( f ) trace ( S )+ < d ( ∇ f )( n ) , n > = − A ( f ) H + A ′ ( f ) A ( f ) , where H is the mean curvature of M . In other words, H = − ( B ( f ) − A ′ ( f ) A ( f )) / A ( f )is a constant along M if the transnormal f is also isoparametric.Conversely, if H is constant along regular level surfaces of the transnor-mal f , then H is a function of f and so ∆ f is a function of f so that f is isoparametric. (cid:3) QUO-SHIN CHI
In particular, he arrived at the same conclusion that over R , theregular level surfaces of an isoparametric function are either all spheres,all cylinders or all planes.This theorem was rediscovered later by Segre in 1924 [52] and Levi-Civita in 1937 [41]. The approach Levi-Civita gave is what we will lookat next. Theorem 2.2. (Levi-Civita, [41])
Over R , a transnormal f is isopara-metric if and only if the two principal curvatures of each regular levelsurface are constant.Proof. Observe first that the integral curves of the unit normal field n = ∇ f / |∇ f | are just line segments. In fact, an integral curve c of n from f = a to f = b assumes the lengthLength of c = Z ba df |∇ f | = Z ba dfA ( f ) . On the other hand, for any curve γ ( t ) , ≤ t ≤ , beginning and endingat the two end points of the given integral curve, we have | df ( γ ( t )) dt | = |h∇ f ( γ ( t )) , γ ′ ( t ) i| ≤ A ( f ( γ ( t )) | γ ′ ( t ) | , so that Length of γ = Z | γ ′ ( t ) | dt ≥ Z A ( f ) dfdt dt = Z ba dfA ( f ) . In other words, the given integral curve c assumes the shortest distanceamong all curves beginning and ending at its end points, i.e., the in-tegral curve is a line segment. In view of this observation, instead ofusing f to parametrize the level surfaces, we might as well use the arclength s of the normal lines of an initial level surface to parametrizeother level surfaces, so that M s := M + s n is now the 1-parameter family of level surfaces of the transnormal f ,where M is the initial level surface with unit normal field n .Let us calculate the mean curvature H s of M s by using the fact that n is still the unit normal to M s . The upshot is [49, p. 209] H s = H − sK − sH + s K , where k and k are the eigenvalues (the principal curvatures) of theshape operator S of M , so that H = ( k + k ) /
2, and K = k k isthe Gaussian curvature of M . Therefore, the mean curvature H s isconstant on M s for all s , i.e., the transnormal f is isoparametric, if and HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 5 only if
H, K are constant on M , if and only if the principal curvatures k , k are constant.Case 1. k = k = 0. M is a sphere.Case 2. k = k = 0. M is a plane.Case 3. k = k . One employs dk = dk = 0 and a bit more surfacegeometry to conclude k k = 0 [49, p. 255], so that one of k , k is zero.Then M is a cylinder. (cid:3) Segre then took up the investigation of generalizing the question to R n in 1938, followed by Cartan’s look into the hyperbolic space H n inthe same year. Theorem 2.3. (Segre, [53])
The same conclusion holds on R n , namely,an isoparametric hypersurface, which is a regular level hypersurface ofan isoparametric function f over R n satisfying |∇ f | = A ( f ) , ∆ f = B ( f ) , is either a hypersphere, a hyperplane, both are totally umbilic ( with oneprincipal curvature ) , or a cylinder S k × R n − − k . Theorem 2.4. (Cartan, [3])
The same conclusion holds on the hy-perbolic space H n of constant curvature − , namely, an isoparametrichypersurface in H n is either a sphere, a hyperbolic H n − , a Euclidean R n − ( i.e., a horosphere ) , all three being totally umbilic, or a cylinder S k × H n − k − .Proof. (sketch) We show again that there are at most two (constant)principal curvatures of the shape operator. Indeed, let λ , · · · , λ n − bethe principal curvatures of an isoparametric hypersurface in a standardspace form of dimension n with constant curvature C . Then we have(2.3) X j = k m j C + λ k λ j λ k − λ j = 0 , summed on j, referred to by Cartan as the “Fundamental Formula”, which was provenby Segre in the Euclidean case and by Cartan in general [2, p. 84]. Here, m j is the multiplicity of λ j and λ i = λ j if i = j .Case 1, C = 0. Let λ k be the smallest positive principal curvature.Note that each term, if nontrivial, in the fundamental formula must benegative, which is a contradiction. Therefore, there are at most twoprincipal curvatures, one of them is zero if there are two.Case 2, C = −
1. It is easy to see that(2.4) C + λ k λ j λ k − λ j < QUO-SHIN CHI if λ j ≤ λ k >
0. Consider those positive principal values. If thereis a 0 < λ l ≤ λ j ) − ≤ λ l for all λ j >
1, we let λ k be thelargest positive principal value ≤
1. It follows that (2.4) is negative forall those 0 < λ j < λ j > λ k . Weconclude that none of the positive λ j other than the reciprocal of λ k exist. Otherwise, there exists some λ j > λ k , which we replace by the smallest principalvalue >
1. Once more, (2.4) is negative for all positive λ j not reciprocalto λ k . We arrive at the same conclusion as in the preceding case. So,we have at most two principal curvatures reciprocal to each other.In the case of two distinct principal curvatures λ and µ , the isopara-metric hypersurface is the product of two simply connected space formsof constant curvatures λ − µ −
1. All these hypersurfaces arehomogeneous.See [51, Theorem 2.5, p. 373] on the product structure in bothcases. (cid:3)
The story now takes a fascinating turn when Cartan directed hisattention to the spherical case. Let us denote by g the number ofprincipal curvatures of an isoparametric hypersurface in the sphere S n .He quickly settled the cases when g ≤
2. For g = 1, the hypersurface isany sphere in the 1-parameter family of such spheres perpendicular tothe axis of the North and South poles of S n ; note that the 1-parameterfamily degenerates to the South and North poles. For g = 2, thehypersurface is any one in the 1-parameter family of the product oftwo subspheres of the form S k × S n − k − whose points are given incoordinates as ( x , · · · , x k , x k +1 , · · · , x n ), where x + · · · + x k = r , x k +1 + · · · + x n = s , r + s = 1 , which degenerates to the two manifolds S k and S n − k − of radius 1 as r approaches 0 or 1. We refer to [51] for a proof. All these hypersurfacesare homogeneous.Then Cartan worked on the case g = 3. A priori , the three principalvalues could potentially carry distinct multiplicities, which was ruledout by him. He went on to classify and was amazed by the beautyof such hypersurfaces, as one can tell from the title of his 1939 pa-pers [4], [5]:
Theorem 2.5.
Let g = 3 . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 7 I. Over the ambient Euclidean space R n +1 ⊃ S n , there is a homo-geneous polynomial F of degree , satisfying |∇ F | = 9 r , r = | x | for x ∈ R n +1 , ∆ F = 0 , (2.5) whose restriction to S n is exactly the isoparametric function f .The range of f is [ − , with ± the only critical values. Thus f − ( c ) , − < c < , form a -parameter family of isoparametrichypersurfaces that degenerates to the two critical sets f − (1) and f − ( − . II.
The three principal values have equal multiplicity m = 1 , , ,or . III.
The two critical sets of f are the real, complex, quaternionic,or octonion projective plane corresponding to the principal mul-tiplicity m = 1 , , , or . Each isoparametric hypersurface inthe family is a tube around the projective plane. IV.
Let F be one of the normed algebras R , C , H , and O . Let X, Y, Z ∈ F and a, b ∈ R . Then F = a − ab + 3 a XX + Y Y − ZZ )+ 3 √ b XX − Y Y ) + 3 √
32 (( XY ) Z + ( XY ) Z ) . (2.6)In particular, all these hypersurfaces are homogeneous.His proof was an algebraic analysis of the homogeneous polynomial F of degree 3, nowadays called the Cartan-M¨unzner polynomial (see thenext section for more expositions on it). He expanded the polynomialat a point p of, say, the critical (or, focal) manifold f − (1) with thecoordinate u parametrizing R p and x , · · · , x n parametrizing ( R p ) ⊥ .He observed that the equations (2.5) dictated that F = u + P ( x ) u + Q ( x ) , where P ( x ) is quadratic and Q ( x ) is cubic homogeneous in x . Sub-stituting this into the equations in (2.5), he found that there was aninteger m such that P = 32 ( x + · · · + x m ) − x m +1 + · · · + x m +1 ) , n = 3 m + 1 . In fact, w := x m +1 , · · · , w m := x m +1 constitute the normal coordinates to the focal manifold in S n at p . Itfollows that the isoparametric hypersurface is of equal multiplicity m QUO-SHIN CHI by identifying any of the three principal spaces of its shape operatorwith the normal space to a point on the associated focal manifold.He then turned to Q ( x ), which he expanded into Q = A + A + A + A , where A i is of degree i in x , · · · , x m and degree 3 − i in w , · · · , w m and deduced that A = A = A = 0 so that Q = Q z + · · · + Q m z m for some homogeneous quadratic polynomials Q , · · · , Q m in x , · · · , x m that are in fact the components of the second fundamental form of thefocal manifold at p . Since the principal curvatures of the focal manifoldare cot( π/
3) and cot(2 π/
3) (see (4.10) below for details), only differingby a sign, he diagonalized Q so that Q = 3 √
32 ( x + · · · + x m − y − · · · − y m ) , by reindexing the coordinates y := x m +1 , · · · , y m := x m . He then set H a := Q a / √ ≤ a ≤ m and concluded H + · · · + H m = ( x + · · · + x m )( y + · · · + y m ) , or, in other words, the map L : ( x, y ) ∈ R m × R m ( H ( x, y ) , · · · , H m ( x, y )) ∈ R m satisfies |L ( x, y ) | = | x || y | , i.e., it is an orthogonal multiplication [34],from which he deduced that m = 1 , , , or, 8.With the expressions of P and Q pinned down, he set v := w andwrote down F = u − uv + 32 u m X i =1 ( x i + y i ) − u m X i =1 w i + 3 √ v m X i =1 ( x i − y i ) + m X i =1 w i Q i ( x, y ) . This is exactly (2.6) when we employ the orthogonal multiplications L ,which give rise to the products of the four normed algebras.Thanks to [36], [37], [18], we have geometric alternatives to the proof,which were inspired, directly or indirectly, by M¨unzner’s fundamentalpapers [46] that entered the stage in the early 1970s.We will derive (2.6) in Subsection 4.3 for the real case, using a Lie-theoretic approach.Next, Cartan worked on the case g = 4 [6]. The situation got muchmore complicated. So, he assumed equal multiplicities m to the four HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 9 principal values and pointed out that the hypersurface satisfies theequations |∇ F | = 16 r , r = | x | for x ∈ R n +1 , ∆ F = 0 . He indicated the classification without proof in the cases when m =1, or 2, by writing down the respective 4th degree Cartan-M¨unznerpolynomials. When m = 1, the polynomial is (4.1) below for k = 3,whereas when m = 2, the polynomial is defined over the Euclideanspace so (5 , R ), where an element Z ∈ so (5 , R ) is written as (cid:0) Z Z Z Z Z (cid:1) by five column vectors, for which the polynomial is(2.7) F ( Z ) := 54 X i | Z i | − X i A hypersurface in S n is isoparametric if all its prin-cipal curvatures are constant with fixed multiplicities. Let M be anisoparametric hypersurface in S n . The number of principal curvaturesof M is denoted by g . The dormancy, 1941-1969 When I was a student at Stanford, I had a few times of working as aTA for David Gilbarg. During a chat he said that he had been trainedas an algebraist working under Emil Artin. But war had a betterdictation over everything. Upon graduation he was designated to workon partial differential equations in the wartime, from that point on it was a point of no return for him. Mathematics more on the “pure”side essentially ground to a halt.Diversified activities awakened and resumed after the war. One ofthe most notable was the vibrancy in the field of algebraic and dif-ferential topology, which led to the profound discoveries in the 1950-60s, from which the stage was set for the revival of the isoparametricstory. At the same time, algebraic geometry also experienced a gigantictransformation in which the theory of schemes via ideals changed itslandscape. 4. The Renaissance, 1970-1979 Nomizu, Takagi and Takahashi’s work on homogeneousisoparametric hypersurfaces. Nomizu wrote two papers to startthis period in the early 1970s, in which he constructed examples [47]that answered the second question of Cartan in the negative and re-viewed Cartan’s work [48].Indeed, consider C k = R k ⊕ R k and write z ∈ C k as z = x + √− y accordingly. Define a homogeneous polynomial of degree 4 on C k by˜ F = ( | x | − | y | ) + 4( h x, y i ) . Then ˜ f := F | S k − is an isoparametric function whose regular level setsform a 1-parameter family of isoparametric hypersurfaces with fourprincipal values and multiplicities { , k − } .The isoparametric hypersurfaces (respectively, all the above isopara-metric hypersurfaces by Cartan) are the principal isotropy orbits of thesymmetric spaces SO ( k + 2) /S (2) × SO ( k ) (respectively, of appropriatesymmetric spaces of rank 2 to be seen in Subsection 4.3).Note that ˜ f has range [0 , f :=1 − f , or rather, by setting(4.1) F := ( | x | + | y | ) − F .F is an isoparametric function such that f has range [ − , f − (1) := { ( x, y ) : | x | = | y | = 1 / , h x, y i = 0 } , the Stiefel manifold of oriented 2-frames, which is orientable, whereas f − ( − 1) = { z ∈ C k : z = e √− θ v, v ∈ S k − , θ ∈ [0 , π ) } , which is doubly covered by S × S k − and is not orientable when k isodd, as was pointed out by Cartan in [6] in the case of multiplicity pair(1 , 1) for k = 3. HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 11 At about the same time Takagi and Takahashi [58] classified homo-geneous isoparametric hypersurfaces in spheres, which are the principalorbits of isotropy representations of simply connected symmetric spacesof rank 2. They also calculated the number g of principal curvatures tobe 1, 2, 3, 4, or 6, and, moreover, verified that there were at most twodistinct multiplicities, in the homogeneous category. We begin with adefinition. Definition 4.1. A connected hypersurface M in a smooth Riemannianmanifold X is homogeneous if I ( X, M ) , the group of isometries of X leaving M invariant, acts transitively on M . It is clear that, for such a hypersurface, the principal curvaturesof its shape operator are everywhere constant, counting multiplicities.Hence, they are isoparametric.Theorems 2.3 and 2.4 classify all isoparametric hypersurfaces in R n and H n to be exactly the homogeneous hypersurfaces in these spaceforms. What is interesting is then the spherical case.Recall a representation ρ : G ֒ → SO ( n +1) acting on R n +1 is effective if every nontrivial element in G displaces some vector in R n +1 . Now,let I ( M ) be the group of isometries of M and let ι : I ( S n , M ) → I ( M )be the restriction map. Let I ( S n , M ) be the connected component ofthe identity of I ( S n , M ) and let G := ι ( I ( S n , M )). Proposition 4.1. [50, II, p. 15] ι : I ( S n , M ) → G is an isomorphism,so that ι − : G ֒ → SO ( n + 1) is an effective representation on R n +1 with M an orbit. Furthermore, M is compact, and so in particular G is compact and hence is a Lie group. Definition 4.2. An effective representation ρ : G ֒ → SO ( n + 1) actingon R n +1 is of cohomogeneity r if the smallest codimension of all orbitsof ρ is r in R n +1 . In particular, the representation ι above of a homogeneous hyper-surface in S n is of cohomogeneity 2. Definition 4.3. Given an effective representation