A new structural approach to isoparametric hypersurfaces in spheres

The classification of isoparametric hypersurfaces in spheres with four or six different principal curvatures is still not complete. In this paper we develop a structural approach that may be helpful for a classification. Instead of working with the isoparametric hypersurface family in the sphere, we consider the associated Lagrangian submanifold of the real Grassmannian of oriented $2$-planes in $\mathbb{R}^{n+2}$. We obtain new geometric insights into classical invariants and identities in terms of the geometry of the Lagrangian submanifold.


Introduction
Originally, isoparametric hypersurfaces were defined to be the level sets of isoparametric functions, i.e., functions on a real space form whose gradient norm and Laplacian are constant on the level sets.This condition translates into the equivalent geometric condition that the principal curvatures of the hypersurfaces are constant.The cases where the ambient space is Euclidean or hyperbolic space were settled by Somigliana [So], Segre [Se], and Cartan [C1]- [C4].In contrast, when the ambient space is a sphere, the number g of distinct principal curvatures can be greater than two, which makes a classification more difficult.Cartan [C1]- [C4] classified isoparametric hypersurfaces with g ≤ 3, and showed that they are all homogeneous, i.e. orbits of isometric group actions on S n+1 .
The problem was picked up again by Münzner [Mz1,Mz2] who showed that the number of distinct principal curvatures g can be only 1, 2, 3, 4, or 6, and gave restrictions for the multiplicities as well.The possible multiplicities of the curvature distributions were classified in [Mz2], [Ab], [St], and coincide with the multiplicities in the known examples.The situation in the case g = 4 is more complex since there exist infinitely many isoparametric hypersurfaces and infinitely many of them are inhomogeneoussee Cecil [Ce] or Thorbergsson [Tb] for a recent survey of this case.In the case g = 6 all multiplicities m coincide and equal either 1 or 2, and precisely two homogeneous examples are known.Dorfmeister and Neher [DN] conjectured that all isoparametric hypersurfaces with g = 6 are homogeneous and in the same paper settled this conjecture in affirmative for m = 1.In the remaining open case m = 2, Miyaoka in [M2] proposed how to establish homogeneity.
In the present paper we develop a new structural approach to isoparametric hypersurfaces in spheres, unifying many of the known geometric properties.We hope that this approach will be helpful in the classification as well.
The basic idea is as follows.Instead of working with the family of parallel surfaces F t : M n → S n+1 , with normal field ν t ∈ Γ(νM n ), one considers the associated submanifold of the real Grassmannian of oriented 2-planes in R n+2 by sending p ∈ M n to the 2-plane spanned by F t (p) and ν t (p).One easily sees that L is independent of t, and in [Pa1], [Pa2] it was observed that for any submanifold of S n+1 the associated submanifold L(M ) ⊂ Gr + 2 (R n+2 ) is Lagrangian with respect to the natural symplectic structure.
We endow the Lagrangian submanifold with a set of invariants which arise naturally: the metric induced via the canonical Kähler metric g Q on Gr + 2 (R n+2 ), i.e., ĝ = L * g Q ; the symmetric tensor α α(X, Y, Z) = g t ((∇ t X A t )Y, Z), where A t denotes the shape operator of F t with respect to ν t , X, Y, Z ∈ Γ(T M ); and where The set of invariants (ĝ, α, B ⊗ B −1 ) depends only on the isoparametric family it is contained in.
Theorem A: The tensor α coincides, up to a factor of two, with the second fundamental form of the Lagrangian submanifold.
The so-called Weyl identities play a crucial role in the theory of isoparametric hypersurfaces in spheres.The classical Weyl identities depend on several indices.In terms of the invariants described above, these multiple identities unify into one tensor identity, see Theorem 3.8.Moreover, the pullback of α under the reflections through each of the focal manifolds coincides with the negative of α.Hereafter these identities are referred to as Symmetry identities.So far, all these considerations are completely general.
With a possible classification of the case g = 6 in mind, we derive several properties which are equivalent to homogeneity.Denote by D k , k ∈ {1, ..., 6}, the eigenspaces of the shape operator of F t with respect to ν t , and by π k the orthogonal projection into D k .Furthermore, in D k and π k the index k is interpreted to be cyclic of order 6.
Theorem B: The homogeneity of isoparametric hypersurfaces with g = 6 is equivalent to the integrability of the direct sums D j ⊕ D j+3 , j ∈ {1, ..., 6}.In addition, homogeneity is equivalent to where R is the curvature tensor of the Lagrangian submanifold L(M n ).
Using this structural approach we can reprove many of the classical results.Most proofs become simpler and render greater geometric insight.This paper is organized as follows: in Section 1 we recall a few preliminary definitions and give a survey of results needed later on.In Section 2 we carry out the translation from the isoparametric hypersurface family in the sphere to the Lagrangian submanifold of the real Grassmannian of oriented 2-planes in R n+2 .In particular we introduce the set (ĝ, α, B ⊗ B −1 ) of structural invariants.Section 3 deals with the fundamental submanifold equations of the Lagrangian submanifold.Moreover, we derive the Weyl identities and the Symmetry identities.Finally, in Section 4 we give several equivalent formulations of homogeneity.
It is a pleasure to thank my thesis advisor, Prof. Dr. Uwe Abresch, for introducing me to isoparametric hypersurfaces and for his generous support.I benefitted enormously from numerous mathematical discussions with him.Furthermore, I am very grateful to Prof. Dr. Wolfgang Ziller for his many valuable comments.

Preliminaries
Throughout this paper M denotes a connected, smooth manifold of dimension n.
Definition 1.1:An embedding F 0 : M ֒→ S n+1 together with a distinguished unit normal vector field ν 0 ∈ Γ(νM ) is called an isoparametric hypersurface in S n+1 if and only if the principal curvature functions are constant.By g ∈ N we denote the number of distinct principal curvatures.
Let A 0 denote the shape operator of F 0 with respect to ν 0 .We denote the constant values of the principal curvature functions of F 0 by λ 0 j , j ∈ {1, ..., g}.We further assume without loss of generality λ 0 1 > ... > λ 0 g and define θ j ∈ − π 2 , π 2 such that λ 0 j = cot(θ j ).Let D j = Eig(A 0 , λ 0 j ) be the j-th curvature distribution and π j the orthogonal projection into D j .It is well-known that D j is integrable and the leaves are small spheres in S n+1 .Finally, m j = trace π j is called the multiplicity of the curvature distribution D j .
1.1.Parallel surfaces.In what follows let F 0 : M ֒→ S n+1 be a fixed isoparametric hypersurface.By slight abuse of notation we also call the image F 0 (M ) an isoparametric hypersurface.We consider the parallel surface F t : M ֒→ S n+1 defined via and endow it with the orientation Below we shall examine the properties of these parallel surfaces.
The map F t induces the following data on M : the Riemannian metric g t = F * t •, • S n+1 , the associated Levi-Civita connection ∇ t and the shape operator A t of the submanifold (M, g t ) ⊂ (S n+1 , •, • S n+1 ) with respect to ν t .
Using the identity dν 0 = −dF 0 A 0 we get and hence rk(dF t|p ) = n for t = θ j and rk(dF t|p ) = n − m j for t = θ j .If t = θ j , the parallel surface F t (M ) is again an isoparametric hypersurface with principal curvatures λ t j = cot(θ j − t) and thus A t = g j=1 λ t j π j .For t 0 = θ j + ℓπ, ℓ ∈ Z 2 , the m j -dimensional eigenspace D j (p) = Eig(A 0|p , λ 0 j ) is the kernel of dF t 0 |p for every p ∈ M .Hence, M j,ℓ := F θ j +ℓπ (M ) is a so-called focal submanifold of dimension (n − m j ).Thus we get Lemma 1.2:In terms of A 0 the shape operator A t is given by where in the cases t = θ j + ℓπ, ℓ ∈ Z 2 , the operator A t id defined on T M \D j . Proof.Since whence the claim.
(3) Thus for each p ∈ M j,ℓ and each pair of orthonormal vectors v 1 , v 2 ∈ ν p M j,ℓ the family is isospectral.We will henceforth refer to L(s) as the linear isospectral family at p ∈ M j,ℓ with respect to (v 1 , v 2 ) ∈ ν p M j,ℓ , or linear isospectral family for short.
The fact that the spectrum of the focal shape operator of the focal submanifold is independent of ν ∈ νM j,ℓ and p ∈ M j,ℓ implies that the eigenvalues λ 0 k , k ∈ {1, ..., g}, are of the form The parameter φ in the formula for θ j encodes the position of F 0 in the isoparametric family.We shall choose the starting hypersurface such that φ = θ 1 = π 2g .Thus the starting isoparametric hypersurface is the one which lies in the middle of the focal submanifolds F −π/2g (M ) and F π/2g (M ).
Using once more that the spectrum of A θ j |νp is independent of ν ∈ νM j,ℓ , Münzner proved that the multiplicities satisfy the equation m i = m i+2 , i ∈ Z g .Therefore at most two distinct values for the multiplicities exist.In particular, if g is odd all multiplicities coincide.1.3.Global structure.The global situation is as follows: is a singular Riemannian foliation, F t (M ) are isoparametric hypersurfaces for all t ∈ (−π/2g, π/2g), F −π/2g (M ) and F π/2g (M ) are submanifolds of codimension at least two in S n+1 .Each normal geodesic γ intersects the focal submanifolds at times t = (2j+1)π/2g, j ∈ Z, alternating between M + := F −π/2g (M ) and M − := F π/2g (M ).In particular there exists exactly two focal submanifolds.The regular set R is the set of times t ∈ R such that γ(t) is not a focal point, Any fixed isoparametric hypersurface F t 0 (M ) ∈ F coincides with either of the tubes Tube d + (M + ) and Tube d − (M − ) of radius d + and d − , respectively, where d ± denotes the distance of M ± to F t 0 (M ).Thus each normal geodesic intersects a given isoparametric hypersurface F t 0 (M ) exactly 2g times before it closes.Furthermore, the focal set of each isoparametric hypersurface F t 0 (M ) is exactly the union of M + and M − .
Figure 1 sketches a normal geodesic in the case g = 3.It intersects each isoparametric hypersurface exactly six times before it closes.The intersection points with one fixed isoparametric hypersurface are marked by solid points.
where the fibers D + and D − have dimensions m 1 + 1 and m 2 + 1, respectively.
This topological fact was used in the papers [Mz2], [Ab] and [St] to classify the number of distinct principal curvatures and their possible multiplicities.

Structural invariants
We assign to each isoparametric hypersurface a set of invariants (ĝ, α, B ⊗B −1 ), which depends only on the isoparametric family it is contained in.Throughout this section let X, Y, Z, W ∈ Γ(T M ).

Notation and basics.
2.1.1.The complex quadric.In the present subsection we introduce the complex quadric.As reference we use the book [GG] of Gasqui and Goldschmidt.
We write •, • C n+2 and •, • h for the standard complex bilinear and the standard hermitian inner product of C n+2 , respectively.Furthermore, we denote by Q n the complex quadric, i.e., the complex hypersurface of CP n+1 given by where z = (z 0 , ..., z n+1 ) denote the standard coordinates of C n+2 .The complex quadric Q n may also be described by where π : C n+2 −{0} → CP n+1 is the natural projection.
1 Ferus, Karcher and Münzner [FKM] used representations of Clifford algebras to produce a class of isoparametric families with four principal curvatures, the so-called isoparametric hypersurfaces of FKM-type.
It is well-known that the complex quadric Q n is diffeomorphic to the real Grassmannian Gr + 2 (R n+2 ) of oriented 2-planes in R n+2 .From now on we shall use this identification whenever convenient.
Let g Q denote the Kähler metric on Q n induced from the Fubini-Study metric g F S on CP n+1 via the inclusion map ι : The associated Levi-Civita connection of Q n is denoted by ∇ Q .For both CP n+1 and Q n the complex structure shall be called J and the associated Kähler form ω.
It is well-known that the projection π : S 2n+3 → CP n+1 is a Riemannian submersion, i.e. the map dπ : H z → T π(z) Q n is an isometry, where In what follows we shall identify the tangent space T q Q n of the complex quadric at a given point q ∈ Q n with C n .Then the complex structure on T q Q n is given by multiplication by i on C n , and the Kähler metric g Q corresponds to the real part of the standard Hermitian inner product on C n .
Recall the following definition.
Definition 2.1: The Kulkarni-Nomizu product (i) of two symmetric (2, 0)-tensors h 1 and h 2 is the (4, 0)-tensor h 1 ∧ h 2 given by (ii) of the skew-symmetric form ω with itself is given by Lemma 2.2 ( [GG]): The Riemann curvature tensor of the quadric Q n is given by 2.2.Construction.We now lift the embeddings F t to the Stiefel manifold St 2 (R n+2 ) and project this horizontal submanifold onto the Grassmannian Gr + 2 (R n+2 ).We assign to each family of isoparametric hypersurfaces in the sphere S n+1 a Lagrangian submanifold of Q n : for t = θ j we define the map Ft via Furthermore, we introduce the map L via which is spanned by F t (p) and ν t|p .Since by definition Ft = e −it F0 , the immersion L does not depend on the parameter t.
The next result was first proved by Palmer ([Pa1], [Pa2]), who showed that every oriented, immersed hypersurface in the sphere naturally leads to a Lagrangian submanifold of the complex quadric.For convenience of the reader we reprove this statement.
Proof.Using the identity dν t = −dF t A t , we obtain In what follows we use the convention Lemma 2.4: Ft (M ) is horizontal with respect to the projection π : Proof.For any p ∈ M in C n+1 .The claim now follows using equation ( 4).
Consequently, we may carry out the calculations in the Stiefel manifold.
Remark 2.5: The above construction was used in [MO] to classify compact homogeneous Lagrangian submanifolds in complex hyperquadrics.
2.3.Invariant metric.The Riemannian metric on M induced from •, • h via Ft henceforth is referred to as ĝ, i.e., ĝ = Re( F * t •, • h ) and by ∇ we denote the associated Levi-Civita connection.By Lemma 2.4 the Riemannian metric ĝ is induced from g Q via L.
We now prove that the metric ĝ is in fact independent of t.
Theorem 2.6: For each p ∈ M and all t ∈ R we have In particular ĝ is independent of t.
Proof.Since Ft = e −it F0 we have By the definition of ĝ and the preceding identity this gives Remark 2.7: The induced metric ĝ is the arithmetic mean of some g t : let φ ∈ (0, π/2g) be given and define the arc Lemma 2.8: The connections ∇ and ∇ t are related by , where the last equality follows from Theorem 2.6.
Proof.α t ist obviously trilinear.Since M is a hypersurface in a constant curvature space, the Codazzi equation says that Next we prove that α t vanishes when we choose two of its entries to be in the same distribution.Since α t is symmetric we can assume without loss of generality that Y, Z ∈ D j for a j ∈ {1, ..., g} and X ∈ Γ(T M ).Thus we get Next we prove that α t is related to the shape operator Â : The next theorem establishes Theorem A of the introduction.
Theorem 2.12: For any t ∈ R, the maps α : In particular the map α t is independent of t ∈ R.
Proof.Throughout the proof fix X, Y, Z ∈ Γ(T M ).By definition of Â and skew symmetry of J we get Since the Weingarten equation is given by and thus we in particular get α 0 (X, Y, Z) = α t (X, Y, Z), which proves the claim.
2.5.The invariant B ⊗ B −1 .In this section we assign to each isoparametric hypersurface (M, g t ) an operator B t , and show that In what follows we denote by ĝ also the complex bilinear extension of ĝ.By the very definition of B t we get the following lemma.
Lemma 2.14: ĝ(B t X, Y ) = −( F * t q)(X, Y ).In other words B t encodes the metric F * t q and thus arises as a natural invariant of the Lagrangian submanifold L(M ) ⊂ Q n .
Lemma 2.15: The operators B t are trace free and satisfy the identities Proof.Every X ∈ D j is an eigenvector of B t with eigenvalue µ t j ∈ C given by µ t j = e 2i(θ j −t) .Using the special form of θ j , we obtain B g t = −e −2git 1l.The second identity is an immediate consequence of the first identity.Moreover, the third equation follows from the definition of B t .Finally, an argument analogous to the proof of Lemma 1.2 gives the identity and hence the fourth identity follows from the very definition of B t .
At first glance it might appear wrong to work with the operator B t since it depends on the parameter t.As it turns out, however, all relevant identities factor through the operator B t ⊗ B −1 t , which is independent of t.
Corollary 2.16: The expression Proof.By the last identity of Lemma 2.15 we get B t = e −2it B 0 , which implies We may easily express the projections on the distributions in terms of B t .
Summarizing the results of the present section, we obtain a set of invariants (ĝ, α, B ⊗ B −1 ) which depends only on the isoparametric family it is contained in.

Weyl and Symmetry identities
3.1.The Codazzi, Gauss and Ricci equations.In this subsection we establish the Gauss equation, the Codazzi equation and the Ricci equation for the submanifold (M, ĝ) For ease of notation, we introduce the (2, 0)-tensors b(X, Y Using Theorem 2.12, the left and side simplifies to Furthermore, the right hand side is given by 2R ), for all X 1 , X 2 ∈ Γ(T M ), an easy calculation yields the result.
In order to prove the second identity, recall that the Gauss equation for (M, ĝ) ⊂ where Π denotes the second fundamental form of (M, ĝ) ⊂ (Q n , g Q ) and the Riemann curvature tensor of Q n is given by R Combining these equalities we obtain One can naturally assign to every ĝ-orthonormal basis (e i ) n i=1 of T M a g Q -orthonormal basis of ν(T M ), namely (Jd F0 e i ) n i=1 .Hence we arrive at the identity ) and Theorem 2.12 we obtain the desired result.
Remark 3.3: The Ricci equation of the Lagrangian submanifold (M, g) 3.2.The Weyl identity.The classical Weyl identities depend on several indices.In terms of the invariants introduced above, these multiple identities can be expressed as a single tensor identity, which we shall call the invariant Weyl identity.This in particular allows one to consider higher derivatives of the Weyl identity which is much more complicated in the classical approaches.

The classical Weyl identities.
The following version of the Weyl identities was first deduced by Karcher [Ka].These identities are henceforth referred to as the classical Weyl identities.
Proposition 3.4: For all i, j ∈ {1, ..., g} with i = j the Weyl identity is satisfied, where v i ∈ D i and v j ∈ D j .
By polarizing the preceding identity twice and expressing the resulting equation in terms of α we obtain the following corollary.
Corollary 3.5: For all i, j ∈ {1, ..., g} with i = j, the identity is equivalent to the Weyl identity, where v i , v i ∈ D i , v j , v j ∈ D j , and tr ′ g 0 denotes the sum over a g 0 -orthonormal basis of T M − D j ⊕ D i .

Invariant Weyl identity.
The following two lemmas are used to derive this identity.
