ISOPARAMETRIC FUNCTIONS IN S⁴ⁿ⁺³

Abstract

In this article, we consider a homogeneous function of degree four in quaternionic vector spaces and $S^{4n+3}$ which is invariant under $S^3$ and U(n + 1)-action. We show it is an isoparametric function providing isoparametric hypersurfaces in $S^{4n+3}$ with g = 4 distinct principal curvatures and isoparametric hypersurfaces in quaternionic projective spaces with g = 5. This extends study of Nomizu on isoparametric function on complex vector spaces and complex projective spaces.

1. INTRODUCTION

A hypersurface Mn embedded in ℝn+1 or a unit sphere Sn+1 is said to be isoparametric if it has constant principal curvatures. Isoparametric hypersurfaces in ℝn+1 must have at most two distinct principal curvatures so that the classification consists of an open subset of a hyperplane and a hypersphere or a spherical cylinder Sk × ℝn−k. On the other hand, isoparametric hypersurfaces in spheres are rather complicated. In 1938-1940, É. Cartan published a series of four remarkable papers [2, 3, 4, 5] about isoparametric hypersurfaces in spheres which also classified isoparametric hypersurfaces in spheres with ɡ = 1, 2 or 3 distinct principal curvatures. More than thirty years later, Münzner showed that isoparametric hypersurfaces in sphere can have only ɡ = 1, 2, 3, 4 or 6 distinct principal curvatures in [10, 11]. After Münzner’s great achievements, many mathematicians strived to classify cases ɡ= 4 and 6. Even though much progress has been made, they are still open.

The concept of isoparametric hypersurfaces in general manifolds is not completely determined. When we consider a compact hypersurface M in a compact symmetric space , we call it isoparametric if all nearby parallel hypersurfaces of M have constant mean curvatures. Wang [15], Kimura [9], Park [13] and Xiao [16] have studied isoparametric hypersurfaces in ℂPn. Here the isoparametric hypersurfaces in S2n+1 and ℂPn are related via the Hopf fibration S2n+1 → ℂPn. Moreover, Park [13] also considered isoparametric hypersurfaces in S4n+3 and ℍPn via the Hopf fibration S4n+3 → ℍPn. Therefrom, we are interested in the isoparametric hypersurfaces in S4n+3 which are invariant under Sp(1) = S3 action of Hopf fibration. In particular, we consider the work of Nomizu [12] which provided an example of ɡ = 4 case in S2n+1 ⊂ ℂn+1 and extend his study to S4n+3 ⊂ ℍn+1. We construct an isoparametric function on S4n+3 ⊂ ℍn+1 which is homogeneous of degree four and invariant under S3 and U(n + 1)-action. By using this function, we obtain a homogeneous isoparametric family of hypersurfaces that presents one example of ɡ = 4 case as an extension of the result of Nomizu [12]. Here, we obtain a family of isoparametric hypersurfaces in S4n+3 of four distinct principal curvatures with multiplicities 2, 2, 2n − 1, 2n − 1. Moreover by applying [13] the family also induces a family of isoparametric hypersurfaces in ℍPn having ɡ = 5 with multiplicities 3, 2, 2, 2n − 4, 2n − 4 via Hopf fibration on S4n+3 to ℍPn, and a family of isoparametric hypersurfaces in ℂP2n+1 having ɡ = 5 with multiplicities 1, 2, 2, 2n − 2, 2n − 2 via Hopf fibration on S4n+3 to ℂP2n+1.

 

2. PRELIMINARIES

In this section, following [7], [12] and [14], we recall the definition of isoparametric function and isoparametric family in a real space form which is an (n + 1)-dimensional, simply connected, complete Riemannian manifolds with constant sectional curvature c (= 1, 0,−1). Here is a unit sphere Sn+1 ⊂ ℝn+2, is ℝn+1, and is the hyperbolic space Hn+1.

Isoparametric family, parallel hypersurfaces, and constant principal curvatures. A non-constant real-valued function F defined on a connected open subset of is called an isoparametric function if it satisfies a system of differential equations

|grad F|2 = T ◦ F, ΔF = S ◦ F,

for some smooth function T and S where ΔF is the Laplacian of F. Moreover, the collection of an 1-parameter hypersurface of which is equal to level sets F−1(t) is called an isoparametric family of .

For each connected oriented hypersurface Mn embedded in with a unit normal vector field ξ on it, we define a map

where ϕ (X, t) is a point in reached after moving the point X of Mn by t along the normal geodesic α(s) in with α(0) = X and α'(0) = ξX (unit normal vector of ξ at X). For each fixed t ∈ ℝ, the image ϕ (Mn, t) of Mn is called parallel hypersurface.

A connected hypersurface Mn in the space form is said to have constant principal curvatures if there are distinct constants λ1 … λɡ representing principal curvatures given by a field of unit normal vector ξ at every point. Here, the multiplicity mi of λi is same throughout Mn and mi = n. If the oriented hypersurface Mn has constant principal curvatures, one can show that each parallel hypersurface also has constant principal curvatures. In particular, it is known that the level hypersurfaces of an isoparametric function F form a family of parallel hypersurfaces with constant principal curvatures. Conversely, for each connected hypersurface Mn of with constant principal curvatures, we can construct an isoparametric function F so that each ϕ(Mn, t) is contained in a level set of F which turns out a level set itself. In conclusion, an isoparametric hypersurface of the isoparametric family is defined as a hypersurface with constant principal curvatures.

Cartan’s work on isoparametric hypersurfaces. Cartan considered an isoparametric hypersurface Mn of with ɡ distinct principal curvatures λ1 … λɡ, having respective multiplicities m1, … mɡ. For ɡ > 1, Cartan showed an important equation known as Cartan’s identity([4])

for each i, 1 ≤ i ≤ ɡ. From this, he was able to determine all isoparametric hypersurfaces in the cases c = 0 and c = −1.

For 1 = Sn+1, Cartan ([3]) provided examples of isoparametric hypersurfaces with ɡ = 1, 2, 3 or 4 distinct principal curvatures. Moreover, he classified isoparametric hypersurfaces with ɡ ≤ 3 as follows.

(i) (ɡ = 1) The isoparametric hypersurface Mn with ɡ = 1 is totally umbilic, thus Mn is an open subset of a great or small hypersphere in Sn+1.

(ii) (ɡ = 2) The isoparametric hypersurface Mn with ɡ = 2 is a standard product of two spheres with radius r1 and r2 in the unit sphere Sn+1(1) in ℝn+2, namely,

Mn = Sp(r1) × Sq(r2) ⊂ Sn+1(1) ⊂ ℝp+1 × ℝq+1 = ℝn+2,

where and n = p + q.

(iii) (ɡ = 3) The isoparametric hypersurface Mn with ɡ = 3 must have the principal curvatures with the same multiplicity m = 1, 2, 4 or 8, and it is a tube of constant radius over a standard embedding of a projective plane 𝔽P2 into S3m+1, where 𝔽 is the division algebra ℝ, ℂ, ℍ and 𝕆, for m = 1, 2, 4 and 8 respectively. Therein, it was showed that any isoparametric family with ɡ distinct principal curvatures of the same multiplicity can be defined by a function F on ℝn+2 satisfying the equation

F|Sn+1 = cos ɡt.

Note the function F is a harmonic homogeneous polynomial of degree ɡ on ℝn+2 with

| gradE F |2 = ɡ2r2ɡ−2,

where the gradE F is the Euclidean gradient and r(X) = |X|.

Münzner’s work on isoparametric hypersurfaces in spheres. In the papers [10][11], an important generalization of Cartan’s work was produced on isoparametric hypersurface in sphere in 1973 by Münzner. Without assuming that the multiplicities are all the same, he proved that the possibilities for the number ɡ are 1, 2, 3, 4 and 6 ([14]). Moreover he obtained the following.

If M is a connected oriented isoparametric hypersurface embedded in Sn+1 with ɡ distinct principal curvatures, there exists a homogeneous polynomial F of degree ɡ on ℝn+2 such that M is an open subset of a level set of the restriction of F to Sn+1 satisfying the following Cartan-Münzner differential equations,

where r(X) = |X|, c = ɡ2(m2 − m1)=2, and m1, m2 are the two possible distinct multiplicities of the principal curvatures. If all the multiplicities are equal, then c = 0 and F is harmonic on ℝn+2. Therefore this generalizes the work of Cartan on the case ɡ = 3 where all multiplicities are equal and F is harmonic. The homogeneous polynomial F satisfying Cartan-Münzer differential equations is called a Cartan-Müunzner polynomial.

Conversely, the level sets of the restriction F|Sn+1 of F satisfying (1) and (2) constitute an isoparametric family of hypersurfaces. Münzner also proved that if M is an isoparametric hypersurfaces with principal curvatures cot θi; 0 < θ1 < … < θɡ < π, with multiplicities mi, then

and the multiplicities satisfy mi = mi+2 (subscripts mod ɡ). Therefore, if ɡ is odd, then all of the multiplicities must be equal and if ɡ is even, there are at most two distinct multiplicities.