ρ : G ֒ → SO ( n + 1) of cohomogeneity r , ρ is maximal if there is no effective representation ρ : G : ֒ → SO ( n + 1) of cohomogeneity r such that G is a propersubgroup of G with ρ ( g ) = ρ ( g ) for all g ∈ G . Proposition 4.2. [50, p. 16](1) The effective representation ι : G ֒ → SO ( n + 1) in Proposi-tion is a maximal effective representation of cohomogeneity . (2) Let ρ : G ֒ → SO ( n + 1) be a maximal effective representationof cohomogeneity . Let M be a G -orbit of codimension in R n +1 . Then ρ ( G ) = I ( S n , M ) . (3) In particular, any maximal effective representation ρ : G ֒ → SO ( n + 1) is obtained as the representation of a homogeneoushypersurface in S n . (4) Two homogeneous hypersurfaces M and N in S n are equivalent,i.e., N = f ( M ) for an f ∈ O ( n + 1) , if and only if I ( S n , M ) ≃ I ( S n , N ) through the isomorphism g f gf − . Therefore, the classification of homogeneous hypersurfaces in S n isequivalent to first classifying maximal effective orthogonal representa-tions ρ : G ֒ → SO ( n + 1) of cohomogeneity 2 and then classifying theirorbits of codimension 2. Hsiang and Lawson classified all maximal or-thogonal representations in [33]. They are closely tied with what arecalled the s -representations of symmetric spaces. We will return to thisin Subsection 4.3.4.2. M¨unzner’s work on the general case. M¨unzner [46] (preprintcirculating in 1973) established a breakthrough result that developedCartan’s work, recorded in Theorem 2.5, in a far-reaching manner: Theorem 4.1. Let M be any isoparametric hypersurface with g prin-cipal curvatures in S n . Then we have the following. (1) There is a homogeneous polynomial F , called the Cartan-M¨unznerpolynomial, of degree g over R n +1 satisfying (4.2) |∇ F | = g r g − , ∆ F = m − − m + g r g − , where r is the radial function over R n +1 . (2) Let f := F | S n . Then the range of f is [ − , . The only criticalvalues of f are ± . Moreover, M ± := f − ( ± are connectedsubmanifolds of codimension m ± + 1 in S n , called focal mani-folds, whose principal curvatures are cot( kπ/g ) , ≤ k ≤ g − . (3) For any c ∈ ( − , , f − ( c ) is an isoparametric hypersurfacewith at most two multiplicities m ± associated with the principalcurvatures. In fact, if we order the principal curvatures λ > · · · > λ g with multiplicities m , · · · , m g , then m i = m i +2 withindex modulo g ; in particular, all multiplicities are equal when g is odd, and when g is even, there are at most two multiplicitiesequal to m ± . (4) Each of the -parameter isoparametric hypersurfaces is a tubearound each of the two focal manifolds, so that S n is obtained HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 13 by gluing two disk bundles over M ± along the isoparametric hy-persurface M := f − (0) . As a consequence, algebraic topologyimplies that the only possible values of g are , , , , or . Indeed, start with an isoparametric hypersurface x : M ֒ → S n whose principal curvatures are set to be λ j = cot( θ j ) , < θ < · · · < θ g < π, with respect to the outward normal field n . Let us consider the paralleltransport of M ,(4.3) x t := cos( t ) x + sin( t ) n, which is the counterpart to the Euclidean parallel transport along thenormal direction. A priori , M t := x t ( M ) is an embedding for small t .Since n t := − sin( t ) x + cos( t ) n is normal to M t , a straightforward calculation derives that the principalcurvatures of M t , with respect to the chosen normal field n t , are λ j ( t ) = cot( θ j − t )(4.4)with the same eigenspace and multiplicity as λ j . On the other hand,for a fixed l , the eigenspace of λ l from point to point defines an inte-grable distribution, called the l -th curvature distribution, on M withspheres of radius | sin( θ l ) | as leaves. This can be directly checked bydifferentiating f l ( x ) := x + v l ( x ) / | v l ( x ) | , v l ( x ) := − x + cot( θ l ) n, to see that f l ( x ) is a constant c l on the l -th curvature leaf through x ;we have(4.5) c l = cos( θ l )(cos( θ l ) x + sin( θ l ) n ) , i.e., the unit vector pointing in the same direction as c l assumes theangle θ l on the unit circle oriented from x to n . Now that the curvatureleaf through x is a sphere of radius | sin( θ l ) | centered at c l , the antipodalpoint to x on this leaf gives the reflection map φ l about c l : φ l ( x ) := x + 2 v l ( x ) / | v l ( x ) | = cos(2 θ l ) x + sin(2 θ l ) n, i.e., φ ( x ) is the point of reflection of x about the line spanned by c l onthe ( x, n )-plane. Therefore, by (4.4), the principal curvatures of M at φ l ( x ) are(4.6) − cot( θ j − θ l ) , ≤ j ≤ g, with the same eigenspaces and multiplicities as x . Note that the signin (4.6) differs from that in (4.4), because the circle x t leaves M at x and enters M at φ ( x ), so that n θ l at φ ( x ) is negative of the chosenoutward normal field n of M at φ ( x ). Since M has constant principalcurvatures, counting multiplicities, we conclude that the following sets(4.7) { cot( θ j ) } , { cot(2 θ l − θ j ) } are identical for all j, l , and two numbers, one from each set, havingthe same index j have the same principal multiplicity m j , regardless ofwhat l is.Now, (4.7) means that the lines L j spanned by c j on the ( x, n )-plane,all through the origin, satisfies the property that the reflection of L j about any L l is another L k . It follows that these lines L j , ≤ j ≤ g, are equally spaced in the ( x, n )-plane so that θ j = ( j − π/g + θ . Thus, the reflections about the lines L j result in m i = m i +2 with indexmodulo g . Accordingly, we denote m and m by m + and m − , respec-tively. (This is reminiscent of a root system and its Weyl chambers.)Having done so, M¨unzner went on to construct a local isoparametricfunction, which is nothing but an appropriate distance function, in aneighborhood of M , already observed by Cartan for g = 3, as follows.Any p in a tubular neighborhood U of M can be written uniquely as p = cos( t ) x + sin( t ) n for some small t . Define µ ( p ) := θ − t, V ( p ) = cos( g µ ( p )) . Extend V ( p ) to a neighborhood of M in the ambient Euclidean spaceby defining F ( rp ) = r g V ( p ) , where r is the Euclidean radial function. Theorem 4.2. F is in fact a homogeneous polynomial of degree g satisfying |∇ F | = g r g − , ∆ F = g m − − m + . Proof. (Sketch) Define G := F − ar g , where a := gg + n − m − − m + . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 15 Then verify that ∆ G = 0 . In general, it is true that for a harmonic function G over R n +1 , we have(4.8) ∆ g |∇ G | = X ( ∂ g +1 G/∂x i · · · ∂x i g +1 ) . On the other hand, for the G engaged in our consideration, a calculationgives |∇ G | = g r g − (1 + a ) − ag r g − F. We therefore find ∆ g − |∇ G | = c with c an appropriate constant. F is thus a homogeneous polynomialby (4.8). (cid:3) Now that F is globally analytic over R n +1 , we set f := F | S n . Acalculation by the formulae |∇ F | = ( ∂F∂r ) + |∇ f | , ∆ f = ∆ F − ∂ F∂r − n ∂F∂r derives, by Theorem 4.2, that |∇ f | = A ( f ) , ∆ f = B ( f ) , where A ( f ) = g (1 − f ) , B ( f ) = − g ( n + g − f + m − − m + g . So, f is an isoparametric function on S n . Note that A ( f ) = 0 only at f = ± 1, so that the range of f is [ − , 1] and ± M ± := f − ( ± 1) be the singular set. S n \ ( M + ∪ M − ) isopen and dense and is diffeomorphic to M c × ( − , 1) for any fixed c ,where M c := f − ( c ) for c ∈ ( − , A priori , M c might not be connected. We claim that this is not thecase. Define d : M × (0 , π/g ) → S n , d ( x, µ ) = cos( θ − µ ) x + sin( θ − µ ) n. Then f ( d ( x, µ )) = cos( gµ )by the analytic nature of f because f | U = V and the identity holds on U ; in particular, M is contained in M c with c = cos( gθ ). But then themap(4.9) d c : M c × (0 , π/g ) → S n , d c ( x, µ ) = cos( θ − µ ) x + sin( θ − µ ) n also satisfies f ( d c ( x, µ )) = cos( gµ ) and d c : M c × (0 , π/g ) → S n \ ( M + ∪ M − ) is a diffeomorphism. From this we see that the map h : M c → S n , x cos( θ ) x + sin( θ ) n maps M c to M + . Observe that h ( x ) points in the same direction as c for the curvature leaf through x whose tangent space is the eigenspacewith principal value cot( θ ), where c is defined in (4.5). It followsthat h : M c → M + is a sphere bundle whose fiber is a curvature leafdiffeomorphic to S m + . Meanwhile, it is easy to check that dh haskernel dimension m + ; at x , the derivative dh preserves eigenspaces ofall principal values other than that of cot( θ ). Therefore, M + is amanifold of dimension dim( M ) − m + , which is of codimension at least2 in S n . Likewise, the codimension of M − is at least 2 in S n .Returning to the map (4.9), we see now S n \ ( M + ∪ M − ) is connectedas M + and M − are of codimension at least 2 in S n . Therefore, that d c is a diffeomorphism ensures that M c is connected, for all c . As aconsequence, M ± are also connected via the map h .Lastly, since h ( x ) = x θ defined in (4.3), we see by (4.4) that theprincipal values of M + , in any normal direction, are(4.10) cot( θ j − θ ) = cot(( j − π/g ) , ≤ j ≤ g. This also holds true for M − .We remark that the sphere bundle property via the map h holdstrue over any complete Riemannian manifold when we only assumetransnormality [62], though the focal manifolds need not be connectedin general. Corollary 4.1. M ± are minimal submanifolds of S n . The minimalitycondition is exactly equation (2.3) , the fundamental formula of Segreand Cartan, when C = 1 .Proof. By the preceding formula, the mean curvature of M + in anynormal direction is g − X j =1 cot( jπ/g ) = 0 , which is the fundamental formula (2.3). (cid:3) Corollary 4.2. There is a unique minimal isoparametric hypersurfacein the -parameter family M t .Proof. By (4.4), the mean curvature of M t is H := g X j =1 cot( θ j − t ) HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 17 for t ∈ (0 , π/g ). H is strictly increasing as the derivative is > 0. Near t = θ < π/g the function is > θ = π/g − θ > < 0. Therefore, there is a unique t ∈ (0 , π/g ) at which H = 0. (cid:3) Now that S n is obtained by gluing two disk bundles over the fo-cal manifolds M ± along an isoparametric hypersurface M , M¨unznerused algebraic topology to express the cohomology ring of M , with Z coefficients, as modules of those of M ± , whose intertwining modulestructures via Steenrod squares then give the remarkably short list that g = 1 , , , , or 6! Thus, it answered the first question of Cartan.4.3. The homogeneous case in retrospect. Now that we have agrand view of the structure of isoparametric hypersurfaces thanks toM¨unzner’s theorem, let us return to the homogeneous case, this timewith more geometric insight. Definition 4.4. An s -representation of rank r is the isotropy rep-resentation of a connected, simply connected semisimple Riemanniansymmetric space of rank r . Here, if the symmetric space is decomposedinto its irreducible components, the rank is the sum of the ranks of thecomponents. An s -representation of rank 2 is either the isotropy representation oftwo irreducible symmetric spaces of rank 1, or of an irreducible symmet-ric space of rank 2. Note that R × M, where M is irreducible symmetricof rank 1, is also of rank 2, although its isotropy representation is notan s -representation.In connection with classifying homogeneous hypersurfaces in S n , weare particularly interested in the isotropy representations of simply con-nected noncompact symmetric spaces of rank 2, because of the theoremof Hsiang and Lawson [33] on the classification of all maximal effectiveorthogonal representations ρ : G ֒ → SO ( n + 1) of cohomogeneity 2: Theorem 4.3. Up to equivalence, the maximal effective orthogonalrepresentations ρ : G ֒ → SO ( n + 1) of cohomogeneity are exactly theisotropy representations of the simply connected noncompact symmetricspaces of rank , i.e., the isotropy representations of (1) R × H n , where the principal orbits are spheres S n − ⊂ S n ⊂ R n +1 , (2) H p × H q , where the principal orbits are S p − × S q − ⊂ S p + q − ⊂ R p + q , and (3) the noncompact irreducible symmetric spaces of rank , whereprincipal orbits are those of s -representations. More generally, Dadok’s classification [19] shows that any polar rep-resentation is orbit equivalent to an s -representation. See also [24], [25]for a more conceptual proof.The principal orbits of the first two items in Theorem 4.3 are easyto visualize. Item (1) is the isotropy representation of O (1 , n ) on thehyperbolic space H n identified with O (1 , n ) /SO ( n ) as a symmetricspace, where O (1 , n ) is the identity component of O (1 , n ) and SO ( n )is identified with K = { (cid:18) A (cid:19) , A ∈ SO ( n ) } . Thus, it gives the standard orthogonal representation SO ( n ) on R n ,whose typical principal orbit is the sphere S n − . The Euclidean factorin item (1) acts trivially, so that a principal orbit of codimension 2 ofthe isotropy representation is S n − ⊂ S n ⊂ R n +1 . In the same vein,a typical principal orbit of the isotropy representation in item (2) is S p − × S q − ⊂ S p + q − ⊂ R p + q .In particular, we have g = 1 or 2 for the homogeneous spaces in thefirst two items of Theorem 4.3. The isoparametric hypersurfaces with g = 1 or 2, classified by Cartan, are exactly the ones in the first twoitems.Let us study item (3) in Theorem 4.3, where the principal orbits giverise to all homogeneous isoparametric hypersurfaces with g ≥ G/K be a noncompact irreducible symmetric space of rank 2(which is automatically simply connected) with the Cartan decompo-sition G = K ⊕ M . Fix a v = 0 ∈ M . We know [32, p. 247] thereis a k ∈ K such that Ad ( k ) · v ∈ A , where A is the maximal abeliansubspace of M . Therefore, we may assume without loss of generalitythat v ∈ A . Proposition 4.3. With the setup above, an orbit Ad ( K ) · v , where v ∈ A , is principal of codimension if and only if v lies in a Weylchamber.Proof. The isotropy subgroup L of Ad ( K ) leaving v fixed has the Liealgebra L := { X ∈ K : [ X, v ] = 0 } . We have the root space decomposition(4.11) 0 = ad h ( Z ) = X λ ∈ Σ λ ( h ) Z λ HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 19 with h ∈ A and Z λ ∈ G λ , and so G = N ⊕ X λ ∈ Σ G λ , where N is the centralizer of A in G .If v belongs to a Weyl chamber, then λ ( v ) = 0 for all λ ∈ Σ, so thatby (4.11) X λ = 0 for all λ ∈ Σ. In other words, X ∈ L if and only if X ∈ M := N ∩ K , the centralizer of A in K .By [32, Lemma 3.6, p. 261], we know M has the same codimensionin K as A in M , i.e.,dim( Ad ( K ) /L ) = dim( K ) − dim( L ) = dim( M ) − dim( A ) . Thus, the isotropy orbit is of codimension dim( A ) = 2.If v lies in a chamber wall, then by (4.11) X ∈ L if and only if X ∈ M ⊕ X λ,λ ( v ) =0 ( G λ ∩ K ) . Therefore, the codimension of the orbit of v is larger than 2. (cid:3) Corollary 4.3. The isotropy representation of an irreducible noncom-pact symmetric space of rank has only two singular orbits and a -parameter family of diffeomorphic principal orbits of codimension degenerating to the two singular orbits.Proof. In the rank 2 case, a Weyl chamber is a sector of the plane ofangle measure π/ A , π/ B , and π/ G . Let us say θ < θ < θ + π/l, l = 3 , , , defines the chamber. Then the pre-ceding proposition says that for any unit v assuming angle θ in thechamber, its isotropic orbit is homogeneous (and hence isoparamet-ric) of codimension 2 and is diffeomorphic to Ad ( K ) /L . So, we havea 1-parameter family of diffeomorphic homogeneous isoparametric hy-persurfaces. At the two chamber walls, i.e., when v assumes the angle θ or θ + π/l , the dimension of the orbit drops. Meanwhile, since thenormalizer of A serves as the Weyl group by Theorem 4.5 below, wesee that the isotropic orbit of v intersecting the chamber plane at somepoints v = v, v , · · · , v l , one in each chamber. So the isotropic repre-sentation has only two singular orbits, even though there are 2 g Weylchambers. All other orbits are principal of codimension 2. (cid:3) Proposition 4.4. With the same setup, Ad ( k )( A ) is the normal planeto the principal orbit Ad ( K )( v ) at Ad ( k )( v ) for v ∈ A .Proof. It suffices to check this at v , where the tangent space of theorbit is T v = { [ h, v ] : h ∈ K} . But then for w ∈ A , we have h w, [ h, v ] i = h [ v, w ] , h i = 0 , since the inner product is proportional to the Killing form, (cid:3) Proposition 4.5. With the same setup, let w be a unit vector perpen-dicular to v in A , and extend it to a global normal field on the principalorbit Ad ( K ) · v, | v | = 1 , in the unit sphere of M . The shape operator S w of the orbit at v satisfies that the eigenvalues are − λ ( w ) /λ ( v ) , where λ are reduced positive roots such that λ/ / ∈ Σ . The eigenspaceassociated with the above eigenvalue is E λ = G λ ⊕ G − λ ⊕ G λ ⊕ G − λ . In particular, g , the number of principal curvatures of the shape opera-tor, is , , or . If we label the principal curvatures by λ > · · · > λ g and their multiplicities by m , · · · , m g , then m i = m i +2 , where the sub-scripts are modulo g . In particular, the multiplicities are all equal when g = 3 . Moreover, if we choose the angles θ i = (2 i − π/ g, i = 1 , · · · , g, to coordinatize the positive roots, then the principal curvatures are λ i = tan( θ − θ i ) , − π/g < θ < π/g, when v assumes the angle θ and w the angle θ + π/ .Proof. As mentioned in the preceding proposition, a vector X tangentto the orbit is of the form X = [ k, v ] = − X λ ∈ Σ λ ( v ) X λ for k ∈ K . Since n = Ad ( K ) · w is a normal vector field to the orbit, the shape operator is S ( X ) := − dn ( X ) = − [ X, w ] = X λ ∈ Σ λ ( w ) X λ . Therefore, we obtain − λ ( v ) S ( X λ ) = λ ( w ) X λ . Since v is regular we have λ ( v ) = 0 for all λ ∈ Σ. It follows that S ( X λ ) = − λ ( w ) /λ ( v ) X λ . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 21 The principal curvatures of S are thus − λ ( w ) /λ ( v ), attained by ± λ, ± λ ,so that the eigenspace E λ with the principal curvature − λ ( w ) /λ ( v ) isthe desired one, where λ need only go through the positive roots λ for which λ/ / ∈ Σ, which form a reduced root system. The number ofpositive roots in the A , B , or G root system is 3, 4, or 6, respectively,which is g .We choose the angles θ i = (2 i − π/ g to coordinatize the positiveroots. We see the Weyl group is generated by(4.12) θ π/g − θ, θ θ + 2 π/g. By Theorem 4.5, the Weyl group preserves the principal curvatures andtheir multiplicities. Hence, m i = m i +2 with index modulo g .Lastly, since v = (cos( θ ) , sin( θ )) , w = ( − sin( θ ) , cos( θ )) , λ i = (cos( θ i ) , sin( θ i )) , we calculate − λ i ( w ) /λ i ( v ) = −h w, λ i i / h v, λ i i = tan( θ − θ i ) . (cid:3) Let us now look at F ( θ ) := sin( gθ ) , − π/ g < θ < π/ g. It is left invariant by the two generators of the Weyl group in (4.12).In fact, F ( θ ) is the restriction to the unit circle of the homogeneouspolynomial of degree g (4.13) F A := [( g − / X (cid:18) g i + 1 (cid:19) ( − i x g − (2 i +1) y i +1 defined by the maximal abelian space A . F A is left invariant by theWeyl group. Theorem 4.4. [38, p. 299] The space of homogeneous polynomials on M left invariant by Ad ( K ) is isomorphic to the space of homogeneouspolynomials on A left invariant by the Weyl group. This theorem is called Chevalley Restriction Theorem. In [38], theproof is given for a compact Lie group, or for a symmetric space ofType II. But the proof there can be modified easily to arrive at thepreceding theorem in view of a characterization of the Weyl group: Theorem 4.5. [32, p. 284, p. 289] Let ( G, K, σ ) be an irreducibleRiemannian symmetric space of noncompact type. Let M := { k ∈ K : Ad ( k ) · v = v, ∀ v ∈ A} , M ′ := { k ∈ K : Ad ( k ) ·A ⊂ A} . Then M ′ /M is the Weyl group. The last ingredient for constructing the Cartan-M¨unzner polynomialin the homogeneous category is the following theorem of Chevalley. Theorem 4.6. [9] The space of homogeneous polynomials on a maxi-mal abelian space A of dimension r left invariant by the Weyl group isgenerated by r algebraically independent polynomials. Since r = 2 in our case and we have found two generators, namely, x + y and F A on A , the space of homogeneous polynomials left invari-ant by Ad ( K ) on M of dimension n is thus generated by ( x ) + · · · +( x n ) and a homogeneous polynomial F of degree g whose restrictionto the circle is F A . F , homogeneous of degree g , thus leaves each isotropic orbit invari-ant. Therefore, we conclude the following. Theorem 4.7. There is a homogeneous polynomial F of degree g ,called the Cartan-M¨unzner polynomial, for g = 3 , , , on M , whoserestriction f to the unit sphere of M satisfies the property that its rangeis [ − , . For each c ∈ ( − , , f − ( c ) is a homogeneous (isoparamet-ric) hypersurface degenerating to two singular submanifolds f − ( ± .The statement is clearly true when g = 1 or . All homogeneous hy-persurfaces in spheres are constructed this way. We remark that for g = 1 in the preceding theorem, the polynomialis F = x n +1 over R n +1 , while for g = 2 the polynomial is F = ( x ) + · · · + ( x r ) − ( x r +1 ) − · · · − ( x r + s ) over R r + s . Example. We find F in the case g = 3 when the symmetric space is SU (3) /SO (3) of Type I and rank 2.Let M be the space of 5-dimensional 3 by 3 real traceless symmetricmatrices. The Cartan decomposition is su (3) = so (3) ⊕ √− M , K = so (3) . M is equipped with the inner product h Y, Y i := tr ( Y Y ) = α + β + γ + x + y + z , which is a multiple of the Killing form of su (3) restricted to M , wherewe write Y := α x/ √ y/ √ x/ √ β z/ √ y/ √ z/ √ γ , α + β + γ = 0 . The isotropic action is the adjoint action Ad ( T ) : V ∈ M 7→ T V T − ∈ M , T ∈ SO (3) . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 23 The diagonal block of M is the maximal abelian subspace A of M .The three positive roots are α := − / √ , α := − / √ , α := − / √ , where α and α are simple roots. We choose the unit angle bisectoras the standard basis element e := (2 α + α ) / √ , and e := α . Then e , e form an orthonormal basis of A , so that an element in M relative to e , e is X := ae + be = a/ √ b/ √ − a/ √ − b/ √ − a/ √ , and the Y above is Y := a/ √ x/ √ y/ √ x/ √ b/ √ − a/ √ z/ √ y/ √ z/ √ − b/ √ − a/ √ . Now, it is clear that det( Y ) is Ad ( SO (3))-invariant. We calculatedet( X ) = 13 √ a − ab ) = 13 √ F A , where F A is given in (4.13), when we set a = cos( π/ − θ ) , b = sin( π/ − θ ) , − π/ < θ < π/ . It follows that F := 3 √ Y )restricts to F A and so F is the Cartan polynomial given in Theorem 4.7.A calculation shows F is exactly the polynomial given in (2.6) by Car-tan in the case when F is R .Note that f , the restriction of F to the unit sphere, has range [ − , f − ( ± 1) are the two singular submanifolds, both being the projec-tive plane. To see this, we set θ = ± π/ 6. Then, respectively, X = / √ − / √ − / √ , / √ / √ − / √ . Let us find the isotropy group L of the isotropy action on X , where L consists of all T ∈ SO (3) commuting with X . We see L is in diagonalblock form. Hence, L ≃ S ( O (1) × O (2)) , so that the singular orbits are Ad ( SO (3)) /L = SO (3) /S ( O (1) × O (2)) = R P . A look at the tables for the symmetric spaces of rank 2 of Types Iand II shows that there are four such spaces with g = 3, which are SU (3) /SO (3) , SU (3) × SU (3) / ∆( SU (3) × SU (3)) ,SU (6) /Sp (3) , E /F , whose Cartan polynomials of their isotropic orbits are the ones givenin (2.6).As in the SU (3) /SO (3) case, the singular orbits of the other threeexamples are, respectively, the complex, quaternionic and octonion pro-jective planes. The principal orbits are tubes around the projectiveplanes.The following grid table is the collection of all symmetric spaces G/K of Types I and II whose isotropy representations give homogeneousisoparametric hypersurfaces M . There are at most two multiplicities( m + , m − ) for the g principal curvatures. G K dim M g ( m + , m − ) S × SO ( n + 1) SO ( n ) n , SO ( p + 1) × SO ( n + 1 − p ) SO ( p ) × SO ( n − p ) n p, n − p ) SU (3) SO (3) 3 3 (1 , SU (3) × SU (3) SU (3) 6 3 (2 , SU (6) Sp (3) 12 3 (4 , E F 24 3 (8 , SO (5) × SO (5) SO (5) 8 4 (2 , SO (10) U (5) 18 4 (4 , SO ( m + 2) , m ≥ SO ( m ) × SO (2) 2 m − , m − SU ( m + 2) , m ≥ S ( U ( m ) × U (2)) 4 m − , m − Sp ( m + 2) , m ≥ Sp ( m ) × Sp (2) 8 m − , m − E ( Spin (10) × SO (2)) / Z 30 4 (6 , G SO (4) 6 6 (1 , G × G G 12 6 (2 , HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 25 Ozeki and Takeuchi’s work on inhomogeneous examplesfor g = 4 . Based on M¨unzner’s work, Ozeki and Takeuchi [50, I] con-structed two classes, each with infinitely many members, of inhomoge-neous isoparametric hypersurfaces with g = 4. This answered Cartan’sthird question in the negative.They also classified all isoparametric hypersurfaces with g = 4 whenone of the multiplicities is 2, which are all homogeneous [50, II].An important ingredient in their work is their expansion formulaof the Cartan-M¨unzner polynomial, which was inspired by Cartan’sapproach to the classification for g = 3 mentioned in Section 2. Thecentral theme is to study the focal manifolds, the singular set of the 1-parameter family of isoparametric hypersurfaces, to recover propertiesof the hypersurface. They fixed a point x on either one of the focalmanifolds, say, M + , and decomposed the ambient Euclidean space by R x , with coordinate t , the tangent space to M at x in S n , where atypical vector is denoted by y , and the normal space to M + at x in S n , where a typical vector is denoted by w with coordinates w i withrespect to a chosen orthonormal basis n , n , · · · , n m + . They expandedthe Cartan-M¨unzner polynomial in t , with undetermined coefficients in y and w , and substituted it into the two equations in (4.2) to result in F ( tx + y + w ) = t + (2 | y | − | w | ) t + 8( m + X a =0 p a w a ) t + | y | − | y | | w | + | w | − m + X a =0 ( p a ) + 8 m + X a =0 q a w a + 2 m + X a,b =0 h∇ p a , ∇ p b i w a w b , (4.14)where p a ( y ) (respectively, q a ( y )) is the a -th component of the 2nd (re-spectively, 3rd) fundamental form of M + at x . Furthermore, p a and q a are subject to ten defining equations [50, I, pp 529-530], of whichthe first three assert that, since S n , the 2nd fundamental matrix of M + in any unit normal direction n , has eigenvalues 1 , − , S n ) = S n for all n . We will return to the ten identities later.Ozeki and Takeuchi then introduced Condition A , where a point p on a focal manifold, say M + , is of Condition A if the shape operators S n at p share the same kernel for all n . Indeed, given a normal basis n , · · · , n m + at x with the associatedshape operators S , · · · , S m + , let E , E , E − be the eigenspaces of S with eigenvalues 0 , , − 1, respectively. Relative to the chosen orthonor-mal bases of E , E , E − , we have the matrix representations of theshape operators(4.16) S = Id − Id 00 0 0 , S a = A a B a A tra C a B tra C tra for 1 ≤ a ≤ m + , where A a : E − → E , B a : E → E and C a : E → E − .Condition A means that B a = C a = 0 for all 1 ≤ a ≤ m + at x . Consider n := ( n a + n b ) / √ S n into (4.15) we canextract(4.17) A a A trb + A b A tra = 2 δ ab Id, A tra A b + A trb A a = 2 δ ab . This implies that the symmetric matrices(4.18) T := (cid:18) I − I (cid:19) , T a := (cid:18) A a A tra (cid:19) , ≤ a ≤ m + , induce a symmetric Clifford C ′ m + -module structure on R m − , so that R m − is decomposed into k irreducible C ′ m + modules for some k , andthus 2 m − = k θ m + = 2 kδ m + , where, following the standard notation, θ s denotes the dimension of anirreducible C ′ s +1 -module while δ s denotes that of an irreducible skew-symmetric Clifford C s − -module satisfying the periodicity δ s +8 = 16 δ s with δ , · · · , δ being 1, 2, 4, 4, 8, 8, 8, 8, respectively. See Definition 5.1and what immediately follows it in the next section for a more detailedaccount.They focused on the case when δ m + = 1 + m + , i.e., the case whenthe dimension of an irreducible module of C m + is 1 + m + . It is wellknown that m + = 1, 3, or 7, which they verified by the above pe-riodicity formula for δ s . As a consequence m − is a multiple of 2, 4,8, respectively. Now, complex multiplication on C give the only ir-reducible representation of C , whereas left and right quaternionic oroctonion multiplications give the only two distinct irreducible C m + -representations in the cases m + = 3, or 7, respectively. Each suchirreducible skew-symmetric Clifford representation A , · · · , A m + recon-structs a symmetric Clifford representation C ′ m + via (4.18), and viceversa. Putting k such symmetric Clifford representations together, we HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 27 recover the second fundamental form, given in (4.16), of M + of codi-mension 1 + m + = 2 , , or 8 in S n .Having determined the second fundamental form, they introducedCondition B. For each tangent vector y decomposed into y = y + y + y − relative to the eigenspaces E , E , E − of the shape operator S ,they stipulated that q a ( y ) = m + X b =1 r ab ( y ) p b ( y ) , ≤ a, b ≤ m + for some 1st-degree polynomials r ab linear in y , to be a candidate forthe polynomial in (4.14). They substituted Condition B into the tendefining equations of an isoparametric hypersurface with four princi-pal curvatures and eventually determined the third fundamental form,and hence the Cartan-M¨uzner polynomial. Since the multiplicity pair( m + , m − ) does not appear on the list of those of homogeneous exam-ples when m + is 3 or 7, they found the first examples of inhomogeneousisoparametric hypersurfaces with four distinct principal curvatures andmultiplicity pair (3 , k ) and (7 , k ) in the sphere.Conditions A and B would play a major role in later development.We note that Takagi [57] classified the case when g = 4 and oneof the multiplicities is 1. They are congruent to the aforementionedexamples of Nomizu, and hence are homogeneous. His method is to alsoexpand the Cartan-M¨unzner polynomial at a focal point and analyzethe algebraic structures constrained by (4.2).5. The enlightenment, 1980-1999 Now that the two questions of Cartan were answered,the next question naturally came down to the possible multiplicity pairs( m + , m − ) of the g principal curvatures. The 1980s started with thepaper of Ferus, Karcher, and M¨unzner that constructed an infinite 2-dimensional array of multiplicity pairs each of which is associated withan isoparametric hypersurface, most of them inhomogeneous, whichinclude the examples of Ozeki and Takeuchi.To motivate their work, let us return to the examples of Nomizu (4.1).Set P := (cid:18) I − I (cid:19) , P := (cid:18) II (cid:19) , u := ( x, y ) tr , where I is the k by k identity matrix. Then F can be rewritten as F = | u | − X i =0 h P i u, u i , P i P j + P j P i = 2 δ ij I. Ferus, Karcher and M¨unzner’s construction is a far-reaching general-ization of this. Definition 5.1. The skew-symmetric (respectively, symmetric) Clif-ford algebra C n (respectively, C ′ n ) over R is the algebra generated bythe standard basis e , · · · , e n of R n subject to the only constraint e i e j + e j e i = − δ ij I (respectively , e i e j + e j e i = 2 δ ij I ) . The classification of the Clifford algebras is known [34]: n C n C H H ⊕ H H (2) C (4) R (8) R (8) ⊕ R (8) R (16) δ n C ′ n R ⊕ R R (2) C (2) H (2) H (2) ⊕ H (2) H (4) C (8) R (16) θ n δ n is the dimension of an irreducible module of C n − , and θ n isthe dimension of an irreducible module of C ′ n +1 . Moreover, C n (respec-tively, C ′ n ) is subject to the periodicity condition C n +8 = C n ⊗ R (16)(respectively, C ′ n +8 = C ′ n ⊗ R (16)). The generators e , · · · , e n actingon each irreducible module of either C n or C ′ n in the grid table giverise to n skew-symmetric or symmetric orthogonal matrices T , · · · , T n satisfying T i T j + T j T i = ± δ ij Id, yielding a representation of C n or C ′ n on the irreducible module, re-spectively. Note that we have θ n = 2 δ n . This is not coincidental. It says that we can construct symmetric rep-resentations of C ′ m +1 from skew-symmetric representations of C m − ,and vice versa. Indeed, let us be given k irreducible representations V , · · · , V k of C m − . Set V := V ⊕ + · · · ⊕ V k ≃ R l , l = kδ m . The representations of e , · · · , e m − on V , · · · , V k result in m − E , · · · , E m − on V . Set P := (cid:18) I − I (cid:19) , P := (cid:18) II (cid:19) , P i = (cid:18) E i − E i (cid:19) , ≤ i ≤ m − . Then P i P j + P j P i = 2 δ ij Id.P , · · · , P m generate a representation of C ′ m +1 on R l . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 29 The Cartan-M¨unzner polynomials of the examples of Ferus, Karcherand M¨unzner are(5.1) F := 2 | u | − m X i =0 ( h P i u, u i ) , u ∈ R l , l = kδ m . Note that we recover Nomizu’s example when m = 1.By a straightforward calculation, we conclude the following [30]. Proposition 5.1. The two multiplicities of the associated isoparamet-ric hypersurface are ( m, kδ m − m − , where m, k ∈ N to make the second entry positive, and the Clifford ac-tion operates on the focal manifold of codimension m in the ambientsphere. Moreover, for m ≡ mod , there are [ k/ 2] + 1 incongruentisoparametric hypersurfaces associated with each multiplicity pair, tobe indicated by [ k/ underlines in the following grid table. k δ m · · · , 2) (6 , 1) – – (9 , · · · , 1) (3 , 4) (4 , 3) (5 , 10) (6 , 9) (7 , 8) (8 , 7) (9 , · · · , 1) (2 , 3) (3 , 8) (4 , 7) (5 , 18) (6 , 17) (7 , 16) (8 , 15) (9 , · · · , 2) (2 , 5) (3 , 12) (4 , 11) (5 , 26) (6 , 25) (7 , 24) (8 , 23) (9 , · · · , 3) (2 , 7) (3 , 16) (4 , 15) (5 , 34) (6 , 33) (7 , 32) (8 , 31) (9 , · · · ... ... ... ... ... ... ... ... ... ... ...Among other things, Ferus, Karcher, and M¨unzner established Theorem 5.1. (1) The multiplicity pairs of the homogeneous isopara-metric hypersurfaces with four principal curvatures are preciselythose listed in the first, second, and fourth columns of the table,together with (9 , and the two pairs (2 , and (4 , not listedon the table. (2) The isoparametric hypersurfaces with multiplicity pairs in thethird and seventh columns are exactly the inhomogeneous ex-amples constructed by Ozeki and Takeuchi. So, except for the first, second and fourth columns, we have infinitelymany families, each with infinitely many members, of inhomogeneousisoparametric hypersurfaces with four principal curvatures. Note that we also have the fact that such a hypersurface with multiplicity pair(1 , l ) or (2 , l ) is congruent to the one with multiplicity pair ( l, 1) or( l, 2) [30, 6.5]. Note also that Cartan classified the cases when themultiplicities are { , } and { , } , both being homogeneous [6].Of particular interest is that the authors proved that the focal man-ifold M − , of the inhomogeneous family with multiplicity pair ( m = m + , m − ) = (3 , k ) constructed by Ozeki and Takeuchi, is homogeneouswhile M + is not. On the other hand, the two inhomogeneous examplesof multiplicity pair ( m = m + , m − ) = (8 , 7) constructed by the authorshave the property that both focal manifolds of the indefinite isopara-metric hypersurface, i.e., the one with P · · · P = ± Id , are inhomoge-neous, whereas for the definite one, i.e., the one with P · · · P = ± Id , M + is homogeneous while M − is not, so that, in particular, the two hy-persurfaces are not congruent; moreover, neither of them is congruentto the one with multiplicity pair ( m = m + , m − ) = (7 , 8) constructed byOzeki and Takeuchi. All of these properties were proved via geometricmethods without resorting to the aforementioned classification in thehomogeneous category. In fact, they showed through geometric meansthat most of the examples on the list are inhomogeneous.Coming next to the arena is the thesis of Abresch [1], in which headded a projective structure, in the case of g = 4 , or 6, to the topo-logical structure of M¨unzner so that now Stiefel-Whitney classes comeinto play with Steenrod squares to obtain the following:Assume m − ≤ m + . If g = 4, then ( A ): m − + m + + 1 is divisible by 2 k := min { σ : 2 σ > m − , σ ∈ N } , or, ( B ): m − is a power of 2 and 2 m − divides m + + 1, or, ( B ): m − is a power of 2 and 3 m − = 2( m + + 1).Moreover, if g = 6, then m − = m + = 1 , or 2 . Based on Abresch’s work, Dorfmeister and Neher [22] succeeded inclassifying the case when g = 6 and m + = m − = 1. It is the homoge-neous space.Grove and Halperin [31] established that when g = 4, m + = m − only when ( m + , m − ) = (1 , , or (2 , , HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 31 formula (2.7), one can show that the element e := ( e + e ) / √ F ( e ) = − e ij ∈ so (5 , R ), i < j , is the matrix whose only nonzero entries are at ( i, j ) and ( j, i )slots with value 1 and -1, respectively.5.2. Tang [59] pursued Abresch’s setup further and ob-tained the refined result for g = 4 that states that no isoparametric hy-persurfaces of type 4 B or 4 B exist if m − = 1 , , , or 8, while Fang [27]followed up to assert that the multiplicity pairs (2 , 2) and (4 , 5) are theonly possibilities in the case of 4 B . For type 4 A , Fang [26] settled alarge portion of the multiplicity problem: Theorem 5.2. Suppose g = 4 and m + ≤ m − . Then m + + m − + 1 isdivisible by δ ( m + ) if m + ≡ , , mod . In particular, the multiplicitypairs ( m + , m − ) , m + ≤ m − , of isoparametric hypersurfaces with fourprincipal curvatures are exactly those in the above grid table of theexamples constructed by Ferus, Karcher, and M¨unzner, provided m + ≡ , , mod . The multiplicity problem was finally settled by the remarkable paperof Stolz [56]: Theorem 5.3. Let g = 4 . The multiplicity pairs ( m + , m − ) , m + ≤ m − , of isoparametric hypersurfaces with four principal curvatures areexactly those in the above grid table of the examples constructed byFerus, Karcher, and M¨unzner, besides the pairs (2 , and (4 , not inthe grid table. He established that if ( m + , m − ) , m + ≤ m − , is neither (2 , 2) nor (4 , m + + m − + 1 is a multiple of 2 φ ( m − − , where φ ( n ) denotes thenumber of numerals s, ≤ s ≤ n, such that s ≡ , , , m + , m − ) are exactlythose in the grid table of Ferus, Karcher, and M¨unzner.His approach is reminiscent of the theorem of Adams: Theorem 5.4. If there are k independent vector fields on S n , then n + 1 is a multiple of φ ( k ) . The core technique Adams developed for proving the above theoremon vector fields was to what Stolz reduced his proof.Fang also showed [28] that, when g = 6, the isoparametric hyper-surface is diffeomorphic (respectively, homotopic) to the homogeneousexample when the equal multiplicity is 1 (respectively, 2); the state-ment in fact holds true in the more general proper Dupin category.Gary Jensen, Tom Cecil, and I started thinking seriously aboutthe classification of isoparametric hypersurfaces with four principal curvatures around 1998-1999. When I saw Stolz’s result, I said tomyself:“Ha-ha! What else could such a hypersurface be, except for theones of Ferus, Karcher, and M¨unzner?”6. The classification We spent quite a lot of time working at understand-ing the underlying geometry of the Ferus-Karcher-M¨unzner examplesin the early 2000s. Let us look at (5.1) more closely, where we set m = m + for convenience. M + is of codimension 1 + m in S n definedby quadrics, M + := { x ∈ S n : h P ( x ) , x i = · · · = h P m ( x ) , x i = 0 } . Its unit normal sphere at x is( U N ) x := { P ( x ) : P = m X i =0 a i P i , m X i =0 a i = 1 } , where P constitute a round sphere S m in the linear space of symmetricmatrices of size ( n + 1) × ( n + 1) equipped with the inner product h A, B i := − tr ( AB ) / 2. The map P ∈ S m P ( x ) ∈ U N x , is an isometry for each x ∈ M + .Consider the unit normal bundle U N of M + with the natural pro-jection π : U N −→ M + . The Levi-Civita connection on M + naturally splits the tangent bundleof U N into horizontal and vertical bundles H and V , respectively, T ( U N ) = V ⊕ H . At each n ∈ U N with base point x , the shape operator S n at x admitsthree eigenspaces E n , E n , E n − with eigenvalues 0 , , − 1. Explicitly, for n = P ( x ), h S n ( v ) , w i = −h P ( v ) , w i , v, w tangent at x, so that, in particular, it is easily checked by a dimension count that(6.1) E n = span { P Q ( x ) : Q ⊥ P in S m } ,E n = { v : P ( v ) = − v, v ⊥ Q ( x ) ∀ Q ∈ S m } ,E n − = { v : P ( v ) = v, v ⊥ Q ( x ) ∀ Q ∈ S m } , which are lifted to U N to further split H into three subbundles E , E , E − ,respectively, so that(6.2) T ( U N ) = V ⊕ E ⊕ E ⊕ E − . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 33 Now, for each P ∈ S m , the set F P := { P ( x ) : x ∈ M + } ⊂ U N, defines the section(6.3) s P : M + −→ U N, x : P ( x ) . The tangent space to F P at P ( x ) is, by (6.1),(6.4) T ( F P ) = G ⊕ E ⊕ E − , where G is the graph of the orthogonal bundle map P : E −→ V , P Q ( x ) Q ( x ) , ∀ Q ⊥ P ∈ S m . In other words, as P varies in S m , we can define the distribution(6.5) ∆ := G ⊕ E ⊕ E − , which is integrable (involutive) with leaves F P .Conversely, let g = 4. Suppose we are given an isoparametric hy-persurface M ⊂ S n with the focal manifolds M ± . We let m := m + and let U N be the unit normal bundle of M + . As above, we have thedecomposition (6.2). Suppose now there is an orthogonal bundle map(6.6) O : E −→ V , which gives rise to a distribution ∆ defined similarly as in (6.5) by thegraph G of O . We wish to find a characterization of the integrabilityof ∆.To this end, we let X a , X p , X α , X µ be an orthonormal frame span-ning, respectively, V , E , E , E − over T ( U N ), where a, p, α, µ denotethe indexes parametrizing the corresponding spaces with(6.7) 1 ≤ a ≤ m = m + , m + 1 ≤ p ≤ m, m + 1 ≤ α ≤ m + m − , m + m − + 1 ≤ µ ≤ m + 2 m − . This convention will be enforced henceforth. We then let θ a , θ p , θ α , θ µ be the dual frame to the orthonormal frame. We set ω ij := h dX j , X i i , and write(6.8) ω ij = X k F ijk θ k , where without the specification in (6.7) indexes are understood to takevalues in all possible index ranges. By (6.8),(6.9) F αpa = − S X a ( X α , X p ) , F µpa = S X a ( X µ , X p ) ,F µαa = S X a ( X α , X µ ) / , the respective components of the shape operator in the normal direction X a , where by a slight abuse of notation we use X p , X α , X µ to also denotetheir pushforwards via π : U N → M + .We will see the geometric meaning of F µαp later.Isoparametricity of M imposes many constraints on F ijk , which is notour concern here (see [7, p.16]).Now the orthogonal bundle map O in (6.6) gives a choice of X a once X p are given, namely, we may specify(6.10) X p − m := −O ( X p ) . With this choice it follows that the distribution ∆ in (6.5) is the kernelof θ a + θ a + m ; differentiating while invoking the structural equations for dθ i (see [7, (5.1), p. 16]), we obtain Proposition 6.1. [10, p. 137] ∆ is integrable if and only if (6.11) F µα p = F µα p − m ,F αa + m b = − F αb + m a ,F µa + m b = − F µb + m a . Let us understand the geometric meaning of this proposition. Eachlocal integral leaf of ∆ now gives rise to a local section(6.12) s : M + −→ U N, x n ( x ) , similar to the one in (6.3). Now, a special feature of g = 4 is that anynormal n ( x ) ∈ π − ( x ) also lives in M + (see the discussions in Section4.2); for clarity of notation, we denote n ( x ) by x ∈ M + . Furthermore,the normal space N x to M + at x is(6.13) N x = R x ⊕ E , where E , E , E − are the eigenspaces of the shape operator S n ( x ) at x with eigenvalues 0 , , and − 1, respectively. We may now find theeigenspaces E , E , E − of the shape operator S x at x ∈ M + to be E = n ( x ) ⊥ ⊂ N x = the normal space at x,E ± = E ± . It becomes evident now, in view of (6.9) and (6.13), that F µα p represents the component S ( X α , X µ ) / of the second fundamen-tal form of M + at x in the normal direction X p ∈ N x , HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 35 for which we have the identity [7, (5.6), p.16](6.14) m X a =1 ( F µα a F νβ a + F µβ a F να a ) = m X p = m +1 ( F µα p F νβ p + F µβ p F να p ) . Note that the local map f s : x ∈ M + −→ M + , x x , that the section s in (6.12) induces is a local isometry on M + , where( f s ) ∗ maps v ± ∈ E ± to ∓ v ± ∈ E ± , and maps v ∈ E to w := O ( v ) ∈ E . Proposition 6.2. [10, p. 138] Assume ∆ is integrable. The localisometries f s of M + extend to ambient isometries of S n for all s if andonly if (6.15) ω ab − ω a + mb + m = X p L pb a ( θ p − m + θ p ) for some smooth functions L pb a . The idea is to show that the local isometries f s , preserving the firstfundamental form, also preserve the second fundamental form and thenormal connection form if and only if the four sets of identities hold.As a result, the analyticity of isoparametricity now implies that thelocal isometries are in fact global ones.We can now fix an x ∈ M + and consider the unit normal m -sphere π − ( x ). Through each n ∈ π − ( x ) there passes a unique section s n of ∆ whose associated local isometry f s n is now a global one inducedby an isometry P n of S n that extends f s n . We thus have an S m -worthof isometries P of the ambient sphere inducing isometries of M + . Ana-lyzing how the S m -worth of the ambient isometries P interact with theexpansion formula (4.14) of Ozeki and Takeuchi confirms the following. Proposition 6.3. [10, 142-154] Assuming (6.11) and (6.15) , the S m -worth of ambient isometries P form a round sphere in the linear spaceof symmetric matrices of size ( n + 1) × ( n + 1) equipped with the stan-dard inner product, so that the isoparametric hypersurface M is oneconstructed by Ferus, Karcher, and M¨unzner. The proof in [7, 33-51] of this proposition was through extremelylong calculations by differential forms with remarkable cancellations!In contrast, the different proof in [10] is considerably shorter and moreconceptual. On the other hand, the former is entirely local and theintuition behind it is also rather clear, in that since isoparametrichypersurfaces are defined by an overdetermined differential system, exterior-differentiating enough times should result in sufficiently manyconstraints for the conclusion.Note that the three equations in (6.11) are algebraic in F ijk whilethe fourth one in (6.15) is a system of PDEs in them. It would bedesirable if (6.15) could be suppressed. This is indeed possible if M + is “sufficiently curved”. More precisely, we