Proof.Due to the last identity of Lemma 2.15 it is sufficient to prove the claim for t = 0.By definition of B 0 we obtain Furthermore, using Lemma 2.8 we get Hence we find We thus obtain the claim using equation the last identity of Lemma 2.15.
Lemma 3.7: We have the identity Proof.Differentiating the equation α(Y, π j Z, π j W ) = 0 we get Consequently we obtain Changing the roles of the pairs (π k X, π k Y ) and (π j Z, π j W ) we get Taking the difference of the two preceding identities the Codazzi equation from Proposition 3.2 completes the proof.
Summing these equations over (j, k) ∈ {1, . . ., g} × {1, . . ., g} results in an identity which no longer depends on θ j and θ k , which is why we call it the invariant Weyl identity.
Theorem 3.8: Proof.Take the sum over j and k of the identities just proved in Lemma 3.7 (from 1 to g) and use the identity The claim then follows from Lemma 3.6.

3.2.3.
The importance of the Weyl identity.In most parts of the literature the Weyl identity does not occur explicitly.They however play a decisive role in all papers concerned with the classification of isoparametric hypersurfaces in spheres.We make the following observations.
(i) Karcher [Ka] was the first to prove the classical Weyl identities, in fact, [Ka] is the only source mentioning them explicitly.Karcher showed that for g = 3 the Weyl identities turn each curvature distribution D j into a normed algebra [Ka] and thus reproved the results of Cartan by a structural approach.
(ii) We have the well-known Cartan identity which one easily sees is equivalent to the minimality of the focal submanifolds [No]: by (3) we obtain tr(A Lemma 3.9: The Weyl identities imply the Cartan identity. Proof.Denote by (f k ) n k=1 an g 0 -orthonormal frame of T M which consists of eigen vector fields of A 0 .Choosing v i = v i = f i and v j = v j = f j in Corollary 3.5, we get .
Hence we obtain where we denote by ′ k,j the sum over those j, k ∈ {1, ..., g} with what is the Cartan identity.
Clearly, the scalar Cartan identity is weaker than the Weyl identity.
(iii) The Gauss equation of the isoparametric hypersurface in the sphere implies the Weyl identities.We denote by ∇ 0 the Levi-Civita connection of the submanifold (M, g 0 ) ⊂ (S n+1 , ., • S n+1 ) and by R 0 the associated Riemann curvature tensor.Then the Gauss equations of this submanifold are given by R 0 = g 0 ∧ g 0 + h 0 ∧ h 0 with is equivalent to the polarized version of the classical Weyl identities derived in Corollary 3.5, where Then the claim follows by polarization.The idea of the proof is the following: we calculate the expression in two ways, first by using the definition of the Riemann curvature tensor, and then the Gauss equations.Thus

By definition, (A
Since both the left hand side of this expression and Applying this identity to the equation for 2R 0 , a lengthy but straightforward computation leads to On the other hand, using the Gauss equation we obtain (iv) As mentioned in Section 1 the spectrum of the focal shape operator A ν is independent of the choice of the normal vector ν ∈ νM f ocal .For each choice of pairs of orthogonal vectors ν 1 , ν 2 ∈ νM f ocal one gets a linear isospectral family L(s) = cos(s)A ν 1 +sin(s)A ν 2 .We would like to point out that the condition that L(s) is isospectral partially encodes the Weyl identity and higher covariant derivatives thereof.We shall make this statement more precise in the case (g, m) = (6, 1) only.
Proposition 3.11: Let (g, m) = (6, 1) and e i ∈ D i unit vector fields.Denote by L 0 and L 1 the shape operator of F θ 6 with respect to ν θ 6 and e 6 , respectively.The fact that L(s) = cos(s)L 0 + sin(s)L 1 is linear isospectral translates into the classical Weyl identity with (i, j) = (3, 6) and the first four covariant derivatives with respect to e 6 ∈ D 6 thereof.
Proof.We only give a sketch of the proof.First we verify .