Remark. Moreover, by [1], [10] and [11] we know (1) if ɡ = 3, then m1 = m2 = m3 = 1, 2, 4 or 8, (2) if ɡ = 4, then m1 = m3, m2 = m4, and m1, m2 are 1 or even, (3) if ɡ = 6, then m1 = m2 = … = m6 = 1 or 2.

 

3. HOMOGENEOUS FUNCTIONS ON QUATERNIONIC VECTOR SPACES

Homogeneous functions on ℝn+2 and Sn+1.In this section, by following [8], we review computation of the gradient and the Laplacian of a homogeneous function F on ℝn+2 and Sn+1 the unit sphere in ℝn+2.

Let F : ℝn+2 → ℝ be a homogeneous function of degree ɡ, that is, F(tX) = tɡF(X) for all nonzero t ∈ ℝ and X ∈ ℝn+2. Note the homogeneous function satisfies the following equation by Euler’s theorem for X ∈ ℝn+2

If F is an isoparametric function defined on a Euclidean space ℝn+2, the restriction of F to the unit sphere Sn+1 is also isoparametric by the following theorem.

Theorem 1. For a homogeneous function F : ℝn+2 → ℝ of degree ɡ, we have

|gradS F|2 = | gradE F|2 − ɡ2F2         ΔS F = ΔE F − ɡ(ɡ − 1)F − ɡ(n + 1)F.

Here, gradS F is the gradient of the restriction of F to the unit sphere Sn+1. Similarly, ΔEF and ΔSF denote the Laplacian of F on ℝn+2 and the unit sphere Sn+1 respectively.

Proof. (1) Let X ∈ Sn+1. Since X is a position vector of the unit sphere Sn+1, gradS F can be written by

Using (3),

|gradS F|2 = ­〈gradE F−〈gradE F,X〉X, gradE F−〈gradE F,X〉X 〉                   = |gradS F|2 − 2ɡF(X) ­〈gradE F,X〉X + ɡ2F2(X) |X|2                   = |gradS F|2 −ɡ2F2(X)

(2) Let ∇E and ∇S denote the Levi-Civita connections on ℝn+2 and Sn+1 respectively. Then ΔSF is the trace of the operator on TXSn+1 given by

For an orthonormal basis {V1,…,Vn+1} for TXSn+1,

Here we use 〈Vi,X〉 = 0 and gradS F = gradE F − ɡFX by (3) and (4). Since X is just an identity map on ℝn+2, we obtain

Therefore,

Thus we have

Now we compute the Laplacian ΔEF for the orthonormal basis {V1,…,Vn+1, X} for TXℝn+2 by applying the above identity and Euler theorem.

Remark. When the homogeneous polynomial F of degree ɡ satisfying the Cartan-Münzner differential equations, we conclude the followings. Notice that Im(F|Sn+1) ⊂ [−1, 1], in fact the image ranges exactly the whole compact connected set [−1, 1]. We can consider a level set Mc of F|Sn+1 defined by

Mc := {X ∈ Sn+1 | F(X) = c} = (F|Sn+1)−1 (c), c ∈ [−1, 1].

Then Mc (c ∈ (−1, 1)) is an isoparametric hypersurface, while M1 = (F|Sn+1)−1 (1) and M−1 = (F|Sn+1)−1 (−1) are focal submanifolds. In other words, we can denote the level set by

where M0 and Mπ/ɡ are two focal submanifolds and for t ∈ Mt is an isoparametric hypersurface.

Quaternionic vector spaces and quaternionic projective spaces. Each element of ℍ quaternions can be represented as

q = a + bi + cj + dk ∈ ℍ

with a, b, c, d ∈ ℝ, and the quaternion multiplication is determined by i2 = j2 = k2 = ijk = −1. The standard conjugate of q which is denoted by is the quaternion number Moreover, we define the norm of q as It is well known that ℍ is one of the composition algebras satisfying |q1q2| = |q1| |q2| for q1, q2 ∈ ℍ. If q ∈ , the complex number, then is the ordinary complex conjugate of q, and if q ∈ ℝ,

On the other hand, if we regard the field of complex number ℂ spanned by {1, k}, we can also present the quaternion number q as

q = z + wi, for z = a + dk, w = b + ck, where z, w ∈ ℂ = span {1, k} :

From this, define another conjugate of q by

where is the conjugate on ℂ = span {1, k}. Moreover, for an n × m matrices A = (aij) ∈ Mn×m (ℍ), we define by Then the conjugation gives us the following lemma.