introduce the followingdefinition. Definition 6.1. [7, p. 19] The unit normal bundle U N of M + satisfiesthe spanning property at some n over base point x , if there is an X in E such that S ( X, · ) : E − −→ n ⊥ is surjective, and there is a Y in E − such that S ( · , Y ) : E −→ n ⊥ is surjective, at x . Here, E , E , E − are the eigenspaces of the shapeoperator S n at x with eigenvalues , , and − , respectively, and n ⊥ isthe orthogonal complement of n in the normal space at x . Equivalently, the spanning property is equivalent to the local condi-tions that the Euclidean vector-valued bilinear form(6.16) B ( X, Y ) := ( X αµ F µα x α y µ , X αµ F µα x α y µ , · · · , X αµ F µα m x α y µ )for α, µ in the specified range in (6.7) satisfies that B ( X, · ) : R m − R m is surjective for some X , and B ( · , Y ) : R m − → R m is surjective for some Y . Proposition 6.4. [7, Proposition 19, p. 28] Suppose the unit normalbundle U N of M + satisfies the spanning property at some n . Thenaround n the first equation in (6.11) , i.e., F µα a = F µα a + m , implies theremaining equations in (6.11) and (6.15) . The proof utilizes the spanning property and various identities [7,pp. 16-17] of covariant derivatives of F ijk .In view of Propositions 6.3 and 6.4, it suffices to find conditions towarrant(6.17) F µα a = F µα a + m , ∀ α, µ, with the spanning property , for the isoparametric hypersurface with four principal curvatures to beone constructed by Ferus, Karcher, and M¨unzner.Note that Takagi’s classification said at the end of Section 4 readilyfollows now. In this case, m = 1, a = 1 and p = 2 = a + m in (6.7).From (6.14), one reads off F µα a = ± F µα a + m HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 37 for all α and µ , where we can assume the sign is positive. Meanwhile,we need to verify that the spanning property holds. Suppose it is nottrue and B ( X, · ) is the zero map for every X in (6.16). Then it mustbe that B = 0 with m = 1 gives that F µα = 0 for all µ, α . Now, with a = 1, (4.16) extracts out of (4.15) the identity(6.18) A A tr + 2 B B tr = Id, A = (cid:0) F µα a (cid:1) = 0 , so that B tr = (cid:0) F α =2 m +1 p a , · · · , F α =2 m + m − p a (cid:1) satisfies B B tr = Id/ 2. This is impossible, whence follows Takagi’sclassification.It is at this point that algebraic geometry comes into play. We referthe reader to [14] for a rather detailed account of the commutativealgebra to be employed in the following. Since the algebro-geometricmethod in [7] is superseded by the simpler and more effective methodthat prevails in [11], [13], [15], to be discussed later, I will only indicatebriefly the inductive steps that are engaged by looking at the case when m = 2 classified by Ozeki and Takeuchi in [50, II]. Let us first look atthe spanning property. Lemma 6.1. [50, II, p. 45] If m − ≥ m + 2 , then each of the m polynomials p a ( x, y ) := X α µ F µα a x α y µ , ≤ a ≤ m, is irreducible.Proof. Note that4 p a ( x, y ) = h (cid:18) xy (cid:19) , (cid:18) A a A tra (cid:19) (cid:18) xy (cid:19) i , where A a is given in (4.16). Hence,rank( p a ) = 2 rank( A a ) , where rank( p a ) is that of the symmetric matrix U := (cid:18) A a A tra (cid:19) . Let V := S a − U (for notational ease, we use U to also denote its augmentation by zerosto match the size of S a ). We have rank( S a ) ≤ rank( U )+rank( V ). Since S a is similar to S , we know rank( S a ) = 2 m − , whereas it is clear that rank( U ) = 2 rank( A a ) and rank( V ) ≤ m . Putting these together, wederive(6.19) 2( m − − m ) ≤ A a ) = rank( p a ) . If p a is reducible, then p a = f g for two linear polynomials f and g ofthe form f = X α a α x α + X µ a µ y µ , g = X α b α x α + X µ b µ y µ . Let a := (cid:0) a α a µ (cid:1) tr , b := (cid:0) b α b µ (cid:1) tr of size 2 m − × 1. Then 4 p a = ( ab tr + ba tr ) / ≤ 2. Thus (6.19)implies m − ≤ m + 1, a contradiction. (cid:3) Lemma 6.2. If m − ≥ m + 1 , then the polynomials p a , ≤ a ≤ m , arelinearly independent.Proof. Suppose they are linearly dependent. There are nonzero con-stants, c , · · · , c m , not all zero, such that0 = c A + · · · + c m A m = q c + · · · + c m A n for some unit normal n . So, we may assume without loss of generalitythat A = 0. But then (6.18) results in B B tr = Id/ 2, which meansthat the row vectors of B of size m − × m are linearly independent, sothat m − ≤ m . This is absurd. (cid:3) Note that both lemmas are equally good if we complexify the poly-nomials p a , as they have real coefficients.In the case of m = 2 in the classification of Ozeki and Takeuchi [50,II], since the two polynomials p and p are irreducible and linearlyindependent, one cannot be a constant multiple of the other. We con-clude that if f p + g p = 0 for two polynomials f and g in x and y ,then p divides f . We can put this succinctly in the language of regularsequences in commutative algebra. Definition 6.2. [14, Definition 1, p. 84] A regular sequence in thecomplex polynomial ring P [ l ] in l variables is a sequence p , · · · , p k in P [ l ] such that firstly the variety defined by p = · · · = p k = 0 in C l is not empty. Moreover, p i is a non-zerodivisor in the quotient ring P [ l ] / ( p , · · · , p i − ) for ≤ i ≤ k ; in other words, any relation p f + · · · + p i − f i − + p i f i = 0 will result in f i being in the form f i = p h i + · · · + p i − h ii − HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 39 for some h i , · · · , h ii − ∈ P [ l ] for ≤ i ≤ k . A regular sequence imposes strong algebraic independence amongstits elements [14, 2.3, p. 91]. Proposition 6.5. (Special case) [7, p. 62] Assume m = 2 and m − ≥ . We use p C and p C to denote the complexification of p and p . Let V C be the variety carved out by p C = p C = 0 . Then dim( V C ) = 2 m − − . In particular, the spanning property is true.Proof. p C and p C are irreducible by Lemma 6.1. It is well known thatthe irreducible p C cuts out an irreducible variety V and the irreducible p C cuts out the variety V of pure codimension 1 in V since they arelinearly independent [54, Theorem 5, p.58]. In particular, the realcounterpart V of V C satisfiesdim( V ) ≤ m − − . We claim the equality holds. To this end, consider the map V ι −→ R m − × R m − π −→ R m − , where ι is the standard embedding and π is the projection onto thefirst summand. Also, for each x ∈ R m − , consider the map S x : y −→ ( p ( x, y ) , p ( x, y )) . Note that(6.20) dim(kernel( S x )) ≥ m − − > , and the kernel of S x is exactly the set { y ∈ R m − : ( x, y ) ∈ ( π ◦ ι ) − ( x ) } . Hence, the map π ◦ ι is surjective. Now, the set Z := { x ∈ R m − : kernel( S x ) assumes the minimum dimension t } is Zariski open. So, by Sard’s theorem, there is an irreducible compo-nent W of V whose image via π ◦ ι contains a regular value x ∈ Z ,so that ( π ◦ ι ) − ( q ) ≃ R t for q in a neighborhood of x ; in other words,( π ◦ ι ) − ( Z ) around x in W is a product. We conclude, by (6.20),dim( W ) = t + m − ≥ m − + m − − m − − . Consequently, the claim(6.21) dim( W ) = 2 m − − t = m − − x , we enjoy the prop-erty dim(( π ◦ ι ) − ( x )) = t = m − − , or, that S x is surjective, so that the spanning property is true for some x . Likewise, the spanning property is true for some y . (cid:3) Let us next investigate the validity of (6.17). In view of (4.16) andthe notation set around it, we denote the components of the secondfundamental form by(6.22) ˜ p i ( v ) := h S i ( v ) , v i , ≤ i ≤ m, and we define D = { z ∈ E + ⊕ E − : | z | = 1 , ˜ p i ( z ) = 0 , i = 0 , · · · , m } . Lemma 6.3. [7, Proposition 25, p. 51] D = ( E + ⊕ E − ) ∩ M + . Proof. This follows when we set t = w = · · · = w m = 0 in (4.14). (cid:3) Notation as around (4.16) and the index range convention (6.7) pre-vailing, similar to the discussions below (6.12), let us denote n by x ∈ M + . Three lines above (6.14) we gave the geometric meaning of F µα p at x vs. F µα a at x . An immediate consequence of the precedinglemma is the following crucial observation. Corollary 6.1. [7, Proposition 28, p. 53] We have that the zero locus V of p a := X α µ F µα a x α y µ , ≤ a ≤ m, and the zero locus of p a := X α µ F µα p x α y µ , m ≤ p ≤ m, are identical in R P m − × R P m − .Proof. The eigenspaces of the shape operator S x at x with eigenvalue ± S n at x witheigenvalues ± 1. The preceding corollary then implies that D = { z = ( x, y ) ∈ E + ⊕ E − : | z | = 1 , X α µ F µα i x α y µ = 0 , ≤ i ≤ m } . Furthermore, observe that in (6.22)˜ p = | x | − | y | = 0on D while | x | + | y | = 1 so that | x | = | y | = 1 / 2. The result followsby projectivization. (cid:3) HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 41 Note that the same conclusion need not necessarily hold on C P m − × C P m − when we complexify the associated polynomials, since the com-plexification of the real irreducible components need not exhaust all thecomplex irreducible components. Nevertheless, the situation is clear inthe complex irreducible case, as follows. Corollary 6.2. [7, p. 63] Notation as in the preceding corollary, if thezero locus V C of p C a , ≤ a ≤ m , is an irreducible variety in C P m − × C P m − and, moreover, the complex dimension of V C equals the realdimension of V , then F µα a = F µα a + m in (6.17) after an orthogonal basischange.Proof. Since the zero locus V C of p C a , ≤ a ≤ m, is irreducible, its ideal I := ( p C , · · · , p C m ) is prime. On the other hand, the preceding corollarysays that p , · · · , p m vanish on V having the same real dimension asthe complex dimension of V C , we conclude that p C , · · · , p C m also vanishon V C so that by Hilbert’s Nullstellensatz, p C a = m X b =1 r ab p C b , ≤ a ≤ m, for some polynomials r ab . We deduce that p a = m X b =1 c ab p b , ≤ a ≤ m, for some constants c ab , since p a and p a are all of bidegree (1 , F µα a + m = m X b =1 c ab F µα b , ≤ a ≤ m, whence (6.14) gives that (cid:0) c ab (cid:1) is indeed an orthogonal matrix. Theconclusion follows by a suitable orthogonal basis change. (cid:3) Let us now return to the classification of Ozeki and Takeuchi inthe case m = 2. By Lemma 6.1, Lemma 6.2, (6.17), (6.21), Propo-sition 6.5, Corollary 6.1, and Corollary 6.2, we can conclude that theisoparametric hypersurface of multiplicity pair ( m = 2 , m − ), m − ≥ , is one constructed by Ferus, Karcher, and M¨unzner, and hence is ho-mogeneous, so long as we can verify that the zero locus of p C and p C isirreducible. This is indeed the case. It relies on the criterion of Serrefor irreducible varieties. Theorem 6.1. [14, Theorem 1, p. 85; Theorem 2, p. 89] Let I k :=( f , · · · , f k ) be the ideal generated by a regular sequence f , · · · , f k , k ≤ l, in the complex polynomial ring P [ l ] in l variables z , · · · , z l , whose zerolocus is V k . Let J k be the subvariety of V k consisting of all points of V k where the Jacobian matrix (6.23) ∂ ( f , · · · , f k ) /∂ ( z , · · · , z l ) is not of full rank k . Suppose the codimension of J k is ≥ in V k . Then V k is reduced, i.e., I k is a radical ideal.Moreover, if V k is connected and the codimension of J k is ≥ in V k , then V k is irreducible, i.e., I k is a prime ideal. In particular, theconnectedness condition is automatically satisfied when f , · · · , f k arehomogeneous polynomials. Notation as above, we know the homogeneous p C and p C form aregular sequence by Lemma 6.1 and the discussion above Definition 6.2.It thus suffices to check that the codimension 2 estimate holds true inTheorem 6.1, where k = 2 and l = 2 m − , for the classification of Ozekiand Takeuchi to go through. An elementary linear algebra argumentestablishes the following. Lemma 6.4. [7, Lemma 49, p. 64] We set m := m + as usual. No-tation is as in (4.16) . There is an orthonormal basis in E and anorthonormal basis in E − such that relative to these bases we have,in (4.16) , (1): B = C = (cid:18) σ (cid:19) with σ = diag( σ , · · · , σ r ) , σ s > , ∀ s , (2): A = (cid:18) I 00 ∆ (cid:19) , where ∆ is an r × r matrix in block form ∆ = ∆ . . . . . . . . . ... ... ... ... ... with ∆ = 0 and ∆ i , i ≥ , nonzero skew-symmetric matricesin block form ∆ i = f i . . . − f i . . . f i . . . − f i . . . ... ... ... ... ... , and , (3): ∆ i = − (1 − σ i ) Id. Clearly, the lemma holds for any fixed a ≥ S a . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 43 Corollary 6.3. dim( Ker ( A a )) = dim(∆ ) ≤ r = rank( B a ) ≤ m forall a ≥ . With m = 2, we first estimate the dimension of the subvariety Z of C m − × C m − at each point of which the Jacobian matrix (6.23) of p C , p C is of rank < 2. At ( x, y ) ∈ Z , the differentials dp C , dp C arelinearly dependent, i.e., there are c , c ∈ C , depending on ( x, y ), suchthat 0 = X a =1 c a dp C a = X α ( X a,µ c a F µαa y µ ) dx α + X µ ( X a,α c a F µαa x α ) dy µ , which requires that the coefficients of dx α be zero and the coefficientsof dy µ be zero. Thus Z = { ( x, y ) ∈ C m − × C m − : ∃ ( c , c ) , X a c a A tra x = X a c a A a y = 0 } . Accordingly, for a fixed ( c , c ) let us define Z ( c ,c ) := { ( x, y ) ∈ C m − × C m − : X c a A tra x = X a c a A a y = 0 } . Consider the incidence space Y in C P × C m − × C m − given by { ([ c : c ] , x, y ) : ( x, y ) ∈ Z ( c ,c ) } . The standard projection of Y to C m − × C m − maps Y onto Z . Let π : Y −→ C P be the standard projection of Y to C P . Then with respect to π wehave(6.24) dim( Z ) ≤ dim( Y ) ≤ dim(base) + dim(fiber) , where dim(fiber) is the maximal dimension of all fibers. It is easier toestimate the dimension of the fibers π − { [ c : c ] } = Z ( c ,c ) . In fact, itcomes down to estimating the dimension of T ( c ,c ) := { y ∈ C m − : X a c a A a y = 0 } for a fixed ( c , c ), because this estimate will also be a valid upperbound for the dimension of { x ∈ C m − : P a c a A tra x = 0 } , thus giving usthe estimate dim( Z ( c ,c ) ) ≤ T ( c ,c ) ) . Case (1) . c , c are either all real or all purely imaginary. This is theeasier case. It is essentially Corollary 6.3. We havedim( T ( c ,c n ) ) ≤ r ≤ m and dim( Z ( c ,c ) ) ≤ T ( c ,c ) ≤ m Case (2) . c , c are not all real and not all purely imaginary. Write c k = α k + √− β k . We may assume without loss of generality that c S e + c S e = ( α + √− β ) S e + √− β S e . By restricting to the A -block in S again we see that β A y = √− α + √− β ) A y ;we may assume both coefficients are nonzero, or else we would be backto Case (1). Hence we are now handling(6.25) ( A − zA ) y = 0for some nonzero z ∈ C . By Lemma 6.4, we may assume A = (cid:18) I 00 ∆ (cid:19) . Write A = (cid:18) Θ ΛΩ Γ (cid:19) of the same block sizes as A . Inspecting the identity A A tr + A A tr + B B tr + B B tr = 0 , extracted out of (4.15) we obtain(6.26) Θ + Θ tr = 0 . With this and r ≤ m = 2, one can eventually come up with the esti-mate [7, p. 71]dim( T ( c ,c ) ) ≤ ( m − + r ) / ≤ ( m − + m − / , whose details are not our concern here. Note that the upper bound( m + m − / 2, instead of the weaker ( m + m ) / 2, holds because weknow m + m is an odd number if 2 ≤ m < m − by M¨unzner [46, II](or, by the more general Abresch [1] mentioned in Subsection 5.1), andas a result(6.27) dim( Z ( c ,c ) ) ≤ T ( c ,c ) ) ≤ m − + m − m − + 1 . On the other hand, the base of π consists of finitely many points sincegeneric [ c : c ] would make Z ( c , c ) null. Putting these together, (6.24)yields(6.28) dim( Z ) ≤ m − + 1 . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 45 To achieve an estimate for dim( J ), where J and V are defined in (6.23)with m = 2, by a well known fact [54, Corollary 5, p. 57], the zerolocus V of p C and p C has the propertydim( V ) ≥ m − − , so that by the fact that J ⊂ Z we achieve the a priori estimatedim( J ) ≤ dim( V ) − m − ≥ 5, in which case the ideal ( p C , p C ) is prime as a result ofSerre’s criterion. The isoparametric hypersurface with the multiplicitypair (2 , m − ), m − ≥ 5, is thus homogeneous. Although this suffices forthe conclusion in the case of m = 2 and m − ≥ m − are all odd [50,II, p. 49], we can perform a general cutting procedure to reach theconclusion for m − ≥ 4. Indeed, consider the surjective map f : C m − × C m − −→ C , ( x, y ) ( p C ( x, y ) , p C ( x, y )) .Z is the set where df is of rank < J = f − (0) ∩ Z .Let W k , 0 ≤ k ≤ , be the subvarieties of Z where df is of rank ≤ − k . We have W ⊃ W . Let X j := W j \ W j +1 . Then Z isstratified into X , X with X j Zariski open in W j , where df is of rank1 − j on X j . Around each ( x, y ) in X , we may assume ∇ ( p C ) is amultiple of ∇ ( p C ) considered as column vectors. Since (cid:0) p C p C (cid:1) = (cid:0) x y (cid:1) (cid:0) ∇ ( p C ) ∇ ( p C ) (cid:1) , we see that near generic ( x, y ) in X , the image of f is of dimension 1.Therefore, a generic line cut through the origin performed in the targetspace of f cuts down the dimension by 1 from Z in the dimensionestimate of J = f − (0) so that dim( J ) ≤ dim( Z ) − ≤ m − by (6.28)and thus dim( J ) ≤ dim( V ) − m − ≥ 4, whence follows theclassification of Ozeki and Takeuchi when m = 2 and m − ≥ m : Proposition 6.6. [7, Proposition 46, p. 61] Notation as in Theo-rem 6.1 above, set m = m + as usual and assume m − ≥ m + 2 . For n ≤ m , let Z n = { ( x, y ) ∈ C m − × C m − : ∃ ( c , · · · , c n ) , n X a =1 c a A tra x = n X a =1 c a A a y = 0 } , and f n : ( x, y ) ∈ C m − × C m − −→ R n , ( x, y ) ( p ( x, y ) , · · · , p n ( x, y )) with J n = f − (0) ∩ Z n . If m − ≥ m , then (6.29) dim( J n ) ≤ dim( V n ) − for all n ≤ m . If m = 2 m − , then dim ( J n ) ≤ dim( V n ) − for all n ≤ m − while dim ( J m ) ≤ dim( V m ) − . As a consequence of the preceding proposition, an argument similarto the one outlined above for the case m = 2 results in the classificationtheorem: Theorem 6.2. [7, Theorem 47, p. 61] If m − ≥ m + − , then anisoparametric hypersurface with four principal curvatures is one con-structed by Ferus, Karcher, and M¨unzner, where the Clifford actionoperates on M + . Note that when m − = 2 m − I m =( p C , · · · , p C m ) is only radical by Theorem 6.1, so that a priori the argu-ments in Corollary 6.2 do not appear to work. However, as in (6.21),the real variety V m cut out by p , · · · , p m is of real dimension 2 m − − m .Let W be an irreducible component of V C m containing an irreduciblecomponent of V m of real dimension 2 m − − m ; the complex dimensionof W is 2 m − − m . Then Nullstellensatz holds locally, around genericpoint z := ( x, y ) of W , for the ideal I m so that, in the notation ofCorollary 6.2,(6.30) F µα a + m x α y µ = m X b =1 c ab F µα b x α y µ , summed on α, µ, for 1 ≤ a ≤ m and some rational functions c ab ( x, y ) smooth around z .Since the projection sending W ⊂ R m − × R m − to x ∈ R m − is surjectivearound z by the local structure of W given in Proposition 6.5, we see(6.31) F µα a + m x α y µ = m X b =1 c ab ( x ) F µα b x α y µ by Taylor-expanding along R m − around x , where c ab ( x ) is the collectionin the analytic c ab ( x, y ) of the terms depending only on x , and so partialdifferentiating with respect to y µ gives F µα a + m x α = m X b =1 c ab ( x ) F µα b x α , summed on α. Taking second order partial derivatives with respect to x , denoted by c ′′ ab , we obtain c ′′ ab F µα b x α = 0for all µ , which implies c ′′ ab = 0 by the surjectivity of the map(6.32) S xm : y ( p ( x, y ) , · · · , p m ( x, y )) , p a = X αµ F µαa x α y µ , HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 47 as detailed in Proposition 6.5. Hence, c ab ( x ) are of degree at most 1in x and hence must be a constant by comparing types in (6.31). Wearrive at the same conclusion as in the case when I m is a prime ideal.In particular, this takes care of the case ( m, m − ) = (2 , 3) in Ozeki andTakeuchi’s classification since their explicit formula [50, II, p. 49] for p and p results in that V C is reduced when m − = 3 [7, Remark 53,p. 73].As good as it gets, Theorem 6.2 exactly reaches the borderline toaccount for all multiplicity pairs except for the four exceptional cases( m = m + , m − ) = (3 , , (4 , , (6 , , and (7 , M + to full generality, where the E -component had been entirelyignored when, inspired by Ozeki and Takeuchi, studying the restrictedform p a = P αµ F µα a x α y µ defined only on E ⊕ E − . Moreover, a glanceat (4.16) showed that every component ˜ p a , ≤ a ≤ m, of the secondfundamental form of M + is irreducible since they are equivalent to˜ p = | x | − | y | after a coordinate change, so long as m − ≥ 2, obtained almost forfree. It seemed reasonable to replace p , · · · , p m by ˜ p , · · · , ˜ p m and findcriteria to warrant that ˜ p , · · · , ˜ p m form a regular sequence by workingthrough the successive codimension 2 estimates in (6.29). The ideapaid off.6.2. From now on we stick to the convention that m := m + ≤ m − . For ease of notation, we denote the components of thesecond fundamental form of M + by p , · · · , p m without the tilde, and,moreover, we drop the superscript C in p C of a real polynomial p when-ever we indicate that p lives in P [ l ], the polynomial ring in l complexvariables, where l := 2 m − + m is the dimension of M + . The case of multiplicity pair (3,4) In the expansion formula (4.14), the components of the second andthird fundamental forms of M + are intertwined in ten convoluted equa-tions. The first three say that the shape operator S n satisfies ( S n ) = S n for any normal direction n , which is agreeable with the fact thatthe eigenvalues of S n are 0 , , − < p a , q b > := h∇ p a , ∇ q b i , ≤ a, b ≤ m. The fourth and fifth combined and the sixth are(6.33) < p a , q b > + < p b , q a > = 0 , (6.34) << p a , p b >, q c > + << p c , p a >, q b > + << p b , p c >, q a > = 0 , for distinct a, b, c . The seventh is(6.35) p q + · · · + p m q m = 0 . Set G := P ma =0 ( p a ) . The last three are(6.36) 16 m X a =0 ( q a ) = 16 G | y | − < G, G >, (6.37) 8 < q a , q a > = 8( < p a , p a > | y | − ( p a ) )+ << p a , p a >, G > − G − m X b =0 < p a , p b > , (6.38) 8 < q a , q b > = 8( < p a , p b > | y | − p a p b )+ << p a , p b >, G > − m X c =0 < p a , p c >< p b , p c >, a, b distinct . (6.35) caught my eye while others seemed dauntingly entangling; it isthe well known syzygy equation in commutative algebra. A propertythe syzygy equation enjoys is that when p , · · · , p m form a regularsequence in P [ l ], l = 2 m − + m , we have(6.39) q a = X b r ab p b , r ab = − r ba for some first degree polynomials r ab [14, Proposition 4]. This is ex-actly Condition B of Ozeki and Takeuchi. With this observation, Iworked out in [11] the a priori codimension 2 estimate in P [ l ] for thecomponents p , · · · , p m of the second fundamental form of M + . Indeed,following the convention in (6.7), let us write(6.40) r ab = X α T αab u α + X µ T µab v µ + X p T pab z p ,p = X α ( u α ) − X µ ( v µ ) ,p a = 4 X αµ F µα a u α v µ − X αp F αp a u α z p + 2 X µp F µp a v µ z p . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 49 Note that y in (6.36) through (6.38) denotes a tangential vector whosecomponents are u α , v µ , and z p . Substituting (6.40) into (6.39) andcomparing polynomial types yields that the coefficient of ( u α ) , denoted q αααa , in the polynomial expression of q a is q αααa = T αa , while the right hand side of (6.36) ensures that there are no ( u α ) -terms. We conclude(6.41) T αa = 0 = − T α a ; likewise, T µa = − T µ a = 0 . In other words, r a consists of only z p -terms in the expansion of q . Meanwhile, it is known that q is homogeneous of degree 1 in u α , v µ and z p by [50, I, p. 537]. We expand the right hand side of (6.39) toascertain that the coefficient of the u α v µ z p -term of q , denoted q αµp , is(6.42) q αµp = 4 X b ≥ T p b F µα b . Here comes something particularly nice. It turns out that q at x encodes information of the second fundamental form at x := n ∈ M + .More precisely, it is shown in [11] that q αµp = 4 F µα p . (See also [14, p. 97] for a different proof using the expansion for-mula (4.14).) It follows that F µα p = X b f pb F µα b , f pb := T p b . We need only verify that (cid:0) f pb (cid:1) is an orthogonal matrix for the firstequation in (6.11) to hold. It is remarkable that this is indeed trueas a consequence of a piece of commutative algebra [39, p. 153] (seealso [14, p. 91]): Proposition 6.7. Let p , · · · , p k be a regular sequence in P [ l ] . Let F ( t , · · · , t k ) be a homogeneous polynomial of degree d in k variableswith coefficients in P [ l ] . Suppose F ( p , · · · , p k ) = 0 . Then all thecoefficients of F belong to I = ( p , · · · , p k ) . The key idea of showing the orthogonality of ( f pb ) is to rewrite (6.37)as a polynomial homogeneous in all p a p b whose coefficients are homo-geneous polynomials of degree 2, so that these coefficients are linear combinations of all p a by the preceding proposition. Specifically, thecoefficient of ( p ) is16 m X a =1 ( r a ) − X α ( u α ) + X µ ( v µ ) + X p ( z p ) ) + 4 < p , p >, which is a linear combination of p , p , · · · , p m . Knowing that r a arefunctions of z p alone by (6.41), we invoke (6.40) and compare variabletypes to conclude(6.43) m X a =1 ( r a ) = m X p = m +1 ( z p ) , which thus asserts the orthogonality of the matrix (cid:0) f pb (cid:1) . We may nowassume(6.44) T a + m b = f a + mb = δ ab , so that F µα a + m = F µα a , and as a result of (6.41) and (6.44) derive r b = X a δ ab z a + m = z b + m . With the Einstein summation convention, we calculate q = r b p b = 2( δ ab z a + m )( S bαµ x α y µ + S bα c + m x α z c + m + S bµ c + m y µ z c + m ) . Hence, we obtain X abcα ( δ ab z a + m ) ( S bα c + m u α z c + m ) = 0 , or equivalently, X ac S aα c + m z c + m z a + m = 0 . In other words, we have(6.45) F αc + m a = − F αa + m c . Similarly,(6.46) F µc + m a = − F µa + m c . In particular, (6.11) are valid now. What is satisfying is that m = m + < m − is the only condition to warrant that (6.15) is true, sothat Proposition 6.3 gives that the isoparametric hypersurface is oneconstructed by Ferus, Karcher, and M¨unzner, as follows. HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 51 Proposition 6.8. [11, Proposition 4] Assume m = m + < m − . Sup-pose Condition B of Ozeki and Takeuchi holds, which is the case whenthe components of the second fundamental form p , p , · · · , p m of M + form a regular sequence in P [ l ] , l = 2 m − + m . Then the isoparametrichypersurface is of the type constructed by Ferus, Karcher, and M¨unznerwith the Clifford action operating on M + . The proof uses another easier spanning property [7, Proposition 7,p. 18], which states that if m < m − then ( F µα , · · · , F µαm ), ∀ α, µ , span R m , to verify that (6.11) implies (6.15) if m < m − , by utilizing certainidentities in [7, Proposition 19].The preceding proposition is pivotal for the classification of the re-maining cases, to be discussed later.In view of the preceding proposition, we need only find conditionsto guarantee that p , · · · , p m form a regular sequence in P [ l ], which iswhere Serre’s criterion Theorem 6.1 of codimension 2 estimate comes inagain. We record the inductive scheme to generate a regular sequence. Lemma 6.5. [7, Proposition 39] Let p , · · · , p m ∈ P [ l ] be linearlyindependent homogeneous polynomials of equal degree ≥ . For each ≤ k ≤ m − , let V k be the variety defined by p = · · · = p k = 0 , andlet J k be the subvariety of V k , where the Jacobian ∂ ( p , · · · , p k ) /∂ ( z , · · · , z l ) is not of full rank k + 1 . If the codimension of J k in V k is ≥ for all ≤ k ≤ m − , then p , · · · , p m form a regular sequence. Let us parametrize C m − + m by points ( u, v, w ), where u, v ∈ C m − and w ∈ C m . For k ≤ m , let V k := { ( u, v, w ) ∈ C m − + m : p ( u, v, w ) = · · · = p k ( u, v, w ) = 0 } be the variety carved out by p , · · · , p k in P [ l ]. We first estimate thedimension of the subvariety X k of C m − + m defined by X k := { ( u, v, w ) ∈ C m − + m : rank of Jacobian of p , · · · , p k < k + 1 } . This amounts to saying that there are constants c , · · · , c k such that(6.47) c dp + · · · + c k dp k = 0 . Since p a = h S a ( x ) , x i , we see dp a = 2 h S a ( x ) , dx i for x = ( u, v, w ) tr ;therefore, by (6.47), X k = { ( u, v, w ) : ( c S + · · · + c k S k ) · ( u, v, w ) tr = 0 } . for [ c : · · · : c k ] ∈ C P k , where h S a ( X ) , Y i = h S ( X, Y ) , n a i is the shapeoperator of the focal manifold M + in the normal direction n a . Since J k = X k ∩ V k , by Lemma 6.5 we wish to establishdim( X k ∩ V k ) ≤ dim( V k ) − k ≤ m − p , p , · · · , p m form a regular sequence.Note that for a fixed λ = [ c : · · · : c k ] ∈ C P k , if we set S λ := { ( u, v, w ) : ( c S + · · · + c k S k ) · ( u, v, w ) tr = 0 } , then we have(6.48) X k = ∪ λ ∈ C P k S λ . Thus, it boils down to estimating the dimension of S λ .We break it into two cases. If c , · · · , c k are either all real or allpurely imaginary, then dim( S λ ) = m, since c S n + · · · + c k S n k = cS n for some unit normal vector n and somenonzero real or purely imaginary constant c , and we know that the nullspace of S n is of dimension m for all normal n .On the other hand, if c , · · · , c k are not all real and not all purelyimaginary, then after a normal basis change, we may assume that(6.49) S λ = { ( u, v, w ) : ( S ∗ − ι λ S ∗ ) · ( u, v, w ) tr = 0 } for some complex number ι λ relative to a new orthonormal normal basis n ∗ , n ∗ , · · · , n ∗ k in the linear span of n , n , · · · , n k by the Gram-Schmidtprocess. In matrix terms, the equation in (6.49) assumes the form(6.50) A BA tr CB tr C tr xyz = ι λ I − I 00 0 0 xyz , where x, y , and z are (complex) eigenvectors of S ∗ with eigenvalues1 , − 1, and 0, respectively.The decomposition Lemma 6.4 ensures that we can normalize thematrix on the left hand side of (6.50) to decompose x, y, z into x =( x , x ) , y = ( y , y ) , z = ( z , z ) with x , y , z ∈ C r λ , where r λ is therank of B , or intrinsically, m − r λ is the dimension of the intersection of the kernels of S ∗ and S ∗ . With respect to this decomposition either x = y = 0, or both arenonzero with(6.51) ι λ = ±√− . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 53 In both cases we have x = − y and can be solved in z so that z maybe chosen to be a free variable. Hence, either x = y = 0, in whichcase dim( S λ ) = m, or both x and y are nonzero, in which case y = ±√− x and so(6.52) dim( S λ ) = m + m − − r λ , where x contributes the dimension count m − − r λ while z does m .Now, by (6.48) we see(6.53) J k = X k ∩ V k = ∪ λ ∈ C P k ( S λ ∩ V k ) , where V k is also defined by p ∗ = · · · = p k ∗ = 0. Let us cut S λ by0 = p ∗ = X α ( x α ) − X µ ( y µ ) to achieve an a priori estimate of dim( J k ).Case 1: x and y are both nonzero. This is the case of nongeneric λ ∈ C P k . We substitute y = ±√− x and x and y in terms of z into p ∗ = 0 to deduce that0 = p ∗ = ( x ) + · · · + ( x m − − r λ ) + z terms;hence, p ∗ = 0 cuts S λ to reduce the dimension by 1. In other words,now by (6.52),(6.54) dim( V k ∩ S λ ) ≤ ( m + m − − r λ ) − ≤ m + m − − . Meanwhile, only a subvariety of λ of dimension k − C P k assumes ι λ = ±√− 1; in fact, this subvariety is the smooth hyperquadric(6.55) Q k − := { λ = [ c : · · · : c k ] : c + · · · + c k = 0 } in C P k . This is because if we write ( c , · · · , c k ) = α + √− β where α and β are real vectors, then ι λ = ±√− h α, β i = 0 and | α | = | β | . In other words, the nongeneric λ ∈ C P k constitute the smooth hyperquadric. Therefore, by (6.53), anirreducible component W of J k over nongeneric λ will satisfy(6.56) dim( W ) ≤ dim( V k ∩ S λ ) + k − ≤ m + m − + k − . (Total dimension ≤ base dimension + fiber dimension.)Case 2: x = y = 0. This is the case of generic λ , where dim( S λ ) = m ,so that an irreducible component V of J k over generic λ will satisfydim( V ) ≤ m + k ≤ m + m − + k − , as we may assume m − ≥ 2, noting that the case m = m − = 1 followsfrom Takagi’s classification for m = 1 as mentioned above.Putting these two cases together, we conclude that(6.57) dim( J k ) = dim( X k ∩ V k ) ≤ m + m − + k − . On the other hand, since V k is cut out by k + 1 equations p = · · · = p k = 0, we have(6.58) dim( V k ) ≥ m + 2 m − − k − . Therefore,(6.59) dim( J k ) ≤ dim( V k ) − k ≤ m − 1, taking m − ≥ m − m − ≥ m − , we have estab-lished (6.59) for k ≤ m − 1, so that the ideal ( p , p , · · · , p k ) is primewhen k ≤ m − p , p , · · · , p m form a regular sequence. It follows by Propo-sition 6.8 that the isoparametric hypersurface is of the type constructedby Ferus, Karcher, and M¨unzner.This approach, done in [11], gives a considerably simpler proof ofTheorem 6.2 above.The extra bonus to this approach is that in [11] I could also classifythe exceptional case when the multiplicity pair is (3 , M + , and the other is the homogeneous one, where theClifford action operates on M − . I also knew that for the homogeneousexample p , p , p , p of M + did not form a regular sequence anymore;otherwise, Proposition 6.8 would give that the hypersurface was theone of Ozeki and Takeuchi.It turned out that Condition A returned in an unexpected way tosettle the case of multiplicity pair (3 , r λ isentirely discarded in (6.54). Condition A enables us to come up witha finer estimate on the right hand side of (6.54) by utilizing r λ .Indeed, if we stratify the hyperquadric Q k − , k ≤ m − , givenin (6.55), of nongeneric λ for which the dimension estimate may createcomplications, into subvarieties L j , of some dimension d j ≤ k − , over which r λ = j , then by (6.54) an irreducible component Θ j of V k ∩ ( ∪ λ ∈L j S λ ) willsatisfy(6.60) dim(Θ j ) ≤ dim( V k ∩ S λ ) + d j ≤ m + m − + d j − − j. HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 55 We run through the same arguments as those following (6.54) to deducethat the codimension 2 estimate (6.59) holds true over L j when(6.61) m − ≥ k + 1 − j − c j , where c j := k − − d j = codimension of L j in Q k − . Note that the inequality dim( V ) ≤ m + k below (6.56) for generic λ in C P k − automatically results in the codimension 2 estimate dim( V ) ≤ dim( V k ) − 2, since m − > m in the remaining four exceptional cases.Thus, it suffices to consider only those λ ∈ Q k − for k ≤ m − m, m − ) = (3 , ≤ k ≤ m − j ≥ 1; the same is also true for all j when k = 1. So, we assume k = 2and j = 0 henceforth.With k = 2 , j = 0, let λ ∈ L be generically chosen; we have r λ = j = 0. Suppose that M + is free of points of Condition A every-where. Let us span λ by the orthonormal n ∗ and n ∗ completed to anorthonormal basis n ∗ , n ∗ , n ∗ , n ∗ . Since r λ = 0, the matrices B = C = 0and A = I in (6.50) for S ∗ . For notational clarity, let us denote the as-sociated B and C blocks of the shape operator matrices S a ∗ by B a ∗ and C a ∗ for the normal basis elements n ∗ , · · · , n ∗ . It follows that p ∗ = 0and p ∗ = 0 cut S λ in the variety { ( x, ±√− x, z ) : X α ( x α ) = 0 } . We may assume ( B ∗ , C ∗ ) is nonzero as M + has no points of ConditionA. Since z is a free variable, p ∗ = 0 will have nontrivial xz -terms0 = p ∗ = X αp S αp x α z p + X µp T µp y µ z p + x α y µ terms= X αp ( S αp ± √− T αp ) x α z p + x α y µ terms , taking y = ±√− x into account, where S αp := h S ( X ∗ α , Z ∗ p ) , n ∗ i and T µp := h S ( Y ∗ µ , Z ∗ p ) , n ∗ i are (real) entries of B ∗ and C ∗ , respectively,and X ∗ α , Y ∗ µ , and Z ∗ p are orthonormal eigenvectors for the eigenspacesof S ∗ with eigenvalues 1 , − , and 0, respectively; hence, the dimen-sion of S λ will be cut down by 2 by p ∗ , p ∗ , p ∗ = 0. In conclusion,modifying (6.54) we havedim( V k ∩ S λ ) ≤ m + m − − , for all λ ∈ L . As a consequence, the right hand side of (6.61), which isno bigger than 5 for j = 0, is now no bigger than 4 with the additional cut p ∗ = 0 so that the codimension 2 estimate goes through for L aswell. It follows by Proposition 6.8 that the isoparametric hypersurfaceis in fact the one of Ozeki and Takeuchi, which thus has points ofCondition A. This is a piece of absurdity to the assumption that M + has no points of Condition A. Therefore, M + does admit points ofCondition A. The result of Dorfmeister and Neher [21] (see also [12])implies that the isoparametric hypersurface is then necessarily of thetype of Ferus, Karcher, and M¨unzner. In particular, it is either theinhomogeneous one by Ozeki and Takeuchi, or the homogeneous one.The classification in the case of multiplicity pair (3 , 4) is now achieved,as was done in [11]. The case of multiplicity pairs (4 , and (6 , , 4) is that, assuming nonexistence of points ofCondition A on M + makes the codimension 2 estimate go through.On the other hand, nonexistence of points of Condition A on M + inthe case of multiplicity pairs (4 , 5) and (6 , 9) always holds true by thediscussions following (4.18). Guided by the (3 , 4) case, I suspected thatnonexistence of points of Condition A on M + could lead to somethingfruitful in the case of multiplicity pairs (4 , 5) and (6 , , 5) is a principalorbit of the action U (5) on so (5 , C ) given by g · Z = gZg, while the homogeneous example of multiplicity pair (6 , M − of an isoparametric hypersurface of the type of Ferus,Karcher, and M¨unzner on which the Clifford action operates, can berealized as the Clifford-Stiefel manifold M − = { ( ζ , η ) ∈ S ⊂ R × R : | ζ | = | η | = 1 / √ , ζ ⊥ η, ˇ J i ( ζ ) ⊥ η, i = 1 , · · · , } , where ˇ J , · · · , ˇ J are the unique (up to equivalence) irreducible rep-resentation of the (anti-symmetric) Clifford algebra C on R con-structed by the octonion algebras as follows. Let e , e , · · · , e be thestandard basis of the octonion algebra O with e the multiplicativeunit. Let J , J , · · · , J be the matrix representations of the octonionmultiplications by e , e , · · · , e on the right over O . Then(6.62) ˇ J i = (cid:18) J i − J i (cid:19) , ≤ i ≤ , ˇ J = (cid:18) I − I (cid:19) . HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 57 In [13, Sections 2.3-2.4], I calculated their second fundamental formsat specifically chosen points of M + , which does not lose generalitybecause of homogeneity, and observed that in the expression (4.16) wehave that B a = C a are of rank 1 for 1 ≤ a ≤ ≤ a ≤ , 5) (respectively, (6 , B a the only nonzero entry is in the last row at column a with value1 / √ 2. This peculiarity is not coincidental and is in fact a consequenceof codimension 2 estimates. Lemma 6.6. Assume ( m, m − ) = (4 , or (6 , . Then either theisoparametric hypersurface is the inhomogeneous one of multiplicitypair (6 , of Ferus, Karcher, and M¨unzner, or r λ = 1 for all λ in Q m − .Proof. It is straightforward to see that when ( m, m − ) = (4 , 5) (respec-tively, ( m, m − ) = (6 , j ≥ 2, the codimension 2 estimate (6.61)goes through for all k ≤ m − k ≤ m − k ≤ m − j . Thus, we may assume k = m − j ≤ λ ∈ Q k − = Q m − ⊂ Q m − .Suppose(6.63) sup λ ∈Q m − r λ ≥ . We may so arrange such that generic λ ∈ Q k − = Q m − assumes r λ ≥ Case 1 . On L where r λ = 1, we have that the codimension 2 estimatestill goes through. This is because (6.60) is now replaced by(6.64) dim(Θ j ) ≤ m + m + k − − j = m + m − + k − j = 1, due to the fact that such λ , being nongeneric in Q k − as r λ ≥ λ , constitute a subvariety of Q k − of dimension atmost k − 2. It follows by (6.58) thatdim(Θ j ) ≤ dim( V k ) − , j = 1 . Case 2 . On L where r λ = 0, (6.64) now readsdim(Θ j ) ≤ m + m − + k − j = 0. We need to cut back one more dimension to make theequality in (6.64) valid. Since r λ = 0, we see B ∗ = C ∗ = 0 and A ∗ = I in (6.50) for S n ∗ , where λ is the 2-plane spanned by n ∗ and n ∗ completedinto the basis n ∗ , n ∗ , n ∗ , · · · , n ∗ m . It follows that p ∗ = 0 and p ∗ = 0 cut S λ in the variety(6.65) { ( x, ±√− x, z ) : X α ( x α ) = 0 } . We may assume ( B ∗ , C ∗ ) is nonzero because of nonexistence of pointsof Condition A on M + . Since z is a free variable in (6.65), p ∗ = 0 willhave nontrivial xz -terms,0 = p ∗ = X αp S αp x α z p + X µp T µp y µ z p + X αµ U αµ x α y µ = X αp ( S αp ± √− T αp ) x α z p , (6.66)taking y = ±√− x into account and remarking that as a result the x α y µ terms are gone because U αµ = − U µα , where U αµ := h S ( X ∗ α , X ∗ µ ) , n ∗ i , S αp := h S ( X ∗ α , Z ∗ p ) , n ∗ i , T µp := h S ( Y ∗ µ , Z ∗ p ) , n ∗ i are entries of A ∗ , B ∗ and C ∗ , respectively, and X ∗ α , ≤ α ≤ m , Y ∗ µ , ≤ µ ≤ m and Z ∗ p , are orthonormal eigenvectors for the eigenspaces of S n ∗ with eigenvalues 1 , − , and 0, respectively, as usual. The skew-symmetry of the matrix U comes from the identity A j A tr + A A trj + 2( B j B tr + B B trj ) = 0 , j = 1 , obtained by inserting (4.16) into (4.15), where ( A , B ) are as given inLemma 6.4 above.Hence the dimension of S λ will be cut down by 2 by p ∗ , p ∗ , p ∗ = 0,so that again(6.67) dim( V k ∩ S λ ) ≤ m + m − − . In conclusion, we deduce(6.68) dim(Θ j ) ≤ dim( V k ∩ S λ ) + k − ≤ m + m − + k − , so that the codimension 2 estimate (6.59) goes through. The codimen-sion 2 estimate holds true verbatim for the case of multiplicity pair(6 , 9) with obvious dimension modifications.Now, the validity of (6.59) implies that the isoparametric hypersur-face is the inhomogeneous one of Ferus, Karcher, and M¨unzner due toProposition 6.8, provided (6.63) holds.We therefore conclude that r λ ≤ λ ∈ Q m − if the hypersur-face is not the inhomogeneous one of Ferus, Karcher, and M¨unzner, inwhich case we claim that generic r λ = 1. Suppose the contrary. Then r λ = 0 for all λ in Q m − . It would follow that B a = 0 for all a ≥ HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 59 Lastly, for a λ with r λ = 0 we have A in (6.50) is the identity matrixby (4.13), so that its rank is full (=5 or 9). It follows that generic λ in Q m − will have the same full rank property. However, for a λ with r λ = 1, the structure of ∆ in Lemma 6.4 implies that ∆ = 0 sothat such A , which are also generic, will be of rank 4 or 8. This is acontradiction. In other words, r λ = 1 for all λ ∈ Q m − . (cid:3) Corollary 6.4. Suppose ( m, m − ) = (4 , or (6 , . Then either theisoparametric hypersurface is the inhomogeneous one of multiplicitypair (6 , of Ferus, Karcher and M¨unzner, or r λ = 1 for all λ ∈ Q m − .In the latter case, the second fundamental form of M + is identical withthat of M + of the homogeneous isoparametric hypersurface of the re-spective multiplicity pair. Here goes the idea. Assume the hypersurface is not the inhomoge-neous one of multiplicity pair (6 , r λ = 1 by the preceding proposi-tion.For a λ in Q m − spanned by n ∗ and n ∗ , we extend them to a smoothlocal orthonormal frame n ∗ , n ∗ , · · · , n ∗ m such that S n ∗ is the squarematrix on the right hand side of (6.50) while S n ∗ is the square oneon the left hand side, where A and B are the respective A and B in Lemma 6.4, for which the skew-symmetric 1-by-1 matrix ∆ = 0 initem (2) and so by item(3) the 1-by-1 matrix σ = 1 / √ M + gives us a nonzeromatrix B ∗ associated with S n ∗ . Modifying (6.65), p ∗ = 0 and p ∗ = 0now cut S λ in the variety(6.69) { ( x , · · · , x , t √ ι λ , ι λ x , · · · , ι λ x , − t √ ι λ , z , · · · , z , t ) : X j =1 ( x j ) = 0 } where ι λ = ±√− x = ( x , x , x , x , x = t/ √ ι λ ) ,y = ( y , · · · , y ) = ( ι λ x , ι λ x , ι λ x , ι λ x , − t/ √ ι λ ) ,z = ( z , z , z , z = t ) . Meanwhile, (6.66) becomes (6.70) 0 = , X α =1 ,p =1 ( S αp ± √− T αp ) x α z p + terms not involving x , · · · , x , z , · · · , z . Since x , · · · , x , z , · · · , z are independent variables, the nontrivialityof the displayed term on the right hand side of the preceding equationimplies that the dimension cut can be reduced by 1 so that we have,by (6.67) and (6.68),dim(Θ j ) ≤ m + m − + k − , j = 1 , for k ≤ 3, so that the codimension 2 estimate (6.59) goes throughin the neighborhood of λ , which is absurd as the hypersurface wouldthen be the inhomogeneous one of the multiplicity pair (6 , 9) in viewof Proposition 6.8. We therefore conclude that ( B j , C j ) , j ≥ , around λ are of the form(6.71) B j = (cid:18) d j b j c j (cid:19) , C j = (cid:18) g j e j f j (cid:19) , ∀ j ≥ , for some real numbers c j and f j with 0 of size 4 × d j = g j = c j = f j = 0 , ∀ j ≥ . So now we have A = (cid:18) I 00 0 (cid:19) , A j = (cid:18) α j 00 0 (cid:19) , j = 2 , , , B j = C j = (cid:18) b j (cid:19) , all of the same block sizes, satisfying α j α k + α k α j = − δ jk I, h b j , b k i = δ jk / , by the identity(6.72) A i A trj + A j A tri + B i B trj + B j B tri = 2 δ ij Id. As a consequence, first of all, we can perform an orthonormal basischange on n ∗ , n ∗ , n ∗ so that the resulting new b j is 1 / √ j thslot and is zero elsewhere. Meanwhile, we can perform an orthonormalbasis change of the E and E − spaces so that I and α j , ≤ j ≤ , areexactly the matrix representations of the right multiplication of 1 , i, j, k on H without affecting the row vectors b j , ≤ j ≤ 4. This is preciselythe second fundamental form of the homogeneous example.Now that the second fundamental form of M + is identical with thatof the homogeneous example of the same multiplicity pair when the HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 61 isoparametric hypersurface is not the inhomogeneous one of Ferus,Karcher and M¨unzner, one can explore the defining equations (6.33)through (6.38) to determine definitively, as was done in [13], that thethird fundamental form is also nothing other than that of the homo-geneous example, most succinctly expressed in the respective quater-nionic and octonion framework. The hypersurface is thus the homoge-neous one and whence follows the classification. The case of multiplicity pair (7 , . In retrospect, we explored twoavenues in the preceding three exceptional cases of multiplicity pairs(3 , , (4 , , 9) to achieve the codimension 2 estimates. Thefirst route was via the right hand side of the a priori estimate (6.61), m − ≥ k + 1 − j − c j , ∀ k ≤ m − , where large rank j and nonzero codimension c j are employed to reducethe situation to j = 0 or 1, to be followed by the second route to handlethe estimate in (6.67), where, by introducing more cuts via p a = 0 ,a ≥ 2, we were able to cut down the upper bound of dim( V k ∩ S λ )from the one in (6.54) to that in (6.67),dim( V k ∩ S λ ) ≤ m + m − − , so that the right hand side of (6.61) was down by 1 to achieve theimproved estimate m − ≥ k − j − c j , ∀ k ≤ m − . The worst case scenario in this procedure is that p a = 0, for all a ≥ 2, always contain S λ so that no dimension cut can be achievedof V k ∩ S λ , which was avoided in the above three cases with the helpof nonexistence of points of Condition A and that the isoparametrichypersurface is not the inhomogeneous one of Ozeki and Takeuchi orof Ferus, Karcher, and M¨unzner.To understand this worst case, I introduced in [15] the notion of r -nullity: Definition 6.3. We say a normal basis n , n , n , · · · , n m is normal-ized if S , · · · , S m are as in (4.16) with S normalized as in Lemma for which B is of rank r . Definition 6.4. Given a normalized normal basis n , · · · , n m , let C m − ≃ C E + , C m − ≃ C E − and C m + ≃ C E be parametrized by x, y and z , re-spectively, where E + , E − and E are the eigenspaces of S with eigen-values , − , and , respectively. Let x := ( x , x ) , y := ( y , y ) and z := ( z , z ) with x , y , z ∈ C r . Let p , · · · , p m be the components ofthe second fundamental form at the base point of n . We say a normal basis element n l , l ≥ , is r -null if p l is identicallyzero when we restrict it to the linear constraints (6.73) y = ι x , y = − x , z = σ − (∆ + ι Id ) x , ι = ±√− . We say the normal basis is r -null if n l are r -null for all l ≥ . Note that the conditions in (6.73) define S λ given in (6.49) when r λ = r . The algebro-geometric definition has a differential-geometric charac-terization. Lemma 6.7. [15, Lemma 3.1] Let n , · · · , n m be a normalized normalbasis. A normal basis element n l , l ≥ , is r -null if and only if theupper left ( m − − r ) -by- ( m + − r ) block of B l and C l of S l are zero. It is now clear by the preceding lemma that Condition A is equiv-alent to that all normalized normal bases are 0-null at the relevantpoint of M + . Moreover, what we showed in Corollary 6.4 for the casesof multiplicity pairs (4 , 5) or (6 , 9) is that, all normalized normal basesof M + are 1-null if the isoparametric hypersurface is not the inhomoge-neous one of Ferus, Karcher, and M¨unzner, from which we determinedbelow (6.71) that the second fundamental form of M + coincides withthat of the homogeneous example with the respective multiplicity pair.What is remarkable is that something similar holds true in the case ofmultiplicity pair (7 , 8) as well, only much more complicated this time. Proposition 6.9. [15, Sections 3-6] Assume ( m, m − ) = (7 , and theisoparametric hypersurface is not the inhomogeneous one constructedby Ozeki and Takeuchi. Then away from points of Condition A, M + isgenerically -null, i.e., generically chosen normalized normal bases are -null. Moreover, we may assume A a and B a are of the form (6.74) A a = (cid:18) z a w a (cid:19) , B a = (cid:18) c a (cid:19) , C a = (cid:18) f a (cid:19) , ≤ a ≤ , (6.75) A a = (cid:18) β a γ a δ a (cid:19) , B a = (cid:18) d a b a c a (cid:19) , C a = (cid:18) g a b a f a (cid:19) , ≤ a ≤ , where A a are of size × , B a and C a are of size × , and the lowerright blocks of all matrices are of size × . The idea is that if we define R := sup λ ∈Q m − r λ , m = 7 , then a generically chosen normalized normal basis is R -null [15, Corol-lary 3.2]. Furthermore, if R ≥ 5, then the codimension 2 estimate goes HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 63 through away from points of Condition A [15, Sections 3-5], so that byProposition 6.8 the isoparametric hypersurface is the one of Ozeki andTakeuchi. Thus, R ≤ n with basepoint x , the normal vector n := x at the “mirror point” x = n of x in M + is also generic spanning the same λ ∈ Q m − , m = 7 , ofnullity R . We have the explicit dictionary to translate the data of theshape operators at x and x . Explicitly,(6.76) A a := (cid:0) S aαµ (cid:1) , B a := (cid:0) S aαp (cid:1) , C a := (cid:0) S aµp (cid:1) , ≤ a ≤ . Let the counterpart matrices at x and their blocks be denoted by thesame notation with an additional A p := (cid:0) S pαµ (cid:1) , B p = (cid:0) S aαp (cid:1) , C p = − (cid:0) S aµp (cid:1) , ≤ p ≤ , following the index convention (6.7). As a consequence, we obtainmany zeros as indicated in (6.74) and (6.75), and in particular, weobtain R = 4 by a resulting Clifford representation [15, Proposition6.1].Though a first glance at (6.74) and (6.75) suggests that it wouldstill be a long way home to determine the second fundamental formof M + , in sharp contrast with the case of multiplicity pairs (4 , 5) and(6 , (cid:0) √ c a w a (cid:1) , ≤ a ≤ , form a Cliffordmultiplication of type [3 , , F : R × R → R , F ( u a , v α ) = the α th row of (cid:0) √ c a w a (cid:1) , for orthonormal bases u a and v α , satisfying | F ( x, y ) | = | x || y | derivedfrom (6.72).A second crucial observation is that, with the conversion (6.74)through (6.77), the first columns of b , · · · , b are, respectively, thefirst, second, third, and fourth columns of c . Similarly, the second(vs. third) columns of b , · · · , b are the respective columns of c (vs. c ). For instance, at x , the normalized c is c = σ σ σ 00 0 0 σ , by Lemma 6.4 due to 4-nullity at x . Hence, the first columns of b , · · · , b are, respectively,( σ , , , tr , (0 , σ , , tr , (0 , , σ , tr , (0 , , , σ ) tr , from which a third crucial observation can be drawn. Indeed, it wasshown in [15, Lemma 7.1] that a generic linear combination b ( x ) := x b + · · · + x b is of rank no more than 2, so that we may in fact assume that all of b , · · · , b have zero third column, by the fact that the Koszul complex0 −→ R x ∧ −→ Λ R x ∧ −→ Λ R x ∧ −→ Λ R x ∧ −→ Λ R → , where R := R [ x , x , x , x ] is the polynomial ring in four variables and x ∧ means taking the wedge product against x , is a free resolution [23,Chapter 17]. The assumption that b ( x ) is generically of rank 2 meansthat the wedge product of the second column v and third column v of b ( x ) lives in the kernel of −→ Λ R x ∧ −→ Λ R , v ∧ v x ∧ ( v ∧ v ) = 0 , so that either v ∧ v = 0, in which case they differ by a constantmultiple, or, v ∧ v = x ∧ w for some w ∈ R , so that we may assumethe first two columns of b ( x ) are both x up to a constant multiple. Asa consequence, the conversion says that we may assume c = 0.In [17, Section 5], F in (6.78) with the constraint c = 0 wereclassified and given by c = ǫ I, c = 0for some ǫ > c is of the form(6.79) c = a Id + b (cid:18) I ± I (cid:19) , I = (cid:18) − 11 0 (cid:19) , b = 0 , for some a and b . By conversion, (6.80) b = ǫ a b 00 0 00 0 0 , b = − b ǫ a 00 0 00 0 0 , b = ǫ a ± b , b = ∓ b ǫ a at x , whose linear combinations are of generic rank 2.In particular, a glance at B a , ≤ a ≤ , in (6.74) shows that theirthird columns are all zero, or equivalently, that there is a commonkernel of generic dimension 1 for all the shape operators S n for all n .This gives us a clear geometric picture: When the isoparametric hypersurface with multiplicities ( m, m − ) =(7 , is not the one constructed by Ozeki and Takeuchi, consider thequadric Q of oriented -planes in the normal space at a generic point x ∈ M + . We know a generic element ( n , n ) in Q is -null, or HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 65 equivalently, the intersection V of the kernels of S n and S n is -dimensional. By the preceding lemma, there is a nonzero unit vector v ∈ V common to all kernels of the shape operators at x . We choosean orthonormal basis e , e , e = v spanning V . When viewed at themirror point x = n ∈ M + , e , e , e are converted to three normalbasis vectors of which the three matrices c , c , c in (6.74) are of theform c = ǫ Id, c = 0 , and c given in (6.79) . By a symmetricreasoning, all this holds true as well at x when both x and x aregeneric. We are ready to see that the isoparametric hypersurface is one ofthe two constructed by Ferus, Karcher, and M¨unzner with the Cliffordaction operating on M − . To this end, for a normal basis n , · · · , n m of M + with base point x , let us set(6.81) x ∗ := ( x + n ) / √ , n ∗ := ( x − n ) / √ .x ∗ is a point on M − and n ∗ is normal to M − at x ∗ . The normal spaceto M − at x ∗ is R n ∗ ⊕ E . Furthermore, the (+1)-eigenspace E ∗ of theshape operator S n ∗ is spanned by n , · · · , n m + , the ( − E ∗− of S n ∗ is E , and the 0-eigenspace E ∗ of S n ∗ is E − .Referring to (4.16), let the counterpart matrices at x ∗ and theirblocks be denoted by the same notation with an additional *. Then,for α = 1 , · · · , m − , (6.82) A ∗ α = −√ (cid:0) S aαp (cid:1) , B ∗ α = − / √ (cid:0) S aαµ (cid:1) , C ∗ α = − / √ (cid:0) S pαµ (cid:1) . There follows from (6.79) important features for the block matricesin (6.74) and (6.75). Lemma 6.8. [15, Lemma 7.6, Corollary 7.1] After a frame changeover E and E − we have the following. (1): The spectral data ( σ, ∆) given in Lemma equal ( Id/ √ , . (2): d a = (cid:18) d a (cid:19) , g a = (cid:18) g a (cid:19) , a = 1 , · · · , , where 0 and d a = g a are of size × . (3): c a = f a , β a = ( γ a ) tr , and δ a is skew-symmetric for all ≤ a ≤ . Proposition 6.3, in which the equations (6.11) and (6.15) are modi-fied with appropriate index changes by the recipe (6.82), characterizes the Ferus-Karcher-M¨unzner examples whose Clifford action operateson M − . It reads, in view of the notation of (6.8) and (6.9),(6.83) A ∗ α = A ∗ µ , ( a, µ ) entry of B ∗ α = − ( a, α ) entry of B ∗ µ , ( p, µ ) entry of C ∗ α = − ( p, α ) entry of C ∗ µ ,ω ij − ω i ′ j ′ = X k L ijk ( θ k + θ k ′ ) , for some smooth functions L ijk , where i, j, k are in the α index rangeand i ′ , j ′ , k ′ are in the µ index range with the respective index values(i.e., i indicates α = i and i ′ indicates µ = i + m − , etc.), recalling (6.7)through (6.9).In fact, employing (6.82), the first three equations in (6.83) for M − take the form(6.84) B a = C a , ∀ a,A a is skew-symmetric , ∀ a,A a is skew-symmetric , ∀ a, over M + , which is exactly Lemma 6.8 after a slight frame change byswapping rows. Note, at x ∗ , we can now change the sign of the lastfour α -rows of A i without affecting the skew-symmetry of δ i and theproperty d i = g i , c i = f i , so that now β i = γ tri , ≤ i ≤ , at x areconverted to satisfy the second and third skew-symmetric conditionsin (6.83) at x ∗ .It remains to establish the fourth equation in (6.83). A slight modi-fication of [11, Lemma 2, p. 11], the last item holds true if either α = i or α = j indexes a basis vector in the image of the linear map(6.85) H : E ∗ + ⊕ E ∗− → E ∗ , ( e a , e p ) X α S aαp e α , which is easily seen to be the direct sum of all e α = l for l = 3 , B a are zero for all 1 ≤ a ≤ i = 3 , j = 4 in the α -range.Referring to the discussions around (6.5), now adopted for M − , sincethe distribution ∆ is the kernel of θ a + θ a + m , a = 1 , · · · , m, we see theright hand side of the fourth equation of (6.83) is automatically zeroover ∆. Thus to establish the identity, it suffices to show that ω − ω ′ ′ on the left hand side annihilates ∆, i.e., for v := e l ′ − e l ∈ F we mustverify ω ( v ) = ω ′ ′ ( v ), or,(6.86) ω ( e l ) = − ω ′ ′ ( e l ′ ) , l = 1 , · · · , , HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 67 since F ijk = 0 whenever exactly two indexes fall in the same α, µ, a , or p range [7, (2.9), p. 9].Now, for x ∈ M + and n in the unit normal sphere to M + at x , themap(6.87) f : ( x, n ) ( x ∗ , n ∗ ) = (( x + n ) / √ , ( x − n ) / √ M + and M − .Fix a point ( x , n ) in the unit normal bundle of M + , consider the twosets S + := { ( x, n ) : x + n = x + n } , S − := { ( x, n ) : x − n = x − n } .S ± are two 8-dimensional spheres, which can be seen by taking deriv-ative of x ± n = c for a constant c , whose typical tangent space to S ± is the eigenspace E ± at ( x, n ), respectively.The diffeomorphism f maps S + to a sphere whose tangent space at( x ∗ , n ∗ ) is the vertical V ∗ of the unit normal bundle of M − because f : ( x, n ) ∈ S + ( c/ √ , c/ √ − √ n ) , so that it is the fiber of the unit normal bundle of M − over x ∗ ; likewise, f maps S − to a sphere whose tangent space at ( x ∗ , n ∗ ) is the horizontal E ∗ because f : ( x, n ) ∈ S − → ( − c/ √ √ x, c/ √ . Thus to calculate the quantities in (6.86), it suffices to observe that (6.85)gives us the informationdim( \ a =1 kernel( B tra )) = 2 , which is a consequence of item (2) of Lemma 6.8. This translates to S + to say that the tangent space to S + at ( x, n ) is identified with E + of the second fundamental form S n , in which there naturally sits a 2-dimensional plane that is the intersection of all kernels of the B trm -blockof S m , with m perpendicular to n at x , which forms a 2-plane bundle P + over S + . By the same token there is a 2-plane bundle P − over S − which comes from the intersection of all kernels of the C trm -block of S m with m perpendicular to n at x . Now, since d a = g a , ≤ a ≤ 7, meansthat P + and P − are parametrized identically in the coordinates, oncewe set up the coordinate system of the ambient Euclidean space by theeigenspace decomposition R x ⊕ R n ⊕ E ⊕ E + ⊕ E − of the shape operator S n at x for ( x, n ) ∈ S + , where the third andfourth rows of B a are zero for all 1 ≤ a ≤ 7. As a consequence, viathe diffeomorphism f in (6.87), a local basis ( e , e ) spanning P + isconverted to one on the image sphere whose tangent space at ( x ∗ , n ∗ )is V , and local basis ( e ′ , e ′ ) spanning P − is converted to one on theimage sphere whose tangent space at ( x ∗ , n ∗ ) is E ∗ . Thus on the imagesphere we derive ω ( e l ) = −h de ( e l ) , e i = h de ′ ( e l ′ ) , e ′ i = ω ′ ′ ( e l ′ ) , which gives (6.86), remarking that the negative sign in the first equalityis a result of the sign convention (6.10).The four equations in (6.83) are satisfied. Thus the isoparamet-ric hypersurface is one of the two constructed by Ferus, Karcher, andM¨unzner, if it is not the one constructed by Ozeki and Takeuchi.Lastly, for g = 6, Miyaoka classified [44, 45] (see also [43]) the casewhen the multiplicity pair is (2 , A few questions There remain a few fundamental questions in the spherical case thatI find especially interesting to be listed here to conclude the article.Is there a geometric way to prove that the number g of principalcurvatures is 1 , , , , or 6 ? M¨unzner’s proof [46, II] is topological. Arecent paper of Fang [29] gave another topological proof of M¨unzner’sresult.On the other hand, is there a geometric proof for the multiplicitypairs ( m + , m − ), m + ≤ m − , when g = 4? Though Stolz’s approach istopological and works for more general compact proper Dupin hyper-surfaces, our classification in [7] exhausts the multiplicity pairs so longas m − ≥ m + − 1. So, the question to ask is whether there is a geometricway to show that the multiplicity pair must be (2 , , (4 , , (3 , , (6 , , or (7 , 8) when m + ≤ m − ≤ m + − 2? Similarly, is there a geometricproof that m + = m − = 1 , or 2 when g = 6?Immervoll [35] gave a different proof of the result in [7] by isopara-metric triple systems that Dorfmeister and Neher developed [20]. Isthere a classification of the exceptional cases by the approach of isopara-metric triple systems?For g = 6, derive an expansion formula for the Cartan-M¨unznerpolynomial similar to the one done by Cartan for g = 3 and by Ozeki HE ISOPARAMETRIC STORY, A HERITAGE OF ´ELIE CARTAN 69 and Takeuchi for g = 4. In addition to Miyaoka’s geometric proof ofhomogeneity of such isoparametric hypersurfaces, is there a proof byutilizing the expansion formula similar to the one given by Cartan forthe case g = 3 that also enjoys equal multiplicity? As Miyaoka’s proofpointed out, Condition A should play a decisive role. 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