Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
33 (2000), pp.31–46

GEOMETRY OF A COMPLEX PROJECTIVESPACE FROM THE VIEWPOINT OF ITSCURVES AND REAL HYPERSURFACES

Sadahiro Maeda and Toshiaki Adachi
Dedicated to Professor Maretsugu Yamasaki for his 60th birthday
Abstract. This expository paper consists of two parts. In the ﬁrst half we study
circles in a complex projective space and give a characterization of homogeneous
real hypersurfaces in a complex projective space by using geometric properties ofcircles. In the latter half we study geodesics of geodesic spheres, which are thesimplest real hypersurfaces in a complex projective space, and investigate theirlength spectrum in detail. Finally we characterize real and complex space forms
from this point of view.

0. Introduction.
The study of circles is one of the interesting objects in diﬀerential geometry.
We here recall the deﬁnition of circles.
A smooth curve γ : R → M in a complete Riemannian manifold M is calleda circle of curvature κ(= 0) if it is parametrized by its arclength s and satisﬁes
the following equation:∇γ̇ ∇γ̇ γ̇(s) = −κ2 γ̇(s),where κ is a constant, which is called the curvature of the circle γ, and ∇γ̇ denotes
the covariant diﬀerentiation along γ with respect to the Riemmanian connection∇ of M . A circle of null curvature is nothing but a geodesic. The notion of
circles is an extension of the notion of geodesics. As a matter of course, a circle
of curvature κ(> 0) in a Euclidean m-space Rm is a circle of radius κ1 in the senseof Euclidean geometry and it is closed. However, in general a circle of positive
curvature in a Riemmanian manifold is not closed.
In section 1, we survey several results on geometric properties of circles in an
n-dimensional complex projective space CP n (c) of constant holomorphic sectional
curvature c (Theorems 1, 2). In this section we ﬁrst show that for each positive
The ﬁrst author talked a part of this paper in the conference of the Second Meeting on
Quaternionic Structures in Mathematics and Physics, Roma, Sept. 6-10, 1999.
31

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SADAHIRO MAEDA AND TOSHIAKI ADACHI

constant κ, there exist inﬁnitely many closed circles of curvature κ and inﬁnitely
many open circles of curvature κ in CP n (c) up to an action of the isometry groupof CP n (c). The main purpose of section 1 is to give an answer to the following
question: If two closed circles γ1 and γ2 in CP n (c) have the same length, arethey congruent with respect to an isometry of CP n (c)?
In section 2, we study real hypersurfaces M 2n−1 in CP n . Typical examplesof real hypersurfaces are homogeneous real hypersurfaces, namely they are given
as orbits under subgroups of the projective unitary group P U (n + 1). Takagi([17]) determined all homogeneous real hypersurfaces in CP n . Due to his work
we ﬁnd that a homogeneous real hypersurface in CP n is locally congruent to oneof the six model spaces of type A1 , A2 , B, C, D and E. They are realized as tubes
of constant radius over compact Hermitian symmetric spaces of rank 1 or rank 2(see Theorem A).
In the study of real hypersurfaces in CP n , there can be the following two
problems:(i) Give a characterization of homogeneous real hypersurfaces.
(ii) Construct non-homogeneous nice real hypersurfaces and characterize such
examples.
In this paper we are devoted to the problem (i). It is well-known that a hypersurface M n isometrically immersed into Rn+1 is locally congruent to a standard
sphere if and only if every geodesic of M is a circle of positive curvature in Rn+1 .
But we remark that there does not exist a real hypersurface M 2n−1 all of whose
geodesics are circles in CP n . The main purpose of section 2 is to give a characterization of all homogeneous real hypersurfaces M 2n−1 in CP n by observing theextrinsic shape of geodesics of M in CP n (see Theorem 5).
In section 3, we restrict ourselves on geodesics of geodesic spheres in CP n .
Geodesic spheres in CP n , which are nothing but homogeneous real hypersurfacesof type A1 , are nice objects in intrinsic geometry as well as extrinsic geometry(that is, submanifold theory).
From the view point of intrinsic geometry, we know that geodesic spheres ofsuﬃciently large radius in CP n are examples of Berger spheres. Namely, these
spheres are homogeneous Riemannian manifolds which are diﬀeomorphic to asphere, whose sectional curvatures lie in the interval [δK, K] for some δ ∈ (0, 19 ),and which have closed geodesics of length less than √2πK . These show that odddimensional version of Klingenberg’s lemma does not hold (for details, see [18]).
From the view point of submanifold theory, they are the simplest real hypersurfaces. In CP n (n = 3), geodesic spheres are the only examples of real
hypersurfaces with at most two distinct principal curvatures at its each point(see [6]). In this context it is natural to study geodesics of these real hypersurfaces. The main purpose of section 3 is to investigate the length spectrum of
geodesic spheres in CP n (see Theorems 7, 8, 9, 10). In this section we supposethat a complex projective space CP n is furnished with the standard metric of

GEOMETRY OF A COMPLEX PROJECTIVE SPACE

33

constant holomorphic sectional curvatute 4.
In section 4, motivated by the discussion in section 3, we characterize realand complex space forms by observing the extrinsic shape of geodesics on their
geodesic spheres.
Real space forms are Riemannian manifolds of constant curvature, which
are locally isometric to either one of standard spheres, Euclidean spaces or real
hyperbolic spaces. In a real space form M , a geodesic sphere Gm (r) = {p ∈M | d(p, m) = r} with center m and radius r is a totally umbilic but not totally
geodesic hypersurface (with parallel second fundamental form). Here, d denotes
the distance function induced by Riemannian metric h , i on M . This fact tells
us that every geodesic on Gm (r) is a circle of positive curvature in the ambientmanifold M .
Next we consider geodesic spheres in complex space forms. Complex space
forms are Kähler manifolds of constant holomorphic sectional curvature. It is
well-known that these are locally complex analytically isometric to either oneof complex projective spaces, complex Euclidean spaces or complex hyperbolic
spaces. For a geodesic sphere Gm (r) of suﬃciently small radius r in these complex
space forms M , we know that both of all geodesics orthogonal to ξ on Gm (r) andall integral curves of ξ are circles of positive curvature in M , where ξ is thecharacteristic vector ﬁeld of Gm (r) in M (for details, see [14]).
Along these contexts we shall give some characterizations of real and complex
space forms (see Theorems 11, 12 and 13).
In this paper we study Riemannian manifolds without boundary.
1. Geometric properties of circles in CP n .
First of all we shall recall the Frenet formula for a smooth Frenet curve in a
Riemannian manifold M with Riemannian metric h , i. A smooth curve γ = γ(s)parametrized by its arclength s is called a Frenet curve of proper order d if there
exist orthonormal frame ﬁelds {V1 = γ̇, · · · , Vd } along γ and positive functions
κ1 (s), · · · , κd−1 (s) satisfying the following system of ