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ISOPARAMETRIC FUNCTIONS IN S4n+3
ISOPARAMETRIC FUNCTIONS IN S4n+3
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2014. Nov, 21(4): 257-270
Copyright © 2014, Korean Society of Mathematical Education
  • Received : July 03, 2014
  • Accepted : September 17, 2014
  • Published : November 30, 2014
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About the Authors
SEO-IN JEE
DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY, SEOUL 120-750, KOREAEmail address:rejinajee@gmail.com
JAE-HYOUK LEE
DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY, SEOUL 120-750, KOREAEmail address:jaehyoukl@ewha.ac.kr

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 ɡ = 4 distinct principal curvatures and isoparametric hypersurfaces in quaternionic projective spaces with ɡ = 5. This extends study of Nomizu on isoparametric function on complex vector spaces and complex projective spaces.
Keywords
1. INTRODUCTION
A hypersurface Mn embedded in ℝ n+1 or a unit sphere S n+1 is said to be isoparametric if it has constant principal curvatures. Isoparametric hypersurfaces in ℝ n+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 × ℝ n−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
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, 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 ℂ Pn . Here the isoparametric hypersurfaces in S 2n+1 and ℂ Pn are related via the Hopf fibration S 2n+1 → ℂ Pn . Moreover, Park [13] also considered isoparametric hypersurfaces in S 4n+3 and ℍ Pn via the Hopf fibration S 4n+3 → ℍ Pn . Therefrom, we are interested in the isoparametric hypersurfaces in S 4n+3 which are invariant under Sp (1) = S 3 action of Hopf fibration. In particular, we consider the work of Nomizu [12] which provided an example of ɡ = 4 case in S 2n+1 ⊂ ℂ n+1 and extend his study to S 4n+3 ⊂ ℍ n+1 . We construct an isoparametric function on S 4n+3 ⊂ ℍ n+1 which is homogeneous of degree four and invariant under S 3 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 S 4n+3 of four distinct principal curvatures with multiplicities 2, 2, 2 n − 1, 2 n − 1. Moreover by applying [13] the family also induces a family of isoparametric hypersurfaces in ℍ Pn having ɡ = 5 with multiplicities 3, 2, 2, 2 n − 4, 2 n − 4 via Hopf fibration on S 4n+3 to ℍ Pn , and a family of isoparametric hypersurfaces in ℂ P 2n+1 having ɡ = 5 with multiplicities 1, 2, 2, 2 n − 2, 2 n − 2 via Hopf fibration on S 4n+3 to ℂ P 2n+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
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which is an ( n + 1)-dimensional, simply connected, complete Riemannian manifolds with constant sectional curvature c (= 1, 0,−1). Here
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is a unit sphere S n+1 ⊂ ℝ n+2 ,
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is ℝ n+1 , and
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is the hyperbolic space H n+1 .
Isoparametric family, parallel hypersurfaces, and constant principal curvatures. A non-constant real-valued function F defined on a connected open subset of
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is called an isoparametric function if it satisfies a system of differential equations
  • |gradF|2=T◦F, ΔF=S◦F,
for some smooth function T and S where Δ F is the Laplacian of F . Moreover, the collection of an 1-parameter hypersurface of
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which is equal to level sets F −1 ( t ) is called an isoparametric family of
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.
For each connected oriented hypersurface Mn embedded in
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with a unit normal vector field ξ on it, we define a map
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where ϕ ( X , t ) is a point in
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reached after moving the point X of Mn by t along the normal geodesic α ( s ) in
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with α (0) = X and α '(0) = ξ X (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
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is said to have constant principal curvatures if there are distinct constants λ 1 … λ ɡ representing principal curvatures given by a field of unit normal vector ξ at every point. Here, the multiplicity mi of λ i is same throughout Mn and
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mi = n . If the oriented hypersurface Mn has constant principal curvatures, one can show that each parallel hypersurface
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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
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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
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with ɡ distinct principal curvatures λ 1 … λ ɡ , having respective multiplicities m 1 , … mɡ . For ɡ > 1, Cartan showed an important equation known as Cartan’s identity( [4] )
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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
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1 = S n+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 S n+1 .
(ii) ( ɡ = 2) The isoparametric hypersurface Mn with ɡ = 2 is a standard product of two spheres with radius r 1 and r 2 in the unit sphere S n+1 (1) in ℝ n+2 , namely,
  • Mn=Sp(r1) ×Sq(r2) ⊂Sn+1(1) ⊂ ℝp+1× ℝq+1= ℝn+2,
where
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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 𝔽 P 2 into S 3m+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 ℝ n+2 satisfying the equation
  • F|Sn+1= cosɡt.
Note the function F is a harmonic homogeneous polynomial of degree ɡ on ℝ n+2 with
  • | gradEF|2=ɡ2r2ɡ−2,
where the grad E 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 S n+1 with ɡ distinct principal curvatures, there exists a homogeneous polynomial F of degree ɡ on ℝ n+2 such that M is an open subset of a level set of the restriction of F to S n+1 satisfying the following Cartan-Münzner differential equations ,
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where r (X) = |X|, c = ɡ 2 ( m 2 m 1 )=2, and m 1 , m 2 are the two possible distinct multiplicities of the principal curvatures. If all the multiplicities are equal, then c = 0 and F is harmonic on ℝ n+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 < θ 1 < … < θɡ < π , with multiplicities mi , then
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and the multiplicities satisfy mi = m i+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 m 1 = m 2 = m 3 = 1, 2, 4 or 8, (2) if ɡ = 4, then m 1 = m 3 , m 2 = m 4 , and m 1 , m 2 are 1 or even, (3) if ɡ = 6, then m 1 = m 2 = … = m 6 = 1 or 2.
3. HOMOGENEOUS FUNCTIONS ON QUATERNIONIC VECTOR SPACES
Homogeneous functions on ℝn+2 and S n+1 .In this section, by following [8] , we review computation of the gradient and the Laplacian of a homogeneous function F on ℝ n+2 and S n+1 the unit sphere in ℝ n+2 .
Let F : ℝ n+2 → ℝ be a homogeneous function of degree ɡ , that is, F ( tX ) = t ɡ F ( X ) for all nonzero t ∈ ℝ and X ∈ ℝ n+2 . Note the homogeneous function satisfies the following equation by Euler’s theorem for X ∈ ℝ n+2
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If F is an isoparametric function defined on a Euclidean space ℝ n+2 , the restriction of F to the unit sphere S n+1 is also isoparametric by the following theorem.
Theorem 1. For a homogeneous function F : ℝ n+2 → ℝ of degree ɡ, we have
  • |gradSF|2= | gradEF|2−ɡ2F2
  •          ΔSF= ΔEF−ɡ(ɡ− 1)F−ɡ(n+ 1)F.
Here , grad S F is the gradient of the restriction of F to the unit sphere S n+1 . Similarly , Δ EF and Δ SF denote the Laplacian of F on ℝn+2 and the unit sphere Sn+1 respectively .
Proof . (1) Let X S n+1 . Since X is a position vector of the unit sphere S n+1 , grad S F can be written by
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Using (3),
  • |gradSF|2= ­〈gradEF−〈gradEF,X〉X, gradEF−〈gradEF,X〉X〉
  •                    = |gradSF|2− 2ɡF(X) ­〈gradEF,X〉X+ɡ2F2(X) |X|2
  •                    = |gradSF|2−ɡ2F2(X)
(2) Let ∇ E and ∇ S denote the Levi-Civita connections on ℝ n+2 and S n+1 respectively. Then Δ SF is the trace of the operator on TX S n+1 given by
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For an orthonormal basis { V 1 ,…, V n+1 } for TX S n+1 ,
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Here we use 〈 Vi , X 〉 = 0 and grad S F = grad E F ɡ FX by (3) and (4). Since X is just an identity map on ℝ n+2 , we obtain
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Therefore,
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Thus we have
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Now we compute the Laplacian Δ EF for the orthonormal basis { V 1 ,…, V n+1 , X } for TX n+2 by applying the above identity and Euler theorem.
