Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematics
56 (2023), pp. 19–34

GEOMETRIC PROPERTIES OF GEODESIC SPHERES IN ACOMPLEX PROJECTIVE SPACESADAHIRO MAEDA
Communicated by Jitsuro Sugie(Received: January 9, 2022)
Abstract. We survey geometric properties of geodesic spheres in a complex
projective space. These spheres can be regarded as the simplest examples in theclass of all real hypersurfaces isometrically immersed into this projective space.
Moreover, geodesic spheres with suﬃciently big radii in this space are well-known
examples of Berger spheres.

1. Two viewpoints on geodesic spheres
We ﬁrst investigate geodesic spheres in a complex projective space from the viewpoint of submanifold theory. We denote by CP n (c) a complex n(≧ 2)-dimensional
complex projective space of constant holomorphic sectional curvature c(> 0), andby M a connected real hypersurface isometrically immersed into CP n (c). Then M
has an almost contact metric structure (ϕ, ξ, η, g) induced from the Kähler structure(J, g) of CP n (c).
Standard examples of connected real hypersurfaces in CP n (c) are homogeneous
real hypersurfaces, i.e., real hypersurfaces given as orbits under the subgroups ofthe full isometry group I(CP n (c)) of the ambient space CP n (c), which is the unitary group U(n + 1). R. Takagi ([28]) classiﬁes homogeneous real hypersurfaces ofCP n (c) and he shows that a homogeneous real hypersurface is locally congruent
to one of the six model spaces of types (A1 ), (A2 ), (B), (C), (D) and (E). Furthermore, we ﬁnd that the number of distinct principal curvatures of homogeneous
real hypersurfaces is 2, 3, 3, 5, 5, 5, respectively (see [29]). Every homogeneous
real hypersurface
geodesic sphere G(r) of radius√ of type (A1 ) is congruent to a √r (0 < r <√π/ c ). Note that G(r) (0 < r < π/ c ) is congruent to a tube ofradius (π/ c − r) over a totally geodesic complex hypersurface CP n−1 (c).
It is known that there exist no totally umbilic real hypersurfaces in CP n (c)and T. Cecil and P. Ryan show that every geodesic sphere is the only example of2010 Mathematics Subject Classiﬁcation. Primary 53B25, Secondary 53C40.
Key words and phrases. Geodesic spheres, complex projective spaces, length spectrum, Berger
spheres, positive sectional curvatures, extrinsic geodesics, ϕ-invariant shape operators, contact
forms.
19

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S. MAEDA

real hypersrfaces in CP n (c) (n ≧ 3) with at most two distinct principal curvaturesat each point (cf. [7]). This implies that a geodesic sphere is the simplest real
hypersurface of CP n (c).
We next study those geodesic spheres from the viewpoint of length spectrum.
Klingenberg ([11]) proved the following: Let M be an even dimensional compact
simply connected Riemannian manifold having the sectional curvature K with 0 <K ≦ L on M , where L√is a constant. Then the length ℓ of every closed geodesic
on M satisﬁes ℓ ≧ 2π/ L .
In this context Berger gave examples of metrics on S 3 for which this inequality
does not hold. This 3-sphere is called a Berger sphere with a Riemannian metricfrom a one-parameter family, which can be obtained from the standard metric byshrinking along ﬁbers of a Hopf ﬁbration. Motivated by this fact, Chavel constructed similar metrics on higher odd-dimensional spheres.
Weinstein ([32]) gave a √description of Berger
and Chavel examples as geodesic√2
spheres G(r) (0 < r < π/ c ) with tan ( c r/2) >√2 in CP n (c), n ≧ 2. Then
it is known that our geodesic sphere G(r) with tan2 ( c r/2)√> 2 admits a closed
geodesic, say γ on G(r) whose length is shorter than 2π/ L , where L is the
maximal sectional curvature of G(r). In the following, we explain this fact indetails. We see that√the sectional curvature K of every geodesic√sphere G(r) of
radius r (0 < r√
< π/ c ) satisﬁes sharp inequalities 0 < (c/4) cot2 ( c r/2) ≦ K ≦2
c + (c/4) cot ( c r/2)(=L) at its each point (cf. [19]). The shape operator A√nof G(r) (0 < r < π/ √c ) (with√a unit normal vector√ ﬁeld N on
√ G(r)) in CP (c)is written as: Aξ = c cot( c r)ξ and AX = ( c /2) cot( c r/2)X for eachX(⊥ ξ), where ξ is the characteristic vector ﬁeld on G(r) deﬁned by ξ = −JN .
The closed geodesic γ = γ(s) is nothing but an integral curve γξ of the characteristic
vector ﬁeld ξ. Since the√curve γ√ξ lies on a holomorphic line CP 1 (c)(= S 2 (c)) as acircle of curvature k = c cot( c r), it is closed with length
√
2π
2π
2π
√ sin( c r).
ℓ= √
=p
=
√
c
k2 + cc cot2 ( c r) + c
Moreover, we see easily that ∇ξ ξ = ϕAξ = 0, so that the curve γξ = γξ (s)√is a
geodesic on G(r), where ∇ is the Riemannian connection on G(r) (0 < r < π/ c )e in the ambient space CP n (c). We set
induced from the Riemannian connection ∇
√
2π
2π
r
 √  > √c sin( c r).
c + 4c cot2 2c r√
Then, solving the above inequality, we have tan2 ( √c r/2) > 2 and √vice versa.
2
In this paper, geodesic spheres G(r) (0 < r < π/ c ) with tan ( c r/2) > 2 in
CP n (c), n ≧ 2 are called Berger spheres (cf. [32]).
2. Comments on type A hypersurfaces
First of all we review fundamental terminologies on the theory of real hypersurfaces (see [21]).

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21

Let M be a connected real hypersurface immersed into CP n (c), n ≧ 2 throughan isometric immersion with a unit normal local vector ﬁeld N . The Riemanniane of CP n (c) and ∇ of M are related by the following formulas ofconnections ∇Gauss and Weingarten:(2.1)

e X Y = ∇X Y + g(AX, Y )N ,
∇

(2.2)

e X N = −AX∇

for arbitrary vector ﬁelds X and Y on M , where g is the Riemannian metric of
M induced from the standard metric of the ambient space CP n (c) and A is theshape operator of M in CP n (c). An eigenvector of the shape operator A is calleda principal curvature vector of M in CP n (c) and an eigenvalue of A is called aprincipal curvature of M in CP n (c). We set Vλ = {v ∈ TM | Av = λv} which iscalled the principal distribution associated to the principal curvature λ.
It is well-known that M has an almost contact metric structure induced from theKähler structure (J, g) of the ambient space CP n (c). That is, we have a quadruple(ϕ, ξ, η, g) deﬁned byg(ϕX, Y ) = g(JX, Y ),

ξ = −JN

and η(X) = g(ξ, X) = g(JX, N ).

Then they satisfy
ϕ2 X = −X + η(X)ξ, η(ξ) = 1 and g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y )for all vectors X, Y ∈ TM . It is known that these equations imply that ϕξ = 0
and η(ϕX) = 0. In the following, we call ϕ, ξ and η the structure tensor, thecharacteristic vector and the contact form on M , respectively.
e = 0 and JX = ϕX + η(X)N that
It follows from (2.1), (2.2), ∇J(2.3)(∇X ϕ)Y = η(Y )AX − g(AX, Y )ξ,(2.4)∇X ξ = ϕAX.

Indeed, for the second equality, we gete X (JN ) + g(AX, JN )N∇X ξ = −∇X (JN ) = −∇e X N + g(AX, JN )N = JAX − g(JAX, N )N = ϕAX.
= −J ∇For the ﬁrst, we see(∇X ϕ)Y = ∇X (ϕY ) − ϕ∇X Y = ∇X (JY − η(Y )N ) − ϕ∇X Y
e X (JY − η(Y )N ) − g(AϕY, X)N − ϕ∇X Y=∇= J(∇X Y + g(AX, Y )N ) − X(η(Y ))ξ + η(Y )AX− g(AϕY, X)N − ϕ∇X Y= ϕ∇X Y + g(∇X Y, ξ)N − g(AX, Y )ξ − g(∇X Y, ξ)N− g(Y, ϕAX)N + η(Y )AX − g(AϕY, X)N − ϕ∇X Y= η(Y )AX − g(AX, Y )ξ.

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S. MAEDA

Denoting the curvature tensor of M by R, we have the equation of Gauss given by(2.5)

g((R(X, Y )Z, W ) = (c/4){g(Y, Z)g(X, W ) − g(X, Z)g(Y, W )+ g(ϕY, Z)g(ϕX, W ) − g(ϕX, Z)g(ϕY, W ) − 2g(ϕX, Y )g(ϕZ, W )}+ g(AY, Z)g(AX, W ) − g(AX, Z)g(AY, W ).

The following is called the equation of Codazzi:(∇X A)Y − (∇Y A)X = (c/4)(η(X)ϕY − η(Y )ϕX − 2g(ϕX, Y )ξ).
Let K be the sectional curvature of M . That is, K is deﬁned by K(X, Y ) =g(R(X, Y )Y, X), where X and Y are orthonormal vectors on M . Then it follows
from (2.5) that
(2.6)
K(X, Y ) = (c/4) 1 + 3g(ϕX, Y )2 + g(AX, X)g(AY, Y ) − g(AX, Y )2 .
We usually call M a Hopf hypersurface if the characteristic vector ξ of M is aprincipal curvature vector at each point of M . Type (A) hypersurfaces are thesimplest examples of Hopf hypersurfaces.
In the following, we study type (A) hypersurfaces. Here, type (A) hypersurfaces
mean either homogeneous real hypersurfaces√of type (A1 ) or type (A2 ). They are(A1 ): a geodesic sphere G(r) (0 < r < π/ √c ) in CP n (c), n ≧ 2,
(A2 ): a tube T ℓ (r) of radius r (0 < r < π/ c ) around a totally geodesic complex
submanifold CP ℓ (c) (1 ≦ ℓ ≦ n − 2) in CP n (c), n ≧ 3.
The shape operator A of type (A1 ) hypersurface is expressed in Section 1. The
tangent bundle TM of√type (A√2 ) hypersurface is decomposed as: TM = {ξ}R ⊕Vλ1 ⊕ Vλ2 , √where Aξ =√ c cot( c r)ξ, dim√Vλ1 = 2n −√2ℓ − 2(≧ 2), dim Vλ2 = 2ℓ(≧2), λ1 = ( c /2) cot( c r/2) and λ2 = −( c /2) tan( c r/2).
Type (A) hypersurfaces have many common geometric properties. For examples,
we know
Theorem 1 ([15, 20, 25]). For a connected real hypersurface M in CP n (c), n ≧ 2,
the following local statements hold:(1) The length of the derivative of the shape operator A of each real hypersurface
M in CP n (c), n ≧ 2 satisﬁes ∥∇A∥2 ≧ (c2 /4)(n − 1). In particular, ∥∇A∥2 =(c2 /4)(n − 1) holds on M if and only if M is of type (A).
(2) A real hypersurface M in CP n (c), n ≧ 2 is of type (A) if and only if theequality ϕA = Aϕ holds on M , where ϕ and A are the structure tensor and theshape operator of M , respectively.
(3) A real hypersurface M in CP n (c), n ≧ 2 is of type (A) if and only if thecharacteristic vector ﬁeld ξ is Killing, that is, Lξ g = 0 holds on M , where L is theLie derivative on M .
(4) A real hypersurface M in CP n (c), n ≧ 2 is of type (A) if and only if every
geodesic γ = γ(s) on M is mapped to a curve having constant the ﬁrst curvaturee γ̇ γ̇∥ along γ, where ∇e is the Riemannian connection on CP n (c). In thisκ1 = ∥∇
case, κ1 depends on the choice of γ.
In some sense Theorem 1 shows that it is not so easy to distinguish between
hypersurfaces of type (A1 ) and type (A2 ). Inspired by this result, we pay particular

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23

attention to geometric properties which discriminate hypersurfaces of type (A1 )from those of type (A2 ). It follows from (2.6) that
Proposition 1 ([19]). For a homogeneous real hypersurface M in CP n (c) we havethe following:(1) M is of type (A1 ) if and only if M has positive sectional curvature K at its
each point. In this case, M has sharp inequalities√√
0 < (c/4) cot2 ( c r/2) ≦ K ≦ c + (c/4) cot2 ( c r/2).
(2) M is of type (A2 ) if and only if the minimal sectional curvature of M is null.
Here, M has sharp inequalities√√
0 ≦ K ≦ c + (c/4)max{cot2 ( c r/2), tan2 ( c r/2)}.
(3) There exist no homogeneous real hypersurfaces all of whose sectional curvatures
are nonpositive.
We next observe the extrinsic shape of geodesics on type (A) hypersurfaces in
CP n (c), n ≧ 2. In this paper for a submanifold M n in a Riemannian manifold
fn+p through an isometric immersion, a curve γ on M n is called an extrinsicMgeodesic if the curve γ, considered as a curve in the ambient space, is a geodesic in
fn+p . Needless to say, such a curve γ is also a geodesic on the submanifold M n .
M
We compute the number of extrinsic geodesics on type (A) hypersurfaces in acomplex projective space.
Proposition 2 ([26]). For a type (A) hypersurface
we ﬁnd the following:√(1) A geodesic sphere G(r) (0 < r√< π/(2 c )) has
noextrinsic geodesics;√(2) A geodesic sphere G(r) (π/(2 c ) ≦ r < π/ c ) has just one congruent classof extrinsic geodesics up to isometries on this sphere;(3) Every type (A2 ) hypersurface M has uncountably inﬁnite congruent classes of
extrinsic geodesics up to isometries on this sphere.
We here recall the congruence theorem for geodesics γ on a type (A) hypersurface
M . For a geodesic γ = γ(s) on M we denote by ργ (s) := g(γ̇(s), ξγ(s) ) the structure
torsion of the curve γ. Note that the structure torsion ργ is constant along γ.
Indeed, from (2.4), Theorem 1(2), the symmetry of A and the skew-symmetry of
ϕ we ﬁnd(2.7)γ̇ργ = γ̇ g(γ̇(s), ξγ(s) ) = g(γ̇, ∇γ̇ ξ) = g(γ̇, ϕAγ̇)= g(γ̇, Aϕγ̇) = −g(ϕAγ̇, γ̇) = 0,
so that ργ is a constant with −1 ≦ ργ ≦ 1. Next, for a geodesic γ = γ(s) on a type(A) hypersurface M we denote by κγ (s) := g(Aγ̇(s), γ̇(s)) the normal curvature of
the curve γ. The normal curvature κγ is also constant. In fact, from Theorem 1(4)we see γ̇κγ = γ̇(g(Aγ̇, γ̇)) = g((∇γ̇ A)γ̇, γ̇) = 0. By using these two invariants ργ
and κγ , the congruence theorem for geodesics can be described as follows:√
Lemma 1 ([26]). (1) Two geodesics γ1 , γ2 on G(r) (0 < r < π/ c ) are congruent
by isometries of this sphere if and only if |ργ1 | = |ργ2 |.

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S. MAEDA(2) Two geodesics γ1 , γ2 on a type (A2 ) hypersurface M are congruent by isometriesof M if and only if |ργ1 | = |ργ2 | and κγ1 = κγ2 .
Using the notion of the structure torsion ργ (s) := g(γ̇(s), ξγ(s) ) for a geodesicγ = γ(s) on a real hypersurface M in CP n (c), n ≧ 2, we give a geometric meaningof the equality ϕA = Aϕ in Theorem 1(2).
Proposition 3. For a connected real hypersurface M isometrically immersed into
CP n (c), n ≧ 2 the following two conditions are mutually equivalent:(1) The equality ϕA = Aϕ holds on M , where ϕ and A are the structure tensor
and the shape operator of M , respectively;(2) The structure torsion ργ of every geodesic γ = γ(s) on M is constant along
the curve γ.
Proof. (1) =⇒ (2): See (2.7).
(2) =⇒ (1): We see that0 = γ̇ργ = γ̇(g(γ̇, ξ)) = g(γ̇, ϕAγ̇) = (1/2)g((ϕA − Aϕ)γ̇, γ̇)holds for each geodesic γ on M , which implies that g((ϕA − Aϕ)X, X) = 0 for
every vector X of M . This, together with a fact that ϕA − Aϕ is symmetric, yields
that g((ϕA − Aϕ)X, Y ) = 0 for every X, Y ∈ TM . Hence we get Condition (1). □At the end of this section we consider the exterior derivative dη of the contact
form η on a real hypersurface M in CP n (c), n ≧ 2, which is given by(2.8) dη(X, Y ) := (1/2){X(η(Y )) − Y (η(X)) − η([X, Y ])} for all X, Y ∈ TM.
By (2.4) and (2.8) we have1dη(X, Y ) = g((ϕA + Aϕ)X, Y ).
2
This implies that dη = 0, i.e., the contact form η is closed if and only if the equality
ϕA + Aϕ = 0 holds on M . However there does not exist such a real hypersurface
M in CP n (c), n ≧ 2 (see [21]).
So it is natural to recall the following:
Lemma 2 ([21]). Suppose that a connected real hypersurface M in CP n (c), n ≧ 2satisﬁes the derivative of the contact form η on M : dη(X, Y ) = kg(ϕX, Y ) for
all X, Y ∈ TM with nonzero constant√ k. Then M is locally congruent to either
a geodesic sphere G(r) (0 < r < π/ c ) or a homogeneous real √hypersurface oftype (B), that is, it is realized as a tube of radius r (0 < r < π/(2 c )) around a
totally real totally geodesic n-dimensional real projective space RP n (c/4) of constantsectional curvature c/4.
3. Characterization of geodesic spheres
We ﬁrst recall the notion of circles in Riemannian geometry (cf. [24]). A smoothreal curve γ = γ(s) parametrized by its arclength s in a Riemannian manifold
M endowed with Riemannian connection ∇ is called a circle of curvature k if itsatisﬁes ∇γ̇ γ̇ = kY, ∇γ̇ Y = −k γ̇ with some nonnegative constant k and a unit

GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE

25

vector ﬁeld Y along γ. We may regard a geodesic as a circle of null curvature. It
is well-known that a smooth real curve γ = γ(s) parametrized by its arclength is a
circle if and only if it satisﬁes ∇γ̇ ∇γ̇ γ̇ = −k 2 γ̇ with some nonnegative constant k
and that the circle γ = γ(s) can be deﬁned for −∞ < s < ∞ when M is complete.
Next, standing on the ambient space CP n (c), n ≧ 2, we observe√ the extrinsic
shape of some geodesics on geodesic spheres G(r), (0 < r < π/ c ). For every
point p of G(r) and each unit vector v(∈ Tp G(r)) orthogonal to the characteristic
vector ξp we take the geodesic γ = γ(s) on G(r) with initial condition that γ(0) = p
and γ̇(0) = v. Then it follows from the constancy of the structure torsion ργ :=g(γ̇, ξ) for the curve γ that γ̇(s) is orthogonal to ξγ(s) for∈ (−∞, ∞),√ each s √so that the shape operator A of G(r) satisﬁes Aγ̇(s) = ( c /2) cot( c r/2)γ̇(s).
e γ̇ γ̇ = (√c /2) cot(√c r/2)N and
This, together with (2.1) and (2.2), yields that ∇e γ̇ N = −(√c /2) cot(√c r/2)γ̇. Hence we get the following:∇√
Proposition 4 ([18]). Every geodesic γ = γ(s) on G(r) (0 < r < π/ c ) satisfying that the initial vector γ̇(0) is perpendicular to the characteristic vector ξγ(0) is
mapped to a circle of the same positive curvature which is independent of the choiceof γ.
Motivated by Proposition 4, we consider the following two conditions on the
extrinsic shape of geodesics on real hypersurfaces M :(ES1) At each point p of M , there exist orthonormal vectors v1 , v2 , . . . , v2n−2 ∈Tp M orthogonal to ξp satisfying the following:(i) geodesics γi = γi (s) (1 ≦ i ≦ 2n − 2) on M with γi (0) = p and γ̇i (0) = vi
are mapped to circles of positive curvature in CP n (c),(ii) geodesics γij = γ√ij (s) (1 ≦ i < j ≦ 2n − 2) on M with γij (0) = p andγ̇ij (0) = (vi +vj )/ 2 are mapped to circles of positive curvature in CP n (c).
(ES2) At each point p of M , there exist orthonormal vectors v1 , v2 , . . . , v2n−2 ∈Tp M orthogonal to ξp satisfying that geodesics γi = γi (s) (1 ≦ i ≦ 2n − 2) on M
with γi (0) = p and γ̇i (0) = vi are mapped to circles of the same positive curvature
in CP n (c).
Weakening Condition (ES2), we also consider the following condition:(ES2’) At each point p of M , there exist orthonormal vectors v1 , v2 , . . . , v2n−2 ∈Tp M orthogonal to ξp satisfying that geodesics γi = γi (s) (1 ≦ i ≦ 2n − 2) on M
with γi (0) = p and γ̇i (0) = vi are mapped to circles of positive curvature in CP n (c).
The diﬀerence between Conditions (ES2) and (ES2’) is that the curvatures of circles
are the same or not. Note that Condition (ES2’) is related to a characterization of
all homogeneous real hypersurfaces of types (A1 ), (A2 ), (B), (C), (D) and (E) in
CP n (c).
Theorem 2 ([2, 6, 10]). For a connected real hypersurface M in CP n (c), n ≧ 2,
M is locally congruent to a homogeneous real hypersurface in this ambient space if
and only if M satisﬁes Condition (ES2′ ).
Combining Conditions (ES1), (ES2), (ES2’) with Propositions 1, 2 and Lemma
2, we establish the following:

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S. MAEDA

Theorem 3 ([16, 18, 26]). For a connected real hypersurface M in CP n (c), n ≧ 2,
the following four conditions are mutually equivalent:√(1) M is locally congruent to a geodesic sphere G(r) (0 < r < π/ c );(2) M satisﬁes Condition (ES2) and has at most one congruent class of extrinsic geodesics with respect to the full isometry group I(M ) of M ;(3) M satisﬁes Condition (ES2′ ) and has positive sectional curvature K on M ;(4) M has nonnegative sectional curvature K on M and the exterior derivativedη of the contact form η on M satisﬁes dη(X, Y ) = kg(ϕX, Y ) for all
X, Y ∈ TM with some nonzero constant k.
As immediate consequences of Theorem 3 we get the following which are characterizations of Berger spheres:
Proposition 5 ([12]). Let M be a connected real hypersurface of CP n (c), n ≧ 2through an isometric immersion. Then M√is locally congruent
to a Berger sphere,√2
namely a geodesic sphere G(r) (0 < r < π/ c ) with tan ( c r/2) > 2, with respect
to the full isometry group U(n + 1) of the ambient space CP n (c) if and only if at
each point p of M there exists an orthonormal basis v1 , v2 , . . . , v2n−2 , ξp of Tp M
such that all geodesics γi = γi (s) (1 ≦ i ≦ 2n − 2) with initial condition thatγi (0) = p and γ̇i (0)√= vi are mapped to circles of the same positive curvature k(p)√with k(p) < c /(2 2 ) in the ambient space CP n (c), where ξp is the characteristic
vector of M at p ∈ √M . In this√case, the function k = k(p) on M is automaticallyconstant with k = ( c /2) cot( c r/2).
Proposition 6 ([12]). Let M be a connected real hypersurface of CP n (c), n ≧ 2through an isometric immersion. Then M is locally congruent to a Berger sphere
if and only if M satisﬁes the following two conditions.
√√(1) There exists a positive constant k with k < c /(2 2 ) such that the exterior derivative dη of the contact form on M satisﬁes either dη(X, Y ) =kg(ϕX, Y ) for all X, Y ∈ TM or dη(X, Y ) = −kg(ϕX, Y ) for all X, Y ∈TM , where g and ϕ are the Riemannian metric and the structure tensor on
M , respectively.
(2) There exists a point x of M satisfying that every sectional curvature of M
at x is positive.
In the statement of Proposition 6, if we remove Condition (2), this proposition
does not hold. The Berger sphere and a certain homogeneous real hypersurface oftype (B) satisfy Proposition 6(1). The following lemma is worth mentioning.
√
Lemma 3. Let G(r) be a geodesic sphere of radius r (0 < r < π/ c ) in
CP n (c), n ≧ 2 and L be the maximal sectional curvature of this sphere. Thenthe following three conditions are mutually equivalent:√(1) The radius r satisﬁes an inequality tan2 ( c r/2) > 2;
(2) The sectional curvature K of G(r) satisﬁes sharp inequalities δL ≦ K ≦ L
for some δ ∈ (0, 1/9) at its each point;(3) The length of every √integral curve of the characteristic vector ﬁeld ξ on G(r)is shorter than 2π/ L .

GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE

27√
Needless to say, for each geodesic sphere G(r) (0 < r < π/ c ) in CP n (c), n ≧ 2every integral curve of the characteristic vector ﬁeld ξ on G(r) is a geodesic.
We here recall the work of Li, Vrancken and and Wang ([14]), which gives acharacterization of three dimensional Berger spheres as Lagrangian submanifoldsof CP 3 . They show the following (for details, see Theorem 1.2 in [14]): Let ι bea Lagrangian isometric immersion from an open part of one of the homogeneous
3-manifolds into a complex space form M3 (c)(= CP 3 (c), CH 3 (c) or C3 ). Thenc > 0 and ι is minimal and M 3 is locally congruent to the Berger sphere.
4. Length spectrum of geodesic spheres
Using the structure torsion ργ for a geodesic γ on G(r) in CP n (c), n ≧ 2, we seethat the curve γ is closed or not. Moreover, when the curve γ is closed, we ﬁnd its
length which is denoted by length(γ).
Theorem√ 4 ([4]).nLet γ be a geodesic on a geodesic sphere G(r) of radius r (0 <r < π/ c ) in CP (c), n ≧ 2.
(1) If the√ structure√ torsion of γ is ±1, then γ is closed and its length is(2π/ c ) sin( c r).
(2) If γ √has null√structure torsion, then γ is also closed and its length is(4π/ c ) sin( c r/2).
(3) When the structure torsion of γ is of the form sin θ (0 < |θ| < π/2), it isclosed if and only ifsin θ =±qp√√sin( c r/2) p2 tan2 ( c r/2) + q 2√
with some relatively prime positive integers p and q with q < p tan2 ( c r/2).
In this case, its length is

√ p
√
√
(4π/ c ) p2 sin2 ( c r/2) + q 2 cos2 ( c r/2)

if pq is even,length(γ) =√√ p 2 2 √
(2π/ c ) p sin ( c r/2) + q 2 cos2 ( c r/2)

if pq is odd.
When we study the length spectrum of geodesics on a Riemannian manifold
N , in order to avoid the inﬂuence of the action of the isometry group of N , weconsider the moduli space of geodesics under the action of isometries. The moduli
space Geod(N ) of geodesics on N is the quotient space of the set of all geodesics
on N under the congruency relation. We call a smooth curve σ open if it is not
closed. For convenience we set length(σ) = ∞ for an open curve σ. We deﬁne thelength spectrum LN : Geod(N ) → R ∪ {∞} of N by LN ([γ]) = length(γ), where[γ] denotes the congruent class containing a geodesic γ. We also call the image
Lspec(N ) = LN (Geod(N )) ∩ R the length spectrum of N . For example, the length
spectrum of a standard unit sphere is Lspec(S m ) = {2π}.

28

S. MAEDA

As a direct consequence of Theorem 4, for a geodesic sphere G(r) of radius r in
CP n (4), we can see that
Lspec(G(r)) = {π sin 2r} ∪ {2π sin r}( q)
p and q are relatively prime2∪ 2π p2 sin r + q 2 cos2 r positive integers which satisfy
pq is even and q < p tan2 r)( q
p and q are relatively prime∪ π p2 sin2 r + q 2 cos2 r positive integers which satisfy .
pq is odd and q < p tan2 r
Therefore we obtain the following.
√
Theorem 5 ([4]). On a geodesic sphere G(r) (0 < r < π/ c ) in CP n (c), there
exist countably inﬁnite congruent classes of closed geodesics. Moreover the length
spectrum Lspec(G(r)) of G(r) is a discrete unbounded subset in the real line R.
For a length spectrum λ ∈ Lspec(N ) we call the cardinality mN (λ) of the set
L−1
N (λ) the multiplicity of λ. When the multiplicity of a length spectrum is 1 wesay it is simple. Clearly for a geodesic sphere G(r) in a complex projective space,
we see by the expression of Lspec(G(r)) that mG(r) (λ) < ∞ at each λ. We here
study the ﬁrst, the second and the third length spectrum, that is, the minimum,
the second minimum and the third minimum of the length spectrum.
√
Proposition 7 ([4]). Let G(r) be a geodesic sphere of radius r (0 < r < π/ c ) in
CP n (c).
√√(1) The ﬁrst length spectrum of G(r) is (2π/ c ) sin( c r), which is the lengthof geodesics with structure torsion ±1. It is simple.
√(2) The second
lengthspectrum
of
G(r)
is
also
simple.
When
0
<
r
≦
π/(2
c ),
√
√
it is (4π/ c ) sin( c r/2),
with null struc√ which is the√length of geodesics√
ture torsion. When π/(2 c ) < r < π/ c√, it is 2π/ c , which is the lengthof geodesics with structure torsion ± cot( c r/2).
√√(3) The third
lengthspectrumisalso
simple.
When
π/(2
c
)
<
r
<
π/
c , it√√is (4π/ c ) sin( √c r/2), which is √the length of√geodesics with null structure
torsion. When 2m − 1 ≦ cot( c r/2) <p 2m + 1 (m = 1, 2, . . .), in√√√
particular, 0 < r ≦ π/(2 c ), it is (2π/ c ) 4m(m + 1) sin2 ( c r/2) + 1,
which is the length
structure torsion
p of geodesics with

√
√
2
2
±1/ sin( c r/2) (2m + 1) tan ( c r/2) + 1 .
Every length spectrum except the ﬁrst one we ﬁnd the following lemma of Klingenberg’s type holds, which is well-known for some geometers.
√
Corollary 1 ([4]). Let G(r) be a geodesic sphere of radius r (0 < r < π/ c )in CP n (c). Except geodesics
torsion ±1, every geodesic γ on G(r)q with structure√2satisﬁes length(γ) > 4π/ c 4 + cot ( c r/2) .

GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE

29

Length spectrum is of course not necessarily simple. For example when M is a
geodesic sphere of radius π/4 in CP n (4), we have
√
√
√
√
√
√
√
Lspec(M ) = {π, 2 π, 5 π, 10 π, 13 π, 17 π, 5π, 26 π, 29 π
√
√
√
√
√
√
√
√
√
34 π, 37 π, 41 π, 50 π, 53 π, 58 π, 61 π, 65 π, 73 π, . . .}√is two; it is the common length of geodesics
and the multiplicity√of 65 π √√ ofstructure torsions 3/ 65 and 7/ 65 . Every spectrum which is smaller than 65 π
is simple.
√
Theorem 6 ([4]). Let G(r) be a geodesic sphere of radius r (0 < r < π/ c ) in
CP n (c).
√(1) If tan2 ( √c r/2) is irrational, then every length spectrum of G(r) is simple.
(2) If tan2 ( c r/2) is rational, then the multiplicity of each length spectrumof G(r) is ﬁnite. But it is not uniformly bounded; lim supλ→∞ mG(r) (λ) =∞. In this case, the growth order of mG(r) is not so rapid. It satisﬁes
limλ→∞ λ−δ mG(r) (λ) = 0 for arbitrary positive δ.
This√ theorem guarantees that on a geodesic sphere of radius r with irrationaltan2 ( c r/2) in a complex projective space, two closed geodesics are √congruent2
if and only if they have the same length. On the other hand, if tan ( c r/2) isrational, this theorem shows that we can not classify congruent classes of geodesics
only by their length.
Finally we make mention of the growth of the number of congruent classes ofgeodesics with respect to their length spectrum for a geodesic sphere in a complex
projective space. For a Riemannian manifold N we denote by nN (λ) the cardinalityof the set {[γ] ∈ Geod(N ) | LN ([γ]) ≦ λ}.
√Theorem 7 ([4]). For a geodesic sphere G(r) of radius r (0 < r < π/ c ) in
CP n (c) we have√nG(r) (λ)
3c c r
√
lim
= 4
.
λ→∞
λ2
8π sin( c r)
The following problem is still open.
Problem 1. Find a geometric meaning of the equalitynG(r) λ3c=.
r→0 λ→∞
λ2
8π 4
lim lim

Remark. On the other hand, it is easy to ﬁnd a geometric meaning of the equalitynG(r) λ=∞λ2r→π/ c λ→∞lim√ lim

because of a fact thatlim√ G(r) = CPr→π/ c

n−1

(c).

30

S. MAEDA

5. Appendixes
Let M denote a connected real hypersurface isometrically immersed into CP n (c),n ≧ 2. Then M has two almost contact metric structures (ϕ, ξ, η, g) associatedto a local unit normal vector N on M and (ϕ, −ξ, −η, g) associated to −N (seeSection 2). So it is natural that we call a real hypersurface M Sasakian if M
satisﬁes either (∇X ϕ)Y = g(X, Y )ξ − η(Y )X for all X, Y ∈ TM or (∇X ϕ)Y =−g(X, Y )ξ + η(Y )X for all X, Y ∈ TM (see Theorem 6.3 in [5]). A Sasakian
manifold is called a Sasakian space form of constant ϕ-sectional curvature c if thesectional curvature K(u, ϕu) := g(R(u, ϕu)ϕu, u) satisﬁes K(u, ϕu) = c for each
unit vector u orthogonal to ξ.
In pp. 114-115 of [5], we ﬁnd a standard construction of Sasakian space forms.
But, in general all Sasakian space forms M (c) (c ̸= 1) can be realized as totally
η-umbilic real hypersurfaces in nonﬂat complex space forms (= CP n (c), CH n (c))(see [4]). The following is the unique existence theorem of Sasakian space forms.
Lemma 4 ([31]). For any two simply connected complete Sasakian manifolds ofconstant ϕ-sectional curvature c, there exists an isomorphism between them which
preserves their almost contact metric structures.
We obtain easily the following:
Proposition 8 ([1]). For a connected real hypersurface M isometrically immersed
into CP n (c), n ≧ 2, the following are mutually equivalent:(1) M is a Sasakian manifold;(2) M is a Sasakian space form of constant ϕ-sectional curvature d, where d is
automatically expressed as d = c + 1;
(3) The shape operator A of M is written either AX = −X + (c/4)η(X)ξ for all
X, Y ∈ TM or AX = X − (c/4)η(X)ξ for all X, Y ∈ TM ;
√
(4)
M
is
locally
congruentageodesic
sphere
G(r)
(0
<
r
<
π/
c ) with√√n( c /2) cot( c r/2) = 1 in CP (c).
By virtue of the works ([3, 4]) and Lemma 4 we ﬁnd
Theorem 8. Let M (c) be a complete simply connected Sasakian space form of
constant ϕ-sectional curvature c(> 1). Suppose that c is irrational. Then for any
closed geodesics γ1 = γ1 (s) and γ2 = γ2 (s) on M (c) they are congruent by some
isometry on this space if and only if they have the same common length.
We next recall a fact that an isometric immersion f of a Kähler manifold withKähler structure J into a Euclidean sphere has parallel second fundamental form
σ if and only if σ is J-invariant , i.e., σ(JX, JY ) = σ(X, Y ) holds for each vector
X, Y on M (cf. [9, 13, 22]). Motivated by this fact, for a real hypersurface M
isometrically immersed into CP n (c), n ≧ 2 we consider the following condition
concerning ϕ-invariance of the shape operator A of M .
The shape operator A of M is called ϕ-invariant if A satisﬁesg(AϕX, ϕY ) = g(AX, Y ), i.e., σ(ϕX, ϕY ) = σ(X, Y )GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE

31

for all vectors X and Y on M . We classify real hypersurfaces M having ϕ-invariant
shape operator A in CP n (c), n ≧ 2 and give a geometric meaning such hypersurfces
M by observing the extrinsic shape of some geodesics and integral curves of ξ on
M.
Theorem 9 ([17]). Let M be a connected real hypersurface of CP n (c), n ≧ 2through an isometric immersion. Then the following conditions (1), (2) and (3)are mutually equivalent:(1) M is locally√congruent to a homogeneous real hypersurface of type (A) of
radius π/(2c ). That is, M is √locally congruent to either a geodesic sphere√ℓG(π/(2 c )) or a tube T (π/(2 c )) around a totally geodesic CP ℓ (c) (1 ≦ℓ ≦ n − 2) in the ambient space CP n (c).
(2) The shape operator A of M is ϕ-invariant.
(3) M satisﬁes both of Condition (ES2) in Section 3 and a condition that there
exists at least one integral curve of the characteristic vector ﬁeld ξ of M
which is mapped to a geodesic in CP n (c).
At the end of this paper, as an application of the theory of real hypersurfaces in
CP n (c) we shall discuss some homogeneous submanifolds in Euclidean sphere. We
shall explain an idea to construct a certain class of (Riemannian) submanifolds in a
sphere. We denote by (M, ιM ) a real hypersurface M of CP n (c), n ≧ 2 through an
isometric immersion ιM : M → CP n (c). In the following, we regard real hypersurfaces M in CP n (c) as submanifolds of the sphere S n(n+2)−1 ((n + 1)c/(2n)) throughf1 ◦ ιM , where f1 is the ﬁrst standard (parallel equivariant) minimal embedding of
CP n (c) into S n(n+2)−1 ((n + 1)c/(2n)).
We here give the deﬁnition and fundamental geometric properties of f1 . The
embedding f1 is deﬁned by eigenfunctions of the ﬁrst eigenvalue of the Laplacian ∆on CP n (c) (for details, see [8, 30]). In submanifold theory, this minimal embedding
f1 is well-known as the only example of a minimal full immersion with parallel
second fundamental form of a complex projective space endowed with Fubini-Study
metric into a Euclidean sphere. The inner product of the ﬁrst normal space of f1
is given by(5.1)⟨σ1 (X, Y ), σ1 (Z, W )⟩ = −(c/(2n))⟨X, Y ⟩⟨Z, W ⟩ + (c/4)(⟨X, W ⟩⟨Y, Z⟩+ ⟨X, Z⟩⟨Y, W ⟩ + ⟨JX, W ⟩⟨JY, Z⟩ + ⟨JX, Z⟩⟨JY, W ⟩).

Here, σ1 is the second fundamental form of the embedding f1 . Equation (5.1) yields
the following properties of f1 :(i) f1 is minimal;(ii) σ1 (JX, JY ) = p
σ1 (X, Y ) for all X, Y ∈ T CP n (c) (namely, σ1 is J-invariant);n(iii) ∥σ1 (X,pX)∥ = (n − 1)c/(2n) for each unit vector X on CP (c) (that is,
f1 is (n − 1)c/(2n) -isotropic (cf. [27])).
Thus we obtain a family of submanifolds {(M 2n−1 , f1 ◦ ιM )} in the sphere. This
class contains some homogeneous submanifolds of S n(n+2)−1 ((n + 1)c/(2n)), that is,they are expressed as orbits of some subgroups of the isometry group SO(n(n + 2))of the ambient sphere. Indeed, if we take a homogeneous real hypersurface M of

32

S. MAEDA

CP n (c), the immersion f1 ◦ ιM gives a homogeneous submanifold M of the sphere,
so that M has constant mean curvature, i.e., the length of the mean curvature
vector of M is constant. However, the second fundamental form of the immersion
f1 ◦ ιM is not parallel for each real hypersurface M of CP n (c) (cf. [9]). So, it isnatural to pose the following problem:
Problem 2. Classify submanifold (M, f1 ◦ιM ) of S n(n+2)−1 ((n+1)c/(2n)) satisfyingthat the immersion f1 ◦ ιM has parallel mean curvature vector with repect to thenormal connection.
The following is an answer to Problem 2:
Lemma 5 ([23]). Let M be a connected real hypersurface of CP n (c), n ≧ 2 throughan isometric immersion ιM and f1 : CP n (c) → S n(n+2)−1 ((n + 1)c/(2n)) be theﬁrst standard minimal √embedding. Then M is locally congruent to the geodesic
sphere G(r) with tan2 ( c r/2) = 2n + 1 in CP n (c) if and only if the immersion
f1 ◦ ιM : M → S n(n+2)−1 ((n + 1)c/(2n)) has parallel mean curvature vector withrespect to the normal connection. Moreover, this submanifold (M, f1 ◦ ιM ) is locallyhomogeneous in this ambient sphere.
In order to establish our Theorem 10 we need the following two lemmas:√
Lemma 6 ([23]). The geodesic sphere G(r) with tan2 ( c r/2) = 2n + 1 is a
Sasakian manifold with respect to the almost contact metric structure induced fromthe Kähler structure (J, g) on CP n (c), n ≧ 2 if and only if c = 8n + 4. Furthermore, this geodesic sphere is a Sasakian space form of constant ϕ-sectional
curvature 8n + 5.
˜
Lemma 7 ([23]). We consider
the following√√ isometric embedding fn of the geodesic2
sphere G(r) (0 < r < π/ c ) with tan ( c r/2) = 2n + 1 in CP (c), n ≧ 2 into
an N (≧ n(n + 2) − 1)-dimensional sphere S N (c̃) of constant sectional curvaturec̃(≦ (n + 1)c/(2n)).
(1) When N > n(n + 2) − 1, f˜ is given by
f1 ◦ιG(r)ιf˜ = ι ◦ (f1 ◦ ιG(r) ) : G(r) −→ S n(n+2)−1 ((n + 1)c/(2n)) −→ S N (c̃),

where ι is a totally umbilic embedding, so that (n + 1)c/(2n) ≧ c̃.
(2) When N = n(n+2)−1, f˜ is nothing but f1 ◦ιG(r) , so that (n+1)c/(2n) = c̃.
Then our geodesic sphere is homogeneous in S N (c̃) and has nonzero parallel mean
curvature vector with respect to the normal connection in this sphere.
Therefore, in view of Lemmas 5, 6 and 7 we establish the following:
Theorem 10 ([23]). (1) For each of c > 0, n(≧ 2), N > n(n + 2) − 1 and c̃ ≦ (n +1)c/(2n), there exists a (2n−1)-dimensional submanifold M in an N -dimensional
sphere S N (c̃) of constant sectional curvature c̃, which has the following properties:(A) M is a homogeneous submanifold which has nonzero parallel mean curvature
vector with respect to the normal connection in S N (c̃);

GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE

33(B) M is a Berger sphere. Here, this means that M is an odd dimensional
compact simply connected Riemannian manifold having the property thatall sectional curvatures of M lie in the interval [δK, K] with some constant√δ ∈ (0, 1/9) but it has a closed geodesic whose length is shorter than 2π/ K,
where K is a positive constant.
(C) When c = 8n + 4, M is a Sasakian space form of constant ϕ-sectional
curvature 8n+5.
(2) For each of c > 0, n(≧ 2), when N = n(n + 2) − 1, there exists also a (2n−1)dimensional submanifold M in an N -dimensional sphere S N (c̃) of constant sectional curvature c̃ = (n + 1)c/(2n), which has the above properties (A), (B), (C).
We ﬁnally pose the following open problem:
Problem 3. Let f1 be a mimimal parallel full immersion of a complex nfn into a Euclidean spheredimensional compact Hermitian symmetric space M
fn satisfying that
S 2n+p (c̃). If there exists a real hypersurface (M 2n−1 , ιM 2n−1 ) of M
2n−1
the corresponding submanifold (M, f1 ◦ιM 2n−1 ) has parallel mean curvature vector with respect to the normal connection in the ambient sphere S 2n+p (c̃), is our
fn holomorphically isometric to a complex projective
Hermitian symmetric space Mnspace CP (c), n ≧ 2 of constant holomorphic sectional curvature c = 2nc̃/(n + 1)
and p = n2 − 1?
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Sadahiro Maeda: Professor emeritus of Saga University and Shimane University, Japan
Email address: sadahiromaeda0801@gmail.com