Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
43 (2010), pp. 1–12

GEOMETRY OF GEODESIC SPHERES IN ACOMPLEX PROJECTIVE SPACESADAHIRO MAEDA
Communicated by Makoto Kimura(Received: June 30, 2009)√
Abstract. A geodesic sphere G(r) of radius r (0 < r < π/ c ) of a complex
projecive space CP n (c) is one of the most interesting objects in diﬀerential geometry. This expository paper consists of two parts. In the ﬁrst half, we study
curve theory on G(r) (see [4, 9]). In the latter half, we investigate G(r) from theviewpoint of submanifold theory ([2, 11]).

1. Introduction√
A geodesic sphere G(r) of radius r (0 < r < π/ c ) in an n-dimensional complex
projective space CP n (c) (n = 2) of constant holomorphic sectional curvature c(> 0)
is impotant in intrinsic geometry as well as extrinsic geometry (i.e., submanifold
theory).
In intrinsic geometry, for example it is known that G(r) is a naturally reductive
Riemannian homogeneous manifold, so that every geodesic of G(r) is a homogeneous curve, namely it is an orbit of a one-parameter subgroupof the isometry√
2group I(G(r)) of G(r) (cf. [12]). Moreover, when tan ( c r/2) > 2, G(r) is a
Berger sphere (see [15]). Inspired
we are interested in geodesics√ by these facts,non G(r) of radius r (0 < r < π/ c ) in CP (c). We ﬁrst investigate the length
spectrum of G(r) in detail (see [4]). We next study non-geodesic homogeneous
curves on G(r). We construct a family of closed non-geodesic homogeneous curves
on G(r) with the same length by using an isometric embedding
)
(
ιG(r)
n+1
f
n
n(n+2)−1c ,(1.1)
f ◦ ιG(r) : G(r) −→ CP (c) −→ S
2nwhere ιG(r) is a natural inclusion mapping of G(r) into CP n (c) and f is so-called
the ﬁrst standard minimal embedding of CP n (c) into an (n(n + 2) − 1)-dimensional
2000 Mathematics Subject Classification. Primary 53B25, 53C22; Secondary 53C40, 53D10.
Key words and phrases. geodesic spheres, complex projective spaces, almost contact metricstructure, geodesics, circles, homogeneous curves, Sasakian manifolds, Sasakian space forms,
Berger spheres, homogeneous submanifolds, parallel mean curvature vector.
1

2

S. MAEDA

sphere S n(n+2)−1 ((n + 1)c/(2n)) of constant sectional curvature (n + 1)c/(2n)(see [9]).
In extrinsic geometry, again by using the above minimal embedding f we
immerse each real hypersurface M 2n−1 of CP n (c) into the ambient sphereS n(n+2)−1 ((n + 1)c/(2n)) as follows:
)
(
n+1
f
ιM
n
n(n+2)−1(1.2)
f ◦ ιM : M −→ CP (c) −→ Sc ,2nwhere ιM is an isometric immersion of M 2n−1 into CP n (c). Note that the isometricimmersion f ◦ ιM does not have parallel second fundamental form for each real
hypersurface M of CP n (c). On the other hand, by direct computation we can seethat f ◦ ιM has parallel mean curvature vector in this√sphere if and only if M islocally congruent to the geodesic sphere G(r) with tan2 ( c r/2) = 2n+1 in CP n (c)(cf. [11]). Needless to say, this geodesic sphere is a Berger sphere. Furthermore, it
has an almost contact metric structure (ϕ, ξ, η, ⟨ , ⟩). In particular, when c = 8n+4,
this geodesic sphere is a Sasakian space form of constant ϕ-sectional curvature
8n + 5 (see [2]). These facts imply that for each of c(> 0) and n(= 2), every N dimensional sphere S N (c̃) of constant sectional curvature c̃ with (n + 1)c/(2n) = c̃and N > n(n + 2) − 1 admits a (2n − 1)-dimensional Riemannian submanifold
M 2n−1 satisfying the following properties.
(1) M is diﬀeomorphic but not isometric to a Euclidean sphere.
(2) M is a homogeneous submanifold of the ambient sphere S N (c̃), i.e., M is anorbit of some subgroup of the isometry group SO(N +1) of S N (c̃). However,
M is not a Riemannian symmetric space.
(3) The mean curvature vector of M in S N (c̃) is nonzero-parallel with respect
the normal connection of M .
(4) M has an almost contact metric structure (ϕ, ξ, η, ⟨ , ⟩). Especially, in thecase of c = 8n + 4, M is a Sasakian space form of constant ϕ-sectional
curvature 8n + 5.
In the latter half of this paper, we clarify these fundamental properties of a certain
geodesic sphere G(r) in CP n (c).
2. Length spectrum of geodesic spheres G(r) in CP n (c)Let M 2n−1 (n = 2) be a real hypersurface with a unit nomal local vector ﬁeld N
in CP n (c). We denote by (ϕ, ξ, η, ⟨ , ⟩) the almost contact metric structure of M
induced from the Kähler structure J of the ambient space CP n (c). That is, thisstructure is deﬁned byξ = −JN , η(X) = ⟨X, ξ⟩ and ϕX = JX − η(JX)ξ for all vectors X on M,
so that it satisﬁes
ϕ2 X = −X + η(X)ξ, η(ξ) = 1 and η(ϕX) = 0 for arbitrary X on TM.

