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.
1. INTRODUCTION
A hypersurface
M^{n}
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
S^{k}
× ℝ
^{n−k}
. On the other hand, isoparametric hypersurfaces in spheres are rather complicated. In 19381940, 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 ℂ
P^{n}
. Here the isoparametric hypersurfaces in
S
^{2n+1}
and ℂ
P^{n}
are related via the Hopf fibration
S
^{2n+1}
→ ℂ
P^{n}
. Moreover, Park
[13]
also considered isoparametric hypersurfaces in
S
^{4n+3}
and ℍ
P^{n}
via the Hopf fibration
S
^{4n+3}
→ ℍ
P^{n}
. 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
P^{n}
having
ɡ
= 5 with multiplicities 3, 2, 2, 2
n
− 4, 2
n
− 4 via Hopf fibration on
S
^{4n+3}
to ℍ
P^{n}
, 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
which is an (
n
+ 1)dimensional, simply connected, complete Riemannian manifolds with constant sectional curvature
c
(= 1, 0,−1). Here
is a unit sphere
S
^{n+1}
⊂ ℝ
^{n+2}
,
is ℝ
^{n+1}
, and
is the hyperbolic space
H
^{n+1}
.
Isoparametric family, parallel hypersurfaces, and constant principal curvatures.
A nonconstant realvalued function
F
defined on a connected open subset of
is called an
isoparametric function
if it satisfies a system of differential equations
for some smooth function
T
and
S
where Δ
F
is the Laplacian of
F
. Moreover, the collection of an 1parameter hypersurface of
which is equal to level sets
F
^{−1}
(
t
) is called an
isoparametric family
of
.
For each connected oriented hypersurface
M^{n}
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
M^{n}
by
t
along the normal geodesic
α
(
s
) in
with
α
(0) =
X
and
α
'(0) =
ξ
_{X}
(unit normal vector of
ξ
at
X
). For each fixed
t
∈ ℝ, the image
ϕ
(
M^{n}
,
t
) of
M^{n}
is called
parallel hypersurface
.
A connected hypersurface
M^{n}
in the space form
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
m_{i}
of λ
_{i}
is same throughout
M^{n}
and
m_{i}
=
n
. If the oriented hypersurface
M^{n}
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
M^{n}
of
with constant principal curvatures, we can construct an isoparametric function
F
so that each
ϕ
(
M^{n}
,
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
M^{n}
of
with
ɡ
distinct principal curvatures λ
_{1}
… λ
ɡ
, having respective multiplicities
m
_{1}
, …
m_{ɡ}
. 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 =
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
M^{n}
with
ɡ
= 1 is totally umbilic, thus
M^{n}
is an open subset of a great or small hypersphere in
S
^{n+1}
.
(ii) (
ɡ
= 2) The isoparametric hypersurface
M^{n}
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
and
n
=
p + q
.
(iii) (
ɡ
= 3) The isoparametric hypersurface
M^{n}
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
Note the function
F
is a harmonic homogeneous polynomial of degree
ɡ
on ℝ
^{n+2}
with
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
CartanMünzner differential equations
,
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 CartanMünzer differential equations is called a
CartanMüunzner 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
θ_{i}
; 0 <
θ
_{1}
< … <
θɡ
<
π
, with multiplicities
m_{i}
, then
and the multiplicities satisfy
m_{i}
=
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 ℝ^{n+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}
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

gradSF2=  gradEF2−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
^{E}F and
Δ
^{S}F denote the Laplacian of F on ℝ^{n+2} and the unit sphere S^{n+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
Using (3),

gradSF2= 〈gradEF−〈gradEF,X〉X, gradEF−〈gradEF,X〉X〉

= gradSF2− 2ɡF(X) 〈gradEF,X〉X+ɡ2F2(X) X2

= gradSF2−ɡ2F2(X)
(2) Let ∇
^{E}
and ∇
^{S}
denote the LeviCivita connections on ℝ
^{n+2}
and
S
^{n+1}
respectively. Then Δ
^{S}F
is the trace of the operator on
T_{X}
S
^{n+1}
given by
For an orthonormal basis {
V
_{1}
,…,
V
_{n+1}
} for
T_{X}
S
^{n+1}
,
Here we use 〈
V_{i}
,
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
Therefore,
Thus we have
Now we compute the Laplacian Δ
^{E}F
for the orthonormal basis {
V
_{1}
,…,
V
_{n+1}
,
X
} for
T_{X}
ℝ
^{n+2}
by applying the above identity and Euler theorem.
Remark.
When the homogeneous polynomial
F
of degree
ɡ
satisfying the CartanMünzner 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
M_{c}
of
F