Lemma 2. For q1, q1 ∈ ℍ,

(1) (2) (3) for A ∈ Mn×m(ℍ), B ∈ Mm×l(ℍ)(4) For A ∈ Mn×m(ℂ) with where

Now we recall the construction of the quaternionic projective space by Hopf fibration. We consider a 4 (n + 1)-dimensional quaternionic space over ℝ

ℍn+1f = {q = (q0,… qn)|qi ∈ ℍ, i = 0,…, n}

which is also a right ℍ-module, that is, for λ ∈ ℍ, q = (q0,…, qn) ∈ ℍn+1,

q · λ := (q0λ,… qnλ) ∈ ℍn+1:

And the unit sphere S4n+3 in ℍn+1 is defined as

The quaternionic projective n-space ℍPn is obtained as the quotient of the unit sphere S4n+3 by the right Sp(1)(= S3)-action, that is, ℍPn ≅ S4n+3/S3. Note that U(n + 1) acts on ℍn+1 and S4n+3 by the matrix multiplication

U(n + 1) × ℍn+1 → ℍn+1                 (A, q)     ↦ A · q := Aq

where q is represented as column matrix. Moreover, U(n + 1) also acts on ℍPn by (A, [q]) ↦ [Aq], where A ∈ U(n + 1), [q] ∈ ℍPn. Here [q] ∈ ℍPn is related to q ∈ n+1 via Hopf fibration.

Homogeneous functions on quaternionic vector spaces. In this subsection we construct a homogeneous function F on ℍn+1 which is invariant under Sp(1) and U(n + 1). Furthermore we will induce from F which is defined on the ℍPn that also invariant under those.

Define a function F on ℍn+1 by

F : ℍn+1 → ℝ              q ↦ F(q) := |q|2 = ,

where the column vector q = (q0,…, qn)t ∈ ℍn+1 and := which is the conjugate transpose of q.

Lemma 3. For the function F defined on ℍn+1,

(1) F is invariant under Sp(1) = S3 and U(n + 1).(2) F is a homogeneous function of degree 4.

Proof (1) Let λ ∈ S3, A ∈ U(n + 1), and q ∈ ℍn+1 the column vector. Using lemma2,

we complete the proof of (1).

(2) For t ∈ ℝ, q ∈ ℍn+1, = and tq = qt obviously. Therefore

Remark. Notice that the restriction of F to the unit sphere S4n+3 is also a homogeneous function of degree 4 and invariant under the action of Sp(1) = S3 and U(n + 1).

Now we induce a homogeneous function on ℍPn with the following diagram.

[q] is corresponding to q ∈ S4n+3. Using the same procedure, we get that is homogeneous with degree 4 and U(n+1)-invariant. Moreover, if q = (q0,…,qn)t ∈ S4n+3,

thus both of the images of the restriction F|S4n+3 and are the closed unit interval [0, 1] in fact exactly [0, 1].

The isoparametric function on S4n+3. In this subsection we prove our homogeneous function F on the unit sphere S4n+3 is isoparametric on S4n+3.

Theorem 4. The homogeneous function on the unit sphere S4n+3 = {q ∈ ℍn+1| |q| = 1} satisfies

|gradS F|2 = 16F(1 − F), ΔSF = 24 − 12F − 4(n + 1)F,

therefore F is isoparametric on the sphere S4n+3.

Proof. The column vector q = (q0,…, qn)t ∈ ℍn+1 can be denoted as

q = z + wi, z,w ∈ ℂn+1

where ℂ = span {1, k}. And we can write

where the standard real inner product in ℂn+1 is given by

〈x, y〉 := Re (x* y), x, y ∈ ℂ.

Here we consider x, y ∈ ℂ as vectors in ℝ2n+2. We also denote Im (x*, y) as −ω (z, w) which is a skew symmetric form on ℝ2n+2, and we write

z*w = (〈a, c〉 + 〈b, d〉) − (〈b, c〉 − 〈a, d〉) k       = Re (z*w) + Im(z*w) k       = 〈z, w〉 − ω (z, w) k, where z = a + dk, w = c + dk, a, b, c, d ∈ ℝn+1.

Then

and

Therefore, we obtain

|gradE F|2 = 16 |q|2 F, ΔEF = 24 |q|2.

Using Theorem 1, we get

|gradS F|2 = |gradE F|2 − 42F2 = 16F (1 − F),          ΔSF = ΔEF − 4 (4 − 1) F − 4 (n + 1) F = 24 − 12F − 4 (n + 1) F

since F is homogeneous of degree 4.            

