Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
45 (2012), pp. 41–48

CHARACTERIZATIONS OF ISOPARAMETRIC
HYPERSURFACES IN A SPHERE
SADAHIRO MAEDA AND HIROMASA TANABE(Received: October 18, 2011)
Abstract. We characterize isoparametric hypersurfaces in a sphere by using
covariant derivatives of their shape operators and by observing the extrinsic
shape of their geodesics.

1. Introduction
It is well-known that a hypersurface M in an (n + 1)-dimensional sphereS (c) (n ≧ 2) of constant sectional curvature c is isoparametric if and only if allof its principal curvatures in the ambient sphere S n+1 (c) are constant. The studyof isoparametric hypersurfaces is one of the most interesting objects in diﬀerential
geometry. The classiﬁcation problem of isoparametric hypersurfaces in a sphere is
still open. However, we know that the number g of distinct principal curvatures of
isoparametric hypersurfaces in a sphere is either g = 1, 2, 3, 4 or 6 (see [6, 7]). Notethat this result is not obtained by classifying all isoparametric hypersurfaces.
We know that every isoparametric hypersurface with g = 1 (totally umbilic
hypersurfaces) or g = 2 (Cliﬀord hypersurfaces) has parallel shape operator but
other isoparametric hypersurfaces do not have parallel shape operator. Moreover,a hypersurface M in S n+1 (c) is totally umbilic if and only if every geodesic of M isa circle (i.e., either a great circle or a small circle of positive curvature) on S n+1 (c).
The main purpose of this paper is to give a characterization of all isoparametric
hypersurfaces in a sphere by weakening these geometric properties (see Theorem
1).
n+1

2. Statements of results
We ﬁrst recall some basic terminology in the theory of hypersurfaces. Let M
be a hypersurface in an (n + 1)-dimensional sphere S n+1 (c) (n ≧ 2) of constant2010 Mathematics Subject Classiﬁcation. Primary 53B25, Secondary 53C40.
Key words and phrases. spheres, isoparametric hypersurfaces, shape operator, principal distribution, geodesics, circles.
The ﬁrst author is partially supported by Grant-in-Aid for Scientiﬁc Research (C)(No. 23540097), Japan Society for the Promotion of Sciences.
41

42

S. MAEDA AND H. TANABE

sectional curvature c and N a unit normal vector ﬁeld on M in S n+1 (c). Then thee of S n+1 (c) and ∇ of M are related by Gauss formula
Riemannian connections ∇(2.1)

e X Y = ∇X Y + ⟨AX, Y ⟩N∇

and Weingarten formula(2.2)

e X N = −AX∇

for arbitrary vector ﬁelds X and Y on M , where ⟨ , ⟩ denotes the Riemannian
metric on M induced from the standard metric on S n+1 (c) and A is the shapeoperator of M in S n+1 (c). The Codazzi equation can be written as(2.3)⟨(∇X A)Y, Z⟩ = ⟨(∇Y A)X, Z⟩

for vector ﬁelds X, Y and Z tangent to M . Eigenvalues and eigenvectors of the
shape operator A are called principal curvatures and principal curvature vectors,
respectively.
Next we recall the deﬁnition of circles in Riemannian geometry. A smooth curve
f parametrized by its arclength s is called aγ = γ(s) in a Riemannian manifold M
circle of curvature k (≧ 0) if it satisﬁes the following ordinary diﬀerential equation:e γ̇ ∇e γ̇ γ̇ = −k 2 γ̇,∇e γ̇ denotes the covariant diﬀerentiation along γ with
where k is constant and ∇e of Mf. Since ∥∇e γ̇ γ̇∥ = k, a circle of null
respect to the Riemannian connection ∇curvature is nothing but a geodesic.
Theorem 1. Let M be a connected hypersurface in an (n + 1)-dimensional sphereS n+1 (c) (n ≧ 2) of constant sectional curvature c. Then the following three conditions are mutually equivalent.
(1) M is locally congruent to an isoparametric hypersurface in S n+1 (c).
(2) The tangent bundle TM of M is decomposed as the direct sum of the principal distributions Vλi = {X ∈ TM |AX = λi X} such that the covariant
derivative of the shape operator A of M in S n+1 (c) satisﬁes (∇X A)Y = 0
for all X, Y ∈ Vλi associated to every principal curvature λi , where ∇ denotes the Riemannian connection of M .
(3) For each point p of M , there exists an orthonormal basis {v1 , . . . , vmp } of theorthogonal complement of ker Ap in Tp M (mp = rank Ap ) such that every
geodesic of M through p with initial vector vi is a small circle of positive
curvature in S n+1 (c).
In our previous paper [4], we proved that the above Conditions (1) and (3) aremutually equivalent. However, for readers we give a complete proof of Theorem 1.
Proof of Theorem 1. We ﬁrst verify that Conditions (1) and (2) are mutually equivalent. Suppose Condition (1). Then the tangent bundle TM of M is decomposedas the direct sum of the principal distributions Vλi = {X ∈ TM |AX = λi X} associated to constant principal curvature λi . For any X, Y ∈ Vλi and any Z ∈ TM ,

CHARACTERIZATIONS OF ISOPARAMETRIC HYPERSURFACES

43

the Codazzi equation (2.3) shows⟨(∇X A)Y, Z⟩ = ⟨(∇Z A)Y, X⟩ = ⟨∇Z (AY ) − A∇Z Y, X⟩= ⟨(λi I − A)∇Z Y, X⟩ = ⟨∇Z Y, (λi I − A)X⟩ = 0.
Hence the covariant derivative of the shape operator A satisﬁes (∇X A)Y = 0 for
any X, Y ∈ Vλi .
We next suppose Condition (2). Taking a nonzero element X in Vλi , we have for
all Z ∈ TM⟨(∇X A)X, Z⟩ = ⟨(∇Z A)X, X⟩ = ⟨∇Z (AX) − A∇Z X, X⟩= ⟨(Zλi )X + (λi I − A)∇Z X, X⟩= (Zλi )∥X∥2 + ⟨∇Z X, (λi I − A)X⟩ = (Zλi )∥X∥2 .
Since (∇X A)X = 0 and X is not zero, we get Zλi = 0 for all Z ∈ TM . Thus we
see that all principal curvatures of M in the ambient sphere S n+1 (c) are constant,so that Conditions (1) and (2) are mutually equivalent.
In the following, we study the relation between Conditions (1) and (3). Suppose
Condition (1). Let M be an isoparametric hypersurface of S n+1 (c) with constantprincipal curvatures λ1 , . . . , λg . Then the tangent bundle TM is decomposed as:
TM = Vλ1 ⊕ Vλ2 ⊕ · · · Vλg . We here recall the fact that every Vλi (1 ≦ i ≦ g) is
integrable and moreover every leaf Lλi of Vλi is totally geodesic in the hypersurface
M . (To do show that, we verify that ∇X Y ∈ Vλi for all X, Y ∈ Vλi . For suchvectors X, Y we have (∇X A)Y = 0 by Condition (2), and hence
A(∇X Y ) = ∇X (AY ) − (∇X A)Y = λi (∇X Y ).)The above fact, together with Gauss formula (2.1), implies that and every leaf Lλi
is totally umbilic in the ambient sphere S n+1 (c). Note that Lλi is nothing but
a sphere S mi (ci ) with mi = dim Vλi and ci = c + λ2i . So, when λi ̸= 0 (resp.
λi = 0), every geodesic γ = γ(s) through p = γ(0) on M with γ̇(0) ∈ Vλi is asmall circle of positive curvature |λi | (resp. a great circle) on S n+1 (c). Therefore,choosing an orthonormal basis {v1 , . . . , vmp } of the orthogonal complement of ker Ap
in Tp M (mp = rank Ap ) as principal curvature vectors of M in S n+1 (c), we obtain
the desired Condition (3).
Conversely, suppose Condition (3). We consider the open dense subset U ={p ∈ M |the multiplicity of each principal curvature of M in S n+1 (c) is constanton some neighborhood Vp (⊂ U) of p} of M . Note that all principal curvaturefunctions are diﬀerentiable on U. In the following, we shall study on a ﬁxed neighborhood Vp . We remark that the shape operator A has constant rank on Vp .
Let γi = γi (s) (1 ≦ i ≦ mp ) be geodesics of M with γi (0) = p and γ̇(0) = vi ,where {v1 , . . . , vmp } is an orthonormal basis of the orthogonal complement of ker Ap
e and ∇ the Riemannian connections of S n+1 (c) and M ,in Tp M . We denote by ∇respectively. Then they satisfy(2.4)

e γ̇ ∇e γ̇ γ̇i = −k 2 γ̇i
∇
i
i
i

44

S. MAEDA AND H. TANABE

for some positive constant ki . Here, without loss of generality we can set k1 ≦ k2 ≦· · · ≦ kmp . It follows from Gauss formula (2.1) and Weingarten formula (2.2) that
(2.5)

e γ̇ ∇e γ̇ γ̇i = −⟨Aγ̇i , γ̇i ⟩Aγ̇i + ⟨(∇γ̇ A)γ̇i , γ̇i ⟩N .
∇i
ii

Comparing the tangential components of (2.4) and (2.5), at s = 0 we obtain⟨Avi , vi ⟩Avi = ki2 vi .
This, together with ki ̸= 0, implies
Avi = ki vi

or Avi = −ki vi(1 ≦ i ≦ mp ),which means that the tangent space Tp M is decomposed as:
Tp M = kerAp ⊕ {v ∈ Tp M |Av = −ki1 v} ⊕ {v ∈ Tp M |Av = ki1 v}⊕ · · · ⊕ {v ∈ Tp M |Av = −kig v} ⊕ {v ∈ Tp M |Av = kig v},
where 0 < ki1 < ki2 < · · · < kig and g is the number of positive distinct kj (j =1, . . . , mp ). Note that every kij is diﬀerentiable on Vp . We shall show the constancy
of kij . We ﬁrst note that vij kij = 0 (see the normal component of Equation (2.5)).
Let {vmp +1 , . . . , vn } be an orthonormal basis of ker A. Then {v1 , . . . , vn } forms an
orthonormal basis of Tp M . For any vℓ (1 ≦ ℓ ̸= ij ≦ n), since A is symmetric, we
see
(2.6)

⟨(∇vij A)vℓ , vij ⟩ = ⟨vℓ , (∇vij A)vij ⟩.

In order to compute Equation (2.6) easily, we extend an orthonormal basis{v1 , . . . , vn } to principal curvature unit vector ﬁelds on some neighborhood Wp (⊂Vp ), say {V1 , . . . , Vn }. Moreover we can choose ∇Vij Vij = 0 at the point p, where(Vij )p = vij . Such a principal curvature unit vector ﬁeld Vij can be obtained asfollows.
We ﬁrst deﬁne a smooth vector ﬁeld Wij on some suﬃciently small neighborhood
Wp (⊂ Vp ) by using parallel displacement for the vector vij along each geodesic withorigin p. We remark that in general Wij is not principal on Wp , but AWij = kij Wij
on the geodesic γ = γ(s) with γ(0) = p and γ̇(0) = vij . We here deﬁne the vector
ﬁeld Uij on Wp as Uij = Πα̸=kij (A−αI)Wij , where α runs over the set of all distinct
principal curvatures of M except for the principal curvature kij . We remark thatUij ̸= 0 on the neighborhood Wp because (Uij )p ̸= 0. Moreover, the vector ﬁeld
Uij satisﬁes AUij = kij Uij on Wp . We deﬁne Vij by normalizing Uij in some sense.
That is, when Πα̸=kij (kij − α)(p) > 0 (resp. Πα̸=kij (kij − α)(p) < 0), we deﬁne
Vij = Uij /∥Uij ∥ (resp. Vij = −Uij /∥Uij ∥). Then we know that AVij = kij Vij on
Wp and (Vij )p = vij . Furthermore, our construction shows that the integral curveof Vij through the point p is a geodesic on M , so that in particular ∇Vij Vij = 0 at
the point p.