Substitute these results into the minimal polynomial equation for L(s).A tedious but straightforward calculation shows that the ideal generated by the resulting equations coincides with the ideal generated by the classical Weyl identity with (i, j) = (3, 6) and the first four covariant derivatives with respect to e 6 ∈ D 6 thereof.
The homogeneity of isoparametric hypersurfaces with g = 6 is equivalent to the property that the kernels of the linear isospectral families L(s) are independent of s [DN], [M2].Although requiring the family L(s) have eigenvalues ± √ 3, ±1/ √ 3 and 0, all with the same multiplicity m, is a very restrictive condition on the symmetric real 5m × 5mmatrices A ν 1 and A ν 2 , so far no one has yet succeeded in classifying such matrices for m ≥ 2. Furthermore, the irreducible representations of SU( 2) provide examples of such linear isospectral families where the kernel of L(s) is not constant when varying s.It thus does not suffice to study the properties of just one linear isospectral family.One also needs to analyze the interaction of the linear isospectral families that show up for the focal projectors attached to the various points on the same normal great circle of M n .This was carried out successfully for the case (g, m) = (6, 1) by Dorfmeister and Neher [DN].
We shall now describe the advantages of the invariant Weyl identity deduced in the previous paragraph compared to the classical Weyl identities.
The discussion in (iv) highlights what important a role the higher covariant derivatives of the Weyl Identities play in the classification.Since the classical Weyl identities depend on several indices, i.e. i and j, taking higher covariant derivatives of these identities would lead to a plethora of different cases.By contrast, in terms of the invariants ĝ, α and B ⊗ B −1 it is entirely possible to consider higher covariant derivatives since the Weyl identities are condensed in a single tensor identity.
We have seen that the condition that L(s) is isospectral encodes only a certain part of the Weyl identity.It is therefore not sufficient to study only the linear isospectral families at one focal submanifold.The invariant Weyl identity however contains all the information.
For g = 3 the Weyl identities turn each curvature distribution D j into a normed algebra [Ka].For g = 4 they reflect parts of the Clifford algebra structure, which is the central underlying structure in this case.An interesting question is if, as for g = 3 and g = 4, there exists a geometric structure for g = 6 captured by the Weyl identities.The existing examples suggest a geometry closely related to G 2 .This approach might lead to a viable strategy for completing classification.
3.3.Symmetry identities.Throughout this section p shall denote a fixed point of the manifold M .Let t ∈ R and k ∈ N be given.The parallel surface map given by F t (p) → F t+2(θ k −t) (p) = F 2θ k −t (p) maps the submanifold F t (M ) ⊂ S n+1 onto itself and flips the sign of ν t .Hence there exist diffeomorphisms τ k : M → M such that Clearly, the maps τ k : M → M are the reflections in the focal submanifolds, and in particular involutions.
In the next theorem we prove the identities which we call Symmetry identities.
Proof.By Theorem 2.12 and F 0 • τ j = F 2θ j we have for all p ∈ M and for all X 1 ∈ T p M .Moreover, the first identity also implies that τ j : (M, g 2θ j ) → (M, g 0 ) is an isometry.Thus we get and thus the first claim.From this the second claim is immediate.
Remark 3.14: Note that the Weyl identities are pointwise identities whereas the Symmetry identities are not.

Several equivalent formulations of homogeneity
4.1.Calculation of α for the homogeneous examples with g = 6.In the case g = 6 only two examples are known, both of which are homogeneous.They are given as orbits of the isotropy representation of G 2 /SO(4) or the compact real Lie group G 2 , respectively.In both cases all six principal curvatures coincide and are given by m = 1 and m = 2, respectively.
For both of the homogeneous examples Miyaoka [M3, M4] where (f i ) 6m i=1 is a g 0 -orthonormal frame with f i ∈ D i and the index in D i is interpreted to be cyclic of order 6.In what follows we use these results to determine α for the homogeneous examples.