GEOMETRY OF GEODESIC SPHERES

3

We here recall the following fundamental equations for M , which are so-called
Gauss formula and Weingarten formula, respectively.
(2.1)

e X Y = ∇X Y + ⟨AX, Y ⟩N∇e X N = −AX,and ∇e and ∇ are the Riemannian connections of CP n (c) and M , respectively,
where ∇and A is the shape operator of M in CP n (c). Then it follows from the fact thate = 0 and (2.1) that
∇J
(2.2)

∇X ξ = ϕAX

and
(2.3)

(∇X ϕ)Y = η(Y )AX − ⟨AX, Y ⟩ξ,where X and Y are any vectors on M .
√In the following, we consider a geodesic sphere G(r) of radius r (0 < r < π/ c )in CP n (c). So we use the commutative condition ϕA = Aϕ without explanation.
We recall the invariance ργ for a geodesic γ = γ(s) on G(r), which is deﬁned by
ργ = ⟨γ̇(s), ξγ(s) ⟩ for −∞ < s < ∞. Equation (2.2) guarantees the constancy of ργwith −1 5 ργ 5 1.
∇γ̇ ργ = ∇γ̇ ⟨γ̇, ξ⟩ = ⟨γ̇, ∇γ̇ ξ⟩= ⟨γ̇, ϕAγ̇⟩ = ⟨γ̇, Aϕγ̇⟩ = ⟨Aγ̇, ϕγ̇⟩= −⟨ϕAγ̇, γ̇⟩ = 0.
This invariance ργ is said to be the structure torsion of a geodesic γ on G(r). We
shall state the congruence theorem on geodesics of G(r) in terms of their structure
torsions. For this purpose we review fundamental notions on congruency for curves
in Riemannian manifolds. Two curves γ1 , γ2 on a Riemannian manifold N are said
to be congruent to each other in the usual sense if there exist an isometry of φ of
N and a constant s0 satisfying γ2 (s) = (φ ◦ γ1 )(s + s0 ) for all s. In the case we cantake s0 = 0, they are said to be strongly congruent to each other. That is, we call
two curves γ1 , γ2 on N strongly congruent to each other if there is an isometry φ of
N with γ2 (s) = (φ ◦ γ1 )(s) for all s. Trivially, a Riemannian manifold N is either
a Euclidean space or a Riemannian symmetric space of rank one if and only if forevery pair of geodesics γ1 , γ2 on N they are strongly congruent to each other. In
this paper, we treat a curve on a Riemannian manifold N is a mapping of the realline R into N .
√Lemma 1 ([4]). On a geodesic sphere G(r) (0 < r < π/ c ) of CP n (c), twogeodesics γ1 , γ2 are strongly congruent to each other if and only if their structure
torsions ργ1 , ργ2 satisfy |ργ1 | = |ργ2 |.
We are now in a position to study lengths of closed geodesics of G(r). It suﬃces
to consider the case of c = 4. Let Π : S 2n+1 (1) → CP n (4) denote the Hopf ﬁbrationof a unit sphere. For a smooth curve γ on CP n (4) a smooth curve γ̃ on S 2n+1 (1)˙˙
is called a horizontal lift of γ if γ̃(s)is a horizontal vector and dΠ(γ̃(s))= γ̇(s) forall s. We note that a curve γ on G(r) is closed if and only if its horizontal lift γ̃ on

4S. MAEDA

S 2n+1 (1) satisﬁes γ̃(s) = eiθ γ̃(s + s0 ) with some cnstants θ ∈ [0, 2π) and s0 (> 0)for each s ∈ (−∞, ∞). The following elementary lemma is a key in our argument.
Lemma 2 ([4]). Let σ be a smooth simple curve on CP n (4). Suppose that ahorizontal lift σ̃ of σ on S 2n+1 (1) is represented as
√

√

√

√

σ̃(s) = Ae −1 as + Be −1 bs + Ce −1 cs + De −1 ds ,which is a curve in Cn+1 with nonzero vectors A, B, C, D ∈ Cn+1 and mutually
distinct real numbers a, b, c, d satisfying a + b + c + d = 0 and a ̸= 0. Then σ isclosed if and only if all the ratios b/a, c/a, d/a are rational. In this case, its length
is
{
}
1
1
1
length(σ) = 2π × L.C.M.
.
,,|b − a| |c − a| |d − a|Here, for positive numbers α1 , α2 , α3 , we denote by L.C.M.{α1 , α2 , α3 } the minimum value of the set {jα1 |j = 1, 2, . . .} ∩ {jα2 |j = 1, 2, . . .} ∩ {jα3 |j = 1, 2, . . .}.
√We remember that every geodesic of G(r) (0 < r < π/ c ) in CP n (c) is ahomogeneous curve (see [12]), so that it is a simple curve. Then by Lemma 2 weobtain the following suﬃcient condition for a geodesic γ = γ(s) on G(r) to beclosed, which can be written by its structure torsion ργ :Theorem√ 1 ([4]).n For a geodesic γ on a geodesic sphere G(r) of radius r (0 <r < π/ c ) in CP (c) we have the following properties according to their structure
torsions:√(1) When ργ = ±1, it is closed and its length is π sin( c r);√(2) When ργ = 0, it is also closed and its length is 2π sin( c r/2);(3) When 0 < |ργ | < 1, it is closed if and only if its structure torsion ργ isgiven by
±q
√
ργ =
√
√
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√()√√
length(r) = 2δ(p, q)π p2 sin2 ( c r/2) + q 2 cos2 ( c r/2) /c .
Here, δ(p, q) takes the value 2 when pq is even and takes the value 1 whenpq is odd.
In consideration of Lemma 1 and Theorem 1 we ﬁnd the following:√
Theorem 2 ([4]). On a geodesic sphere G(r) (0 < r < π/ c ) in CP n (c), there
exist countably inﬁnite congruence classes of closed geodesics. The length spectrum
LSpec(G(r)) of G(r) is a discrete unbounded subset in the real line R.
We here investigate the ﬁrst length spectrum λ1 , the second length spectrum λ2and the third length spectrum λ3 of G(r) in detail.
√
Proposition 1 ([4]). A geodesic sphere G(r) (0 < r < π/ c ) in CP n (c) has thefollowing properties on the lengths of closed geodesics.

GEOMETRY OF GEODESIC SPHERES

5(1) λ1 , λ2 and √λ3 are simple,
that is, each of their multiplicities is one.
√(2) λ1 = (2π/ c ) sin( c r), which is the length of the geodesics of structure
torsion ±1.
√√√(3) When 0 < r 5 π/(2 c ), we have λ2 = (4π/ c ) sin( c r/2), which is thelength of the√geodesics with√ null structure torsion.
√
When π/(2 c ) < r < π/ c , we have λ2 √= 2π/ c , which is the length of
the geodesics√ with structure√ torsion ± cot( c r/2).
√√(4) When π/(2 c ) < r < π/ c we have λ3 = (4π/ c ) sin( c r/2), which is
the length of the geodesics
with null structure√√ torsion.
√
When 0 < r 5 π/(2 c ) anditsatisﬁes
2k − 1 5 cot r < 2k + 1 (k =√√1, 2, . . .), we have λ3 = 2π {4k(k + 1) sin2 ( c r/2) + 1}/c , which is thelengthgeodesics with structure torsion )( of the√√√±1/ sin( c r/2) (2k + 1)2 tan2 ( c r/2) + 1 .
√We remark that the sectional
curvature K of G(r)(0 < r < π/ c ) lies in the√√closed interval [(c/4) cot2 ( c r/2),√c + (c/4) cot2 ( c r/2)]. Hence, as mentioned in
Introduction, in the case of tan2 ( c r/2) > 2 we ﬁnd that it is an example of aso-called Berger sphere. But for all lengths except the bottom λ1 of LSpec(G(r)),
we ﬁnd that the following inequality of Klingerberg’s type holds.
Corollary 1. Except geodesics withtorsion ±1, every geodesic γ of a
√ structurengeodesic sphere G(r)
(0
<
r
<
π/
c
)
in
CP
(c)
satisﬁes√√
2
length(r) > 4π/ c (4 + cot ( c r/2)) .
Each element of LSpec(G(r)) is not necessarily simple. For example, for G(π/4)
in CP n (4) we have
{ √
√
√
√
√
√
√
√
LSpec(G(π/4)) = 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, . . . .
√Though each element√from λ1 = π to λ16 = 61 π is simple, we ﬁnd that themultiplicity of λ17 = √ 65 π is two.
√ It is the common length of the geodesics withstructure torsions 3/ 65 and 7/ 65 .
√Theorem 3 ([4]). For a geodesic sphere G(r) (0 < r < π/ c ) in CP n (c) we
obtain the following:√(1) If tan2 (√ c r/2) is irrational, every element of LSpec(G(r)) is simple.
(2) If tan2 ( c r/2) is rational, the multiplicity of each element of LSpec(G(r))
is ﬁnite, but not uniformly bounded and satisﬁes lim supλ→∞ m(λ) = ∞. Its
growth is less than polynomial growth. It satisﬁes limλ→∞ λ−δ m(λ) = 0 for
arbitrary positive δ.
(3) We denote by n(λ) the number of congruency classes of closed geodesics
whose length is not longer than λ. Its growth is polynomial√ order of thesecond degree and satisﬁes limλ→∞ λ−2 n(λ) = 3cr/ (4π 4 sin( c r)).

6

S. MAEDA√This theorem√guarantees that on a geodesic sphere G(r) (0 < r < π/ c ) withirrational tan2 ( c r/2) in CP n (c) two closed geodesics are√congruent if and only
if they have the same length. On the other hand, if tan2 ( c r/2) is rational, we
cannot classify congruency classes of closed geodesics only by their lengths.
3. Non-geodesic homogeneous curves of G(r) in CP n (c)We are interested√ in ﬁnding a nice family of curves including all geodesics ofG(r) (0 < r < π/ c ). To do this, we recall the notion of Sasakian curves.
f, J) a smooth curve γ is said
On a real hypersurface N in a Kähler manifold (Mto be a Sasakian curve if it satisﬁes(3.1)∇γ̇ γ̇ = kϕγ̇with some constant k, where ϕ is the structure tensor of N induced by J. Needlessto say, for an arbitrary constant k and a unit vector v at each point x ∈ N ,there exists the unique Sasakian curve γ satisfying (3.1) with initial condition thatγ(0) = x and γ̇(0) = v. Sasakian curves on a manifold admitting an almost contact
metric structure can be considered as correspondences of Kähler circles on Kähler
manifolds.
We here recall two invariances for a Sasakian curve γ = γ(s) on G(r) in CP n (c).
One is the structure torsion ργ = ⟨γ̇(s), ξγ(s) ⟩. The other is the normal curvatueκγ = ⟨Aγ̇(s), γ̇(s)⟩. By the same computation as above and (3.1) we ﬁnd theconstancy of ργ :∇γ̇ ργ = ⟨∇γ̇ γ̇, ξ⟩ + ⟨γ̇, ∇γ̇ ξ⟩ = k⟨ϕγ̇, ξ⟩ + ⟨γ̇, ϕAγ̇⟩= ⟨γ̇, ϕAγ̇⟩ = ⟨γ̇, Aϕγ̇⟩ = −⟨ϕAγ̇, γ̇⟩ = 0.
Furthermore, by the fact ⟨(∇X A)X, X⟩ = 0 for all vectors X on G(r), we see theconstancy of κγ . We remark that for a Sasakian curve(3.1) on G(r) the√γ satisfying
2
ﬁrst curvature κ1 = ∥∇γ̇ γ̇∥ of γ is given by κ1 = |k| 1 − ργ , so that it is constantalong γ. Hence, in the following we say the constant k to be the coeﬃcient ofa Sasakian curve γ satisfying (3.1). As a matter of course we treat geodesics as
Sasakian curves in a trivial sense.
√
For about Sasakian curves on G(r) (0 < r < π/ c ) in CP n (c), the following is
obtained in [3].
Lemma 3. Let γi (i = 1, 2) be Sasakian curves of coeﬃcients
κi and structure√
torsions ργi on a geodesic sphere G(r) (0 < r < π/ c ) in CP n (c). They arestrongly congruent to each other if and only if one of the following conditions holds:i) |ργ1 | = |ργ2 | = 1;ii) ργ1 = ργ2 = 0 and |κ1 | = |κ2 |;iii) 0 < |ργ1 | = |ργ2 | < 1 and κ1 ργ1 = κ2 ργ2 .
As immediate consequences of this lemma we obtain the following corollary on
the homogeneity of Sasakian curves on G(r).

GEOMETRY OF GEODESIC SPHERES

7√
Corollary 2. Every Sasakian curve on G(r) (0 < r < π/ c ) in CP n (c) is homogeneous. That is, it is an orbit of a one-parameter subgroup of the isometry group
I(G(r)) of G(r).
In order to get√geometric properties of Sasakian curves on a geodesic sphereG(r) (0 < r < π/ c ) in CP n (c), we study them through the isometric embeddingf ◦ ιG(r) given by (1.1). As mentioned in Introduction, this embedding f ◦ ιG(r)does not have parallel second fundamental form but is equivariant. So we canhence treat our geodesic sphere G(r) as a homogeneous submanifold in this spherethrough this embedding f ◦ ιG(r) .
Proposition 1 tells us that every integral curve γ of the characteristic
vector ﬁeld ξ√on G(r) is the shortest closed geodesic for each radius r ∈ (0, π/ c ). Furthermore,
its shape ιG(r)√◦ γ throughthe inclusion ιG(r) : G(r) → CP n (c) is a Kähler circle√
of curvature c cot( c r) in CP n (c), namely this integral curve γ satisﬁes eithere γ̇ γ̇ = √c cot(√c r)J γ̇ or ∇e γ̇ γ̇ = −√c cot(√c r)J γ̇. If we see this curve through∇the embedding f ◦ ιG(r) , we ﬁnd the curve f ◦ ιG(r) ◦ γ is a small circle on a sphereS n(n+2)−1 ((n+1)c/(2n)) by the following lemma.
Lemma 4 ([6]). A smooth curve µ on CP n (c) is a Kähler circle of curvature κ
if and only√if the curve f ◦ µ on S n(n+2)−1 ((n+1)c/(2n)) is a circle of positive
curvature κ2 + ((n−1)c/(2n)).
In this context, we naturally come to the position to pose the following problem:
Problem 1. Find and classify smooth curves γ on G(r) whose shape f ◦ ιG(r)( ◦γn(n+2)−1
through )the equivariant isometric embedding f ◦ ιG(r) are circles in S(n+1)c/(2n) .
By virtue of Lemma 4 this problem is equivalent to the problem to ﬁnd and toclassify curves on G(r) which are mapped to Kähler circles in CP n (c) through theinclusion ιG(r) . For a smooth curve γ on G(r) we get by the Gauss formula thate γ̇ γ̇ = ∇γ̇ γ̇ + ⟨Aγ̇, γ̇⟩N and J γ̇ = ϕγ̇ + ργ N . We can hence obtain the following.
∇
Lemma 5 ([10]). A smooth curve γ on G(r) can be seen as a Kähler circle ofcurvature κ on CP n (c) through the inclusion ι if and only if it satisﬁes both of theequations ∇γ̇ γ̇ = ±κϕγ̇ and ⟨Aγ̇, γ̇⟩ = ±κργ , where double signs take the same
signatures.
By use of this lemma we can get the following answer to our problem. The
answer depends on the radius of a geodesic sphere.
√
Theorem 4. Let G(r) be a geodesic sphere of radius 0 < r 5 π/(2 c ) in CP n (c).
{}√(1) For 0 5 k < c cot2 ( c r)+(n−1)/(2n) , there are no curves(on G(r) whose)n(n+2)−1(n+1)c/(2n).
shape through{f ◦ ιG(r) is a circle of curvature
k
on
S
}
√
2
2
(2) When k = c cot ( c r) + (n−1)/(2n) , the shape of a curve γ on G(r)through f ◦ ιG(r) is a circle of curvature k if and only if it is a geodesic withstructure torsion ργ = ±1, which is an integral curve of ξ on G(r).

8

S. MAEDA{}√(3) When k 2 > c cot2 ( c r) + (n−1)/(2n) , the shape of a curve γ on G(r)through f ◦ ιG(r) is a √circle of curvature k if and only if it is a Sasakian
curve of coeﬃcient ± k 2 − (n−1)c/(2n) whose structure torsion is√{√}√ργ = ±c−1/2
k 2 + (n+1)c/(2n) − k 2 − (n+1)c/(2n) cot( c r/2),
where double signs take the same signatures.
√
Trivially these curves in (2), (3) are closed with length 2π/ k 2 + (n+1)c/(2n) .
√√Theorem 5. Let G(r) be a geodesic sphere of radius r with π/(2 c ) < r < π/ c
in CP n (c).
√(1) For 0 5 k < (n−1)c/(2n) , there are no curves on (G(r) whose shape)thorough f ◦ ιG(r) is a circle of curvature k on S n(n+2)−1 (n+1)c/(2n) .
√(2) When k = (n−1)c/(2n) , the shape of a curve on G(r) through f ◦ ιG(r) isa circle of curvature
k if and only if it is a geodesic with structure torsion√
ργ = ±√
cot( c r/2).
{}√(3) When (n−1)c/(2n) < k < c cot2 ( c r) + (n−1)/(2n) , the shape of acurve γ on G(r) through f ◦ ιG(r) is a√circle of curvature k if and only if
it is a Sasakian curve of coeﬃcient ± k 2 − (n−1)c/(2n) whose structure
torsion is
{√
}
√
√
−1/2
2
2
ργ = ±ck + (n+1)c/(2n) − k − (n+1)c/(2n) cot( c r/2)
or

}
{ √
√
√
ργ = ±c−1/2 − k 2 + (n+1)c/(2n) − k 2 − (n+1)c/(2n) cot( c r/2),
where double {signs take}√ the same signatures.
2(4) When k = c cot ( c r) + (n − 1)/(2n) , the shape of a curve on G(r)through f ◦ ιG(r) is a circle of curvature k if and only if it is a geodesic withstructure torsion{ ργ√= ±1.
}(5) When k > c cot2 ( c r) + (n − 1)/(2n) , the shape of a curve γ on G(r)through f ◦ ιG(r) is a √circle of curvature k if and only if it is a Sasakian
curve of coeﬃcient ± k 2 − (n−1)c/(2n) whose structure torsion is{√}√√ργ = ±c−1/2
k 2 + (n+1)c/(2n) − k 2 − (n+1)c/(2n) cot( c r/2),
where double signs take the same signatures.
√
Trivially these curves in (2),(3),(4),(5) are closed with length 2π/ k 2 +(n+1)c/(2n).
Remark 1.
(1) For each curve γ in Theorems 4 and 5, the curve ιG(r) ◦ γ isa homogeneous curve on totally geodesic CP 1 (c)(= S 2 (c)) of CP n (c) (see[3]). This fact shows that each curve in Theorems 4 and 5 is an orbit of aone-parameter subgroup of SO(3).
(2) For each Sasakian curve γ on G(r) the curve ιG(r) ◦ γ is a homogeneous
curve on totally geodesic CP 2 (c) of CP n (c) (see [1]). Hence every Sasakian
curve on G(r) is an orbit of a one-parameter subgroup of SU(3).

GEOMETRY OF GEODESIC SPHERES

9(3) Curves in Theorem 4(3), Theorem 5(3) and Theorem 5(5) are non-geodesic
Sasakian curves.
(4) There exist two non-geodesic Sasakian curves in Theorem 5(3) which are
not congruent to each other with respect to I(G(r)),
but they() are mappedn(n+2)−1
to a circle of the same curvature k on S(n+1)c/(2n) . Hence thesecurves, considered as curves on this sphere, are congruent to each otherwith respect to the isometry group SO(n(n + 2)) of the sphere.
4. Properties of certain geodesic spheres G(r) in CP n (c) insubmanifold theory()
We ﬁrst investigate the minimal embedding f : CP n (c) → S n(n+2)−1 (n+1)c/(2n)which is deﬁned by eigenfunctions of the ﬁrst eigenvalue of the Laplacian ∆ on
CP n (c). The inner product of the ﬁrst normal space of f is given by((4.1)⟨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 denotes the second fundamental form of f . Equation (4.1) shows thefollowing properties of f :i) f1 is minimal;ii) σ1 (JX, JY ) = √σ1 (X, Y ) for ∀X, Y ∈ T CP n (c) (namely, σ is J-invariant);niii) ∥σ1 (X,√ X)∥ = (n − 1)c/(2n) for each unit vector X on CP (c) (that is,
f is (n − 1)c/(2n) -isotropic (cf. [14])).
We remark that σ1 is J-invariant is equivalent to saying that the second fundamental form σ1 of our embedding f is parallel. As mentioned in Introduction, theembedding f is usually called the ﬁrst standard minimal embedding.
In this (section, we immerse
all real hypersurfaces M of CP n (c) into the sphere)n(n+2)−1S(n + 1)c/(2n) (see (1.2)). Note that for every real hypersurface M(, thesecond fundamental
form of the isometric immersion f ◦ ιM : M → S n(n+2)−1 (n +)1)c/(2n) is not parallel. However, in this class {(M, f ◦ ιM )|ιM : M→ CP n (c) is()
an isometric immersion} of all submanifolds in the sphere S n(n+2)−1 (n+1)c/(2n) ,there exist nonzero-constant mean curvature submanifolds. For √example, direct
calculation tells us that the mean curvature Hr (0 < r < π/ c ) deﬁned by
the length( of the mean) curvature vector of the embedding f ◦ ιG(r) : G(r) →n(n+2)−1S(n + 1)c/(2n) given by (1.1) is expressed as{( √c )( √c ) −4n2 + 4n − 2 }
c
2
2
2
2
Hr =
(2n − 1) cot
r + tanr +̸= 0.
4(2n − 1)22
2nIn this context, it is natural to pose the following problem:
Problem 2. Classify submanifolds (M, f ◦ ιM ) given by (1.2) having parallel mean(
curvature) vector with respect to the normal connection in the sphere S n(n+2)−1 (n+1)c/(2n) .
The following proposition plays as a key in this section.

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S. MAEDAn
Proposition 2. Let M 2n−1 be a real hypersurface( of CP (c) )through an isometnn(n+2)−1
ric immersion ιM and f : CP (c) → S(n + 1)c/(2n) the ﬁrst standard
minimal√ embedding. Then M isn locally congruent to the geodesic sphere G(r) withtan2 ( c r/2)
= 2n + 1 )in CP (c) if and only if the immersion f ◦ ιM : M →(n(n+2)−1S(n + 1)c/(2n) has parallel mean curvature vector with respect to thenormal connection. Moreover, this submanifold (M, f ◦ ιM ) is homogeneous in thisambient sphere.

Remark√ 2. The geodesic sphere G(r) in Proposition 2 is a Berger sphere, since2tan ( c r/2) = 2n + 1 > 2.
We next study the almost contact structure of our geodesic sphere in Propsition 2.
For this purpose we review fundamental notions in contact geometry. Let M 2m+1
be an almost contact metric manifold endowed with an almost contact metricstructure (ϕ, ξ, η, ⟨ , ⟩). That is, this structure satisﬁes the following identities:
ϕ2 X = −X + η(X)ξ, ϕξ = 0, η(ξ) = 1, ⟨ϕX, ϕY ⟩ = ⟨X, Y ⟩ − η(X)η(Y )for all vectors X, Y on M . M is called a Sasakian manifold if the structure tensor
ϕ of M satisﬁes the following diﬀerential equation:(4.2)(∇X ϕ)Y = ⟨X, Y ⟩ξ − η(Y )Xfor ∀X, Y ∈ TM,

where ∇ denotes the Riemannian connection of the Riemannian metric ⟨ , ⟩ of M .
A Sasakian manifold is called a Sasakian space form of constant ϕ-sectional curvature c if the sectional curvature K(u, ϕu) := ⟨R(u, ϕu)ϕu, u⟩ = c holds for every
unit vector u orthogonal to ξ, where R is its curvature tensor. For construction of
Sasakian space forms, see pp. 99-100 in [5].
In the following, we shall consider case that a real hypersurface M of CP n (c)is a Sasakian manifold with respect to the alomost contact metric structure(ϕ, ξ, η, ⟨ , ⟩) induced from the Kähler structure J of the ambient space CP n (c).
Then it follows from (4.2) and (2.3) that ξ is principal. Hence, again by using(4.2) and (2.3) we ﬁnd that Au = −u for each vector u orthogonal to ξ, so that
our real hypersurface M is a member of totally η-umbilic hypersurfaces in CP n (c).
Hence, using the classiﬁcation theorem of totally η-umbilic hypersurfaces in CP n (c)(see [13]), we see that the shape operator A of our Sasakian manifold M in CP n (c)is written as(4.3)

AX = −X + (c/4)η(X)ξ

for each vector X ∈ TM.

Conversely, it follows from (2.3) and (4.3) that Equation (4.2) holds. Thus weknow that M is a Sasakian manifold if and only if M has the shape operator A
satisfying (4.3). Furthermore, M has constant ϕ-sectional curvature c + 1.
Therefore, from the discussion here and Proposition 2 we obtain the following:√
Proposition 3 ([2]). The geodesic sphere G(r) with tan2 ( c r/2) = 2n + 1 in
CP n (c) is a Sasakian manifold with respect to the almost contact metric structure
induced from the ambient space CP n (c) if and only if c = 8n + 4. Moreover, thisgeodesic sphere is a Sasakian space form of constant ϕ-sectional curvature c + 1.
By virtue of our discussion we establish the following:

GEOMETRY OF GEODESIC SPHERES

11

Theorem 6 ([11]). For each of c > 0, n(= 2) and N > n(n + 2) − 1, there
exists a (2n−1)-dimensional Riemannian submanifold M 2n−1 in an N -dimensional
sphere S N (c̃) of constant sectional curvature c̃ with (n + 1)c/(2n) = c̃ satisfying thefollowing three conditions:(1) M 2n−1 is a homogeneous submanifold with nonzero parallel mean curvature
vector with respect to the normal connection in the ambient sphere S N (c̃);(2) M 2n−1 is a Bereger sphere;(3) M 2n−1 has an almost contact metric structure (ϕ, ξ, η, ⟨ , ⟩). In particular,when c = 8n + 4, this submanifold M is a Sasakian space form of constantϕ-sectional curvature 8n + 5.
Moreover, for each of c > 0 and n(= 2), when N = n(n+2)−1, there exists a (2n−1)-dimensional Riemannian submanifold M 2n−1 in an N -dimensional sphere S N (c̃)
of constant sectional curvature c̃ = (n + 1)c/(2n) satisfying the above conditions(1), (2), (3).
References[1] T. Adachi, Trajectories on geodesic spheres in a non-flat complex space form, J. Geom. 90(2008), 1–29.
[2] T. Adachi, M. Kameda and S. Maeda, Geometric meaning of Sasakian space forms from theviewpoint of submanifold theory, to appear in Kodai Math. J.
[3] T. Adachi and S. Maeda, A congruence theorem of geodesics on some naturally reductive
Riemannian homogeneous manifolds, C. R. Math. Rep. Acad. Sci. Canada 26 (2004), 11–17.
[4] T. Adachi, S. Maeda and M. Yamagishi, Length spectrum of geodesic spheres in a non-flatcomplex space form, J. Math. Soc. Japan 54 (2002), 373–408.
[5] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509 (1976),
Springer, Berlin.
[6] B.Y. Chen and S. Maeda, Extrinsic characterizations of circles in a complex projective space
imbedded into a Euclidan space, Tokyo J. Math. 19 (1996), 169–185.
[7] M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurface in P n (C),Math. Ann. 276 (1987), 487–497.
[8] M. Kimura and S. Maeda, On real hypersurfaces of a complex projective space, Math. Z. 202(1989), 299–311.
[9] S. Maeda and T. Adachi, Sasakian curves on hypersurface of type (A) in a nonflat complex
space form, Results in Math. 56 (2009), 489–499.
[10] S. Maeda, T. Adachi and Y.H. Kim, Geodesic spheres in a nonflat complex space form andtheir integral curves of chracteristic vector fields, Hokkaido Math. J. 36 (2007), 353–363.
[11] S. Maeda and S. Udagawa, Construction of a certain homogeneous submanifold with nonzero
parallel mean curvature vector in Euclidean sphere, preprint.
[12] S. Nagai, The classification of naturally reductive homogeneous real hypersurfaces in complex
projective space, Arch. Math. (Basel) 69 (1997), 523–528.
[13] R. Niebergall and P.J. Ryan, Real hypersurfaces in complex space forms, in: Tight and Taut
submanifolds, T.E. Cecil and S.S. Chern (eds.), Cambridge Univ. Press, 1998, 233–305.
[14] B. O’Neill, Isotropic immersions and Kaehler immersions, Canad. J. Math. 17 (1965), 905–
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[15] A. Weinstein, Distance spheres in complex projective spaces, Proc. Amer. Math. Soc. 39(1973), 649–650.

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S. MAEDA

Sadahiro Maeda: Department of Mathematics, Saga University, 1 Honzyo, Saga
840-8502, Japan
E-mail address: smaeda@ms.saga-u.ac.jp