Remark. Theorem 4 extends Nomizu’s work ([12]) on construction of isoparametric function on S2n+1 = { z ∈ ℂn+1| |z| = 1 } by

h(z) = | ztz|2 = ( |x|2 − |y|2)2 + 4 〈x, y〉2 , for z = x + iy, x, y ∈ ℝn+1.

The function h (z) is invariant under the actions of U (1) = S1 and O (n + 1). Moreover, it is an isoparametric function indeed since it satisfies

|gradS h|2 = 16h (1 − h), ΔSh = 16 − 12h − 4 (n + 1) h,

by correcting the computation of the Laplacian of h in [12]. In [12], he showed that h induces isoparametric hypersurfaces in S2n+1 of ɡ = 4 with multiplicities 1, 1, n − 1, n − 1.

By above Theorem4, the homogeneous function does not satisfy Cartan-Münzner differential equations. In the following theorem, we show that the isoparametric function gives a family of isoparametric hypersurfaces with ɡ = 4.

Theorem 5. For the above isoparametric function F, the level sets Mt forms the isoparametric family of hypersurfaces in S4n+3 with four distinct principal curvatures with multiplicities 2, 2, 2n − 1, 2n − 1.

Proof. By Münzner [10], the hypersurface Mt in the sphere can have only ɡ = 1, 2, 3, 4 or 6 distinct principal curvatures. We consider the preimage of F at zero which is a focal set in S4n+3. For the preimage of F at zero, we have

and that we get |z| = |w| and z*w = 0. Since |q|2 =|z|2+|w|2 = 1, F−1(0) consists of ordered orthogonal complex vectors in ℂn+1 of length 1/2. In other words, F−1(0) in S4n+3 is equivalently a complex Stiefel manifold of all orthogonal pairs of vectors in ℂn+1 which has dimension 4n.

Therefore each isoparametric hypersurface of dimension 4n+2 has one principal curvature with multiplicity 2. By dimension counting according to Remark 2, we conclude ɡ = 4. In particular, the four distinct principal curvature of isoparametric hypersurfaces in consideration have multiplicities 2, 2, 2n − 1, 2n − 1.

Remark.

1. In [13], Park showed that the possible g in S4n+3 are only 2 and 4. Since we show that a focal set F−1(0) is not a sphere, we exclude the case ɡ = 2 so that we conclude ɡ = 4.2. The preimage F−1(1) which is the other focal set and the isoparametric hypersurface F−1(t), t ∈ (0, 1) are homogeneous spaces. Identifying these spaces is an interesting question related to Veronese imbedding of complex projective spaces. We will explain it in another paper.

A complex projective space ℂPn is obtained from the Hopf fibration π of the unit sphere S2n+1 by the unit sphere S1. The isoparametric hypersurface M in ℂPn has constant principal curvatures if and only if M is homogeneous([9]). Moreover, a hypersurface M in ℂPn is isoparametric if and only if its inverse image π−1(M) under the well known Hopf map is isoparametric in S2n+1([15]). Similar study on quaternionic projective space ℍPn such as [13] is also very interesting. By applying the study in [13], we also obtain the following corollary.

Corollary 6. By Hopf fibration on S4n+3 to ℍPn, the isoparametric hypersurfaces in S4n+3 given by F induce a family of isoparametric hypersurfaces in ℍPn having ɡ = 5 with multiplicities 3, 2, 2, 2n − 4, 2n − 4. By Hopf fibration on S4n+3 to ℂP2n+1, the isoparametric hypersurfaces in S4n+3 given by F give a family of isoparametric hypersurfaces in ℂP2n+1 having ɡ = 5 with multiplicities 1, 2, 2, 2n − 2, 2n − 2.

Proof. According to the study in [13] , we can easily conclude F induces a family of isoparametric hypersurfaces in ℍPn having ɡ = 5 with multiplicities 3, 2, 2, 2n− 4, 2n − 4 because F gives family of isoparametric hypersurfaces in S4n+3 with ɡ = 4 of multiplicities 2, 2, 2n − 1, 2n − 1. Here, we observe S3-action on S4n+3 is nontrivial for the principal distributions with the multiplicities with 2n − 1 and trivial for the principal distributions with the multiplicities with 2. Since the S1(⊂ S3)-action of Hopf fibration also has the similar properties, we get F induces a family of isoparametric hypersurfaces in ℂP2n+1 having ɡ = 5 with multiplicities 1, 2, 2, 2n − 2, 2n − 2.

Remark. With similar argument, we also know that the homogeneous function S2n+1 of Nomizu in Remark in the Therorem 4 produces a family of isoparametric hypersurfaces in ℂPn having ɡ = 5 with multiplicities 1, 1, 1, n − 2, n − 2.