_{Sn+1}
defined by

Mc:= {X∈Sn+1F(X) =c} = (FSn+1)−1(c),c∈ [−1, 1].
Then
M_{c}
(
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
where
M
_{0}
and
M
_{π/ɡ}
are two focal submanifolds and for
t
∈
M_{t}
is an isoparametric hypersurface.
Quaternionic vector spaces and quaternionic projective spaces.
Each element of ℍ quaternions can be represented as
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
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 
q
_{1}
q
_{2}
 = 
q
_{1}
 
q
_{2}
 for
q
_{1}
,
q
_{2}
∈ 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, forz=a+dk,w=b+ck, wherez,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
= (
a_{ij}
) ∈
M
_{n×m}
(ℍ), we define
by
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 ℍmodule, that is, for λ ∈ ℍ,
q
= (
q
_{0}
,…,
q_{n}
) ∈ ℍ
^{n+1}
,

q· λ := (q0λ,…qnλ) ∈ ℍn+1:
And the unit sphere
S
^{4n+3}
in ℍ
^{n+1}
is defined as
The
quaternionic projective nspace
ℍ
P^{n}
is obtained as the quotient of the unit sphere
S
^{4n+3}
by the right
Sp
(1)(=
S
^{3}
)action, that is, ℍ
P^{n}
≅
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
P^{n}
by (
A
, [
q
]) ↦ [
Aq
], where
A
∈
U
(
n
+ 1), [
q
] ∈ ℍ
P^{n}
. Here [
q
] ∈ ℍ
P^{n}
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 ℍ
^{n+1}
which is invariant under
Sp
(1) and
U
(
n
+ 1). Furthermore we will induce
from
F
which is defined on the ℍ
P^{n}
that also invariant under those.
Define a function
F
on ℍ
^{n+1}
by

F: ℍn+1→ ℝ

q↦F(q) := q2=,
where the column vector
q
= (
q
_{0}
,…,
q_{n}
)
^{t}
∈ ℍ
^{n+1}
and
:=
which is the conjugate transpose of
q
.
Lemma 3.
For the function F defined on
ℍ
^{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,
we complete the proof of (1).
(2) For
t
∈ ℝ,
q
∈ ℍ
^{n+1}
,
=
and
tq
=
qt
obviously. Therefore
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
on ℍ
P^{n}
with the following diagram.
[
q
] is corresponding to
q
∈
S
^{4n+3}
. Using the same procedure, we get that
is homogeneous with degree 4 and
U
(
n
+1)invariant. Moreover, if
q
= (
q
_{0}
,…,
q_{n}
)
^{t}
∈
S
^{4n+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
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
on the unit sphere
S
^{4n+3}
= {
q
∈ ℍ
^{n+1}
 
q
 = 1}
satisfies

gradSF2= 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}
,…,
q_{n}
)
^{t}
∈ ℍ
^{n+1}
can be denoted as
where ℂ =
span
{1,
k
}. And we can write
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
and
Therefore, we obtain

gradEF2= 16 q2F, ΔEF= 24 q2.
Using Theorem 1, we get

gradSF2= gradEF2− 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
S
^{2n+1}
= {
z
∈ ℂ
^{n+1}
 
z
 = 1 } by

h(z) = ztz2= ( x2− y2)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

gradSh2= 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 CartanMünzner 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 M_{t} 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
M_{t}
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
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
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.
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
P^{n}
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
P^{n}
has constant principal curvatures if and only if
M
is homogeneous(
[9]
). Moreover, a hypersurface
M
in ℂ
P^{n}
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
P^{n}
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
ℍ
P^{n}
,
the isoparametric hypersurfaces in
S
^{4n+3}
given by F induce a family of isoparametric hypersurfaces in
ℍ
P^{n}
having
ɡ
= 5
with multiplicities
3, 2, 2, 2
n
− 4, 2
n
− 4.
By Hopf fibration on
S
^{4n+3}
to
ℂ
P
^{2n+1}
,
the isoparametric hypersurfaces in
S
^{4n+3}
given by F give a family of isoparametric hypersurfaces in
ℂ
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
P^{n}
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.
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
P^{n}
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.2201333780012).
U. Abresch
1983
Isoparametric hypersurfaces with four or six distinct principal curvatures
Math. Ann.
264
283 
302
DOI : 10.1007/BF01459125
Cartan É.
1938
Familles de surfaces isoparamétriques dans les espaces à courbure constante
Annali di Mat.
17
177 
191
DOI : 10.1007/BF02410700
Cartan E.
1939
Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques
Math. Z.
45
335 
367
DOI : 10.1007/BF01580289
Cartan E.
1939
Sur quelques familles remarquables d’hypersurfaces
C.R. Congrès Math. Liège
30 
41
Cartan E.
1940
Sur des familles d’hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions
Revista Univ. Tucumàn
1
5 
22
Cecil T.E.
1974
Cecil: A characterization of metric spheres in hyperbolic space by Morse theory
Tôhoku Math. J.
26
(2)
341 
351
DOI : 10.2748/tmj/1178241128
Cecil T.E.
2008
Isoparametric and Dupin hypersurfaces. Symmetry, Integrability and Geometry
Symmetry, Integrability and Geometry: Methods and Applications
4
(062)
1 
28
Cecil T.E.
,
Ryan P.J.
1985
Research Notes in Math
Pitman
Boston
Tight and taut immersions of manifolds
Münzner H.F.
1981
Isoparametrische Hyperflächen in Sphären. II, Über die Zerlegung der Sphäre in Ballbundel
Math. Ann.
256
215 
232
DOI : 10.1007/BF01450799
Thorbergsson G.
2000
Handbook of differential geometry
NorthHolland
Amsterdam
A survey on isoparametric hypersurfaces and their generalizations
963 
995
Wang Q.M.
1982
Isoparametric hypersurfaces in complex projective spaces
Diff. geom. and diff. equ., Proc. 1980 Beijing Sympos.
3
1509 
1523