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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 M 1 = ( 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
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where M 0 and M π/ɡ are two focal submanifolds and for t
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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 i 2 = j 2 = k 2 = ijk = −1. The standard conjugate of q which is denoted by
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is the quaternion number
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Moreover, we define the norm of q as
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It is well known that ℍ is one of the composition algebras satisfying | q 1 q 2 | = | q 1 | | q 2 | for q 1 , q 2 ∈ ℍ. If q
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, the complex number, then
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is the ordinary complex conjugate of q , and if q ∈ ℝ,
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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, forz=a+dk,w=b+ck, wherez,w∈ ℂ =span{1,k} :
From this, define another conjugate
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of q by
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where
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is the conjugate on ℂ = span {1, k }. Moreover, for an n × m matrices A = ( aij ) ∈ M n×m (ℍ), we define
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by
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Then the conjugation gives us the following lemma.
Lemma 2. For q 1 , q 1 ∈ ℍ,
  • (1)
  • (2)
  • (3)forA∈Mn×m(ℍ),B∈Mm×l(ℍ)
  • (4)ForA∈Mn×m(ℂ)withwhere
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 = ( q 0 ,…, qn ) ∈ ℍ n+1 ,
  • q· λ := (q0λ,…qnλ) ∈ ℍn+1:
And the unit sphere S 4n+3 in ℍ n+1 is defined as
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The quaternionic projective n-space Pn is obtained as the quotient of the unit sphere S 4n+3 by the right Sp (1)(= S 3 )-action, that is, ℍ Pn S 4n+3 / S 3 . Note that U ( n + 1) acts on ℍ n+1 and S 4n+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 ℍ Pn by ( A , [ q ]) ↦ [ Aq ], where A U ( n + 1), [ q ] ∈ ℍ Pn . Here [ q ] ∈ ℍ Pn is related to q
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n+1 via Hopf fibration.
Homogeneous functions on quaternionic vector spaces. In this subsection we construct a homogeneous function F on ℍ n+1 which is invariant under Sp (1) and U ( n + 1). Furthermore we will induce
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from F which is defined on the ℍ Pn that also invariant under those.
Define a function F on ℍ n+1 by
  • F: ℍn+1→ ℝ
  •               q↦F(q) := |q|2=,
where the column vector q = ( q 0 ,…, qn ) t ∈ ℍ n+1 and
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:=
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which is the conjugate transpose of q .
Lemma 3. For the function F defined on n+1 ,
  • (1)F is invariant under Sp(1) =S3and U(n+ 1).
  • (2)F is a homogeneous function of degree4.
Proof (1) Let λ ∈ S 3 , A U ( n + 1), and q ∈ ℍ n+1 the column vector. Using lemma2,
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we complete the proof of (1).
(2) For t ∈ ℝ, q ∈ ℍ n+1 ,
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=
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and tq = qt obviously. Therefore
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Remark. Notice that the restriction of F to the unit sphere S 4n+3 is also a homogeneous function of degree 4 and invariant under the action of Sp (1) = S 3 and U ( n + 1).
Now we induce a homogeneous function
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on ℍ Pn with the following diagram.
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[ q ] is corresponding to q S 4n+3 . Using the same procedure, we get that
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is homogeneous with degree 4 and U ( n +1)-invariant. Moreover, if q = ( q 0 ,…, qn ) t S 4n+3 ,
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thus both of the images of the restriction F | S4n+3 and
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are the closed unit interval [0, 1] in fact exactly [0, 1].
The isoparametric function on S 4n+3 . In this subsection we prove our homogeneous function F on the unit sphere S 4n+3 is isoparametric on S 4n+3 .
Theorem 4. The homogeneous function
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on the unit sphere S 4n+3 = { q ∈ ℍ n+1 | | q | = 1} satisfies
  • |gradSF|2= 16F(1 −F), ΔSF= 24 − 12F− 4(n+ 1)F,
therefore F is isoparametric on the sphere S 4n+3 .
Proof . The column vector q = ( q 0 ,…, qn ) t ∈ ℍ n+1 can be denoted as
  • q=z + wi, z,w∈ ℂn+1
where ℂ = span {1, k }. And we can write
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where the standard real inner product in ℂ n+1 is given by
  • 〈x,y〉 := Re (x*y),x,y∈ ℂ.
Here we consider x , y ∈ ℂ as vectors in ℝ 2n+2 . We also denote Im ( x * , y ) as − ω ( z, w ) which is a skew symmetric form on ℝ 2n+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, wherez=a+dk,w=c+dk, a, b, c, d∈ ℝn+1.
Then
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and
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Therefore, we obtain
  • |gradEF|2= 16 |q|2F, ΔEF= 24 |q|2.
Using Theorem 1, we get
  • |gradSF|2= |gradEF|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.            
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Remark. Theorem 4 extends Nomizu’s work ( [12] ) on construction of isoparametric function on S 2n+1 = { z ∈ ℂ n+1 | | z | = 1 } by
  • h(z) = |ztz|2= ( |x|2− |y|2)2+ 4 〈x,y〉2, forz=x+iy,x,y∈ ℝn+1.
The function h ( z ) is invariant under the actions of U (1) = S 1 and O ( n + 1). Moreover, it is an isoparametric function indeed since it satisfies
  • |gradSh|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 S 2n+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 S 4n+3 with four distinct principal curvatures with multiplicities 2, 2, 2 n − 1, 2 n − 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 S 4n+3 . For the preimage of F at zero, we have
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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 ℂ n+1 of length 1/2. In other words, F −1 (0) in S 4n+3 is equivalently
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a complex Stiefel manifold of all orthogonal pairs of vectors in ℂ n+1 which has dimension 4 n .
Therefore each isoparametric hypersurface of dimension 4 n +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, 2 n − 1, 2 n − 1.
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Remark.
  • 1. In[13], Park showed that the possible g inS4n+3are only 2 and 4. Since we show that a focal setF−1(0) is not a sphere, we exclude the caseɡ= 2 so that we concludeɡ= 4.
  • 2. The preimageF−1(1) which is the other focal set and the isoparametric hypersurfaceF−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 ℂ Pn is obtained from the Hopf fibration π of the unit sphere S 2n+1 by the unit sphere S 1 . The isoparametric hypersurface M in ℂ Pn has constant principal curvatures if and only if M is homogeneous( [9] ). Moreover, a hypersurface M in ℂ Pn is isoparametric if and only if its inverse image π −1 ( M ) under the well known Hopf map is isoparametric in S 2n+1 ( [15] ). Similar study on quaternionic projective space ℍ Pn 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 S 4n+3 to Pn , the isoparametric hypersurfaces in S 4n+3 given by F induce a family of isoparametric hypersurfaces in Pn having ɡ = 5 with multiplicities 3, 2, 2, 2 n − 4, 2 n − 4. By Hopf fibration on S 4n+3 to P 2n+1 , the isoparametric hypersurfaces in S 4n+3 given by F give a family of isoparametric hypersurfaces in P 2n+1 having ɡ = 5 with multiplicities 1, 2, 2, 2 n − 2, 2 n − 2.
Proof . According to the study in [13] , we can easily conclude F induces a family of isoparametric hypersurfaces in ℍ Pn having ɡ = 5 with multiplicities 3, 2, 2, 2 n − 4, 2 n − 4 because F gives family of isoparametric hypersurfaces in S 4n+3 with ɡ = 4 of multiplicities 2, 2, 2 n − 1, 2 n − 1. Here, we observe S 3 -action on S 4n+3 is nontrivial for the principal distributions with the multiplicities with 2 n − 1 and trivial for the principal distributions with the multiplicities with 2. Since the S 1 (⊂ S 3 )-action of Hopf fibration also has the similar properties, we get F induces a family of isoparametric hypersurfaces in ℂ P 2n+1 having ɡ = 5 with multiplicities 1, 2, 2, 2 n − 2, 2 n − 2.
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Remark. With similar argument, we also know that the homogeneous function S 2n+1 of Nomizu in Remark in the Therorem 4 produces a family of isoparametric hypersurfaces in ℂ Pn having ɡ = 5 with multiplicities 1, 1, 1, n − 2, n − 2.
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(No.2-2013-3378-001-2).
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