By taking the covariant derivative of
where the index in λ 0 i is interpreted to be cyclic of order 6.Thus for j = k we get Instead of calculating α(f i , f j , f k ) we determine α(e i , e j , e k ), where (e i ) 6m i=1 denotes the ĝorthonormal basis with e i ∈ D i , which is associated to the g 0 -orthonormal basis (f i ) 6m i=1 , i.e., Substituting the Christoffel symbols [M3] into the above equation we obtain the following lemma.
Lemma 4.1: For the homogeneous isoparametric hypersurfaces with (g, m) = (6, 1) the components α i j k := α(e i , e j , e k ) are given by All other α i j k with i ≤ j ≤ k vanish.
Lemma 4.2: For the homogeneous isoparametric hypersurfaces with (g, m) = (6, 2) the components α i j k := α(e i , e j , e k ) are given by All other α i j k with i ≤ j ≤ k vanish.
We use the following two lemmas to prove an extended version of Theorem B.
Lemma 4.3: Let g = 6.For each i ∈ {1, ..., 6} the identity holds, he index of the projections is interpreted to be cyclic of order 6.
Proof.An easy calculation yields (ĝ Proof.By Section 2.5 we obtain for j ∈ {1, ..., 6}.Hence, using B 6 θ j = 1l we get the identity (B 2 θ j − 1l) Ŷ = 0 and thus (∇ X B 2 θ j ) Ŷ = (1l − B 2 θ j )∇ X Ŷ .By interchanging the roles of X and Y and subtracting the resulting equation from the preceding equation we obtain By definition of B t we get (1l − B 2 θ j )Z = 0 if and only if Z ∈ D j ⊕ D j+3 .Combining this with the previous identity yields the desired result.
Theorem 4.5: Each of the following statements is equivalent to the homogeneity of isoparametric hypersurfaces in spheres with g = 6.
(v) For j ∈ {1, ..., 6} the direct sum D j ⊕ D j+3 is integrable.(vi) The kernel of each linear isospectral family L(s) is independent of s ∈ R.
Proof.It is well-known that the sixth statement is equivalent to the homogeneity of isoparametric hypersurfaces in spheres with g = 6 [DN], [M2].Hence it is sufficient to prove that the six statements are equivalent to each other.
Theorem 4.5 suggests a new strategy for establishing homogeneity of isoparametric surfaces in spheres with g = 6: we hope that a detailed study of the geometry of the Lagrangian submanifold in the complex quadric might be helpful.
where ∇ St and d denote the Levi-Civita connection of the Stiefel manifold and Euclidean space, respectively.Plugging in and thus the claim follows from Proposition 3.2.Lemma 4.4: Let j ∈ {1, ..., 6}.For each vector field Z ∈ Γ(T M ) introduce the vector field Ẑ := π D j ⊕D j+3 Z.The direct sum D j ⊕ D j+3 is integrable if and only if j B k θ ℓ ⊗ B j θ ℓ ,where we made use of the last identity of Lemma 2.15 and θ ℓ+3 = θ ℓ + π 2 to obtain the last equality.By Lemma 2.15 again we j ξ (j+k)ℓ B k −π/12 ⊗ B j −π/12 , where we introduced ξ = e −i π3 j ξ (j+k)ℓ B k −π/12 ⊗ B j −πj B j 0 ⊗ B −j 0 ,where we made use of Lemma 2.15 to get the last equality.Combined we get Topology.Each isoparametric hypersurface F t 0 (M ) separates the sphere S n+1 into two connected components B + and B − , i.e.F t 0 (M ) = B + ∩ B − and S n+1 = B + ∪ B − , such that these components are disk bundles over the focal manifolds.Assume without loss of generality that M + has codimension m 1 +1 and that M − has codimension m 2 +1.Thus we have the disk bundles

Table 1 .
Classification results of isoparametric hypersurfaces in spheres Classification results.In Table1the known classification results for isoparametric hypersurfaces in spheres with g different principal curvatures are summarized.