CHARACTERIZATIONS OF ISOPARAMETRIC HYPERSURFACES

45

Thanks to the Codazzi equation, at the point p we have(the left hand side of (2.6)) = ⟨(∇vℓ A)vij , vij ⟩ = ⟨(∇Vℓ A)Vij , Vij ⟩p= ⟨∇Vℓ (kij Vij ) − A∇Vℓ Vij , Vij ⟩p= ⟨(Vℓ kij )Vij + (kij I − A)∇Vℓ Vij , Vij ⟩p = vℓ kij
and(the right hand side of (2.6)) = ⟨Vℓ , (∇Vij A)Vij ⟩p= ⟨Vℓ , ∇Vij (kij Vij ) − A∇Vij Vij ⟩p= ⟨vℓ , (vij kij )vij ⟩ = 0.
Thus we can see that the diﬀerential dkij of kij vanishes at the point p, whichshows that every kij (> 0) is constant on Wp , since we can take the point p as anarbitrarily ﬁxed point of Wp . So the principal curvature function kij is constantlocally on the open dense subset U of M . This, combined with the continuityof kij and the connectivity of M , yields that kij is constant on the hypersurface
M . Hence all nonzero principal curvatures of M are constant, so that we obtain
Condition (1).
□As an immediate consequence of Conditions (1) and (3) in Theorem 1 we havethe following:
Theorem 2 ([4]). Let M be a connected hypersurface in an (n + 1)-dimensional
sphere S n+1 (c) (n ≧ 2) of constant sectional curvature c. Then M is locally congruent to an isoparametric hypersurface with nozero principal curvatures in S n+1 (c) if
and only if for each point p of M there exists an orthonormal basis {v1 , . . . , vn } of
Tp M such that every geodesic of M through p with initial vector vi is a small circleof positive curvature in S n+1 (c).
Isoparametric hypersurfaces in S n+1 (c) with two distinct constant principal curvatures are called Cliﬀord hypersurfaces. For a pair (c1 , c2 ) of positive constants
satisfying 1/c1 + 1/c2 = 1/c and a positive integer r with 1 ≦ r ≦ n − 1, we denote
by Mr,n−r = Mr,n−r (c1 , c2 ) a naturally embedded hypersurface in S n+1 (c) whichis congruent to S r (c1√
) × S n−r (c2 ). It has two distinct
constant constant principal√
curvatures λ1 = c1 / c1 + c2 and λ2 = −c2 / c1 + c2 with multiplicities r andn − r, respectively. Let TMr,n−r = Vλ1 ⊕ Vλ2 be the decomposition into principal
distributions coresponding to principal curvatures λ1 , λ2 .
Proposition 1 ([1]). Let γ be a geodesic on a Cliﬀord hypersurface Mr,n−r (c1 , c2 )in S n+1 (c). Then
(1) The curve γ is a geodesic in S n+1 (c)√ if and only if the initial vector is of the√√form γ̇(0) = ( c2 w1 + c1 w2 )/ c1 + c2 with unit vectors wi ∈ Vλi (i =1, 2).
(2) If the initial vector γ̇(0) is neither principal nor of the form in (1), then γ
is a circle (namely, a great circle or a small circle of positive curvature) on
S n+1 (c).

46

S. MAEDA AND H. TANABE

Proof. Since Mr,n−r has parallel shape operator, we ﬁndd⟨Aγ̇(s), γ̇(s)⟩ = ⟨(∇γ̇ A)γ̇(s), γ̇(s)⟩ = 0.
ds
Thus we may study geodesics at its initial point. We set γ̇(0) = a1 w1 + a2 w2
with unit vectors wi ∈ Vλi (i = 1, 2) and nonnegative constants a1 , a2 satisfying
a21 + a22 = 1. In this case we have ⟨Aγ̇(0), γ̇(0)⟩= a21 λ1 + a22 λ2 . Hence
we can√ √√ √see that ⟨Aγ̇, γ̇⟩ ≡ 0 if and only if a1 = c2 / c1 + c2 and a2 = c1 / c1 + c2 .
Therefore, from Gauss formula (2.1), we get Statement (1).
Statement (2) is an immediate consequence of the proof of Theorem 1 and theabove Statement (1).
□Paying attention to Proposition 1, we characterize all Cliﬀord hypersurfaces
Mr,n−r (c1 , c2 ) in S n+1 (c).
Theorem 3 ([1]). A connected hypersurface M in S n+1 (c) is locally congruent toa Cliﬀord hypersurface Mr,n−r (c1 , c2 ) with some r (1 ≦ r ≦ n − 1) if and only ifthere exist a function d : M → {1, 2, . . . , n − 1}, a constant α (0 < α < 1) and an
orthonormal basis {v1 , . . . , vn } of Tp M at each point p ∈ M satisfying the following
two conditions:(1) Every geodesic on M through p with initial vector vi (1 ≦ i ≦ n) is a small
circle of positive curvature in S n+1 (c);√(2) Every geodesic γij on M through p with initial vector αvi + 1 − α2 vj (1 ≦i ≦ d(p) < j ≦ n) is a great circle in S n+1 (c).
In this case d is a constant function with d ≡ r and
M = Mn,n−r (c/α2 , c/(1 − α2 )).
Proof. The “only if” part follows from Theorem 2 and Proposition 1. So, weshall prove the “if” part. By Condition (1) we see that our real hypersurface M
is isoparametric with nonzero principal curvatures in S n+1 (c) (see Theorem 2 ).
Consider a ﬁxed point p0 . Setting Avi = λi vi (1 ≦ i ≦ n) at this point p0 , we cansee that(2.7)

α2 λi + (1 − α2 )λj = 0 for 1 ≦ i ≦ d(p0 ) < j ≦ n.

Therefore M has just two distinct constant principal curvatures, so that M is locallycongruent to √some Mr,n−r (c1 , c2 ). √Moreover, from the equalities 1/c1 + 1/c2 =
1/c, λi = c1 / c1 + c2 , λj = −c2 / c1 + c2 and (2.7) we can see that c1 = c/α2
and c2 = c/(1 − α2 ).
□√Remark 1. In Theorem 3, setting α = r/n , we obtain a characterization of all
minimal Cliﬀord hypersurfaces M = Mr,n−r (nc/r, nc/(n − r)) in S n+1 (c) (see [5]).
We ﬁnally study minimal isoparametric hypersurfaces with three distinct principal curvatures in S n+1 (c) from the viewpoint of Theorem 1. Isoparametric
hypersurfaces with three distinct principal curvatures are usually called Cartan
hypersurfaces. If we denote by mi the multiplicity of a principal curvature λi ,then we ﬁnd that these three principal curvatures have the same multiplicity (i.e.,

CHARACTERIZATIONS OF ISOPARAMETRIC HYPERSURFACES

47

m1 = m2 = m3 ) (see [6, 7]). When a Cartan hypersurface is minimal, it is congruent to one of the following hypersurfaces:
M 3 = SO(3)/(Z2 + Z2 ) → S 4 (c),M 6 = SU(3)/T 2 → S 7 (c),M 12 = Sp(3)/Sp(1) × Sp(1) × Sp(1) → S 13 (c),M 24 = F4 /Spin(8) → S 25 (c).
Principal curvatures of a Cartan minimal hypersurface are[2, 3]).

√√
3c , 0, − 3c (seeTheorem 4 ([1]). Let M be a connected hypersurface of S n+1 (c). Suppose that at
each point p ∈ M there exists an orthogonal basis {v1 , . . . , vmp } of the orthogonal
complement of ker Ap in Tp M (mp = rank Ap ) such that
(1) all geodesics through p with initial vector vi (1 ≦ i ≦ mp ) are small circles
of positive curvature in S n+1 (c),(2) they have the same curvature kp .
Then kp = k (constant) on M√and M is locally congruent either a totally umbilc
hypersurface S n (c1 ) with k = c1 − c , a Cliﬀord hypersurface Mr,n−r (2c,√ 2c) (1 ≦√r ≦ n − 1) with k = c , or a Cartan minimal hypersurface with k = 3c .
Proof. A totally umbilic hypersurface satisﬁes the hypothesis of Theorem 4 trivially.
By the discussion in the proof of Theorem 1 we see that a hypersurface satisfying the
hypothesis of Theorem 4 is isoparametric with at most three principal curvaturesk, −k, 0 in S n+1 (c). Thus we get the desired result.
□
Remark 2.
(1) In Theorems 3 and 4, we only need Condition (2) at some pointp0 ∈ M .
(2) If we add a condition that M is complete to assumptions of Theorems 1, 2,
3 and 4, then these theorems are global results. So, we can delete “locally”
in these statements.
(3) In the assumptions of Theorems 1, 2, 3 and 4, we do not need to take thevectors {vi } as a local smooth ﬁeld of orthonormal frames on M .
At the end of this paper, we pose the following open problem related to Theorem
1:Problem. Let M be a connected hypersurface in an (n + 1)-dimensional sphereS n+1 (c) (n ≧ 2) of constant sectional curvature c. If for each point p of M , there
exists an orthonormal basis {v1 , . . . , vn } of Tp M such that every geodesic of M
through p with initial vector vi is a circle (i.e., either a small circle of positive curvature or a great circle) in S n+1 (c), then is M locally congruent to an isoparametric
hypersurface in this ambient sphere?
References[1] T. Adachi and S. Maeda, Isoparametric hypersurfaces with Less than four principal curvaturesin a sphere, Colloquium Math. 105 (2006), 143–148.

48

S. MAEDA AND H. TANABE[2] É. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann.
Mat. Pura Appl 17 (1938), 177–191.
[3] É. Cartan, Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces
sphériques, Math. Z. 45 (1939), 335–367.
[4] M. Kimura and S. Maeda, Geometric meaning of isoparametric hypersurfaces in a real space
form, Canadian Math. Bull 43 (2000), 74–78.
[5] S. Maeda, A characterization of Cliﬀord minimal hypersurfaces in terms of their geodesics,
Trends in Diﬀerential Geometry, Complex Analysis and Mathematical Physics, Soﬁa, Bulgaria, 25-29 August 2008, World Sci.
[6] H. Münzner, Isoparametrische hyperﬂächen in sphären I, Math. Ann. 251 (1980), 57–71.
[7] H. Münzner, Isoparametrische hyperﬂächen in sphären II, Math. Ann. 256 (1981), 215–232.
[8] S.T. Yau, Open problems in geometry, Proc. Sympos. Pure Math. 54 (1993), Part I, 1–28.
Sadahiro Maeda: Department of Mathematics, Saga University, Saga 840-8502,
Japan
E-mail address: smaeda@ms.saga-u.ac.jp
Hiromasa Tanabe: Department of Science, Matsue College of Technology,
Matsue, Shimane 690-8518, Japan
E-mail address: h-tanabe@matsue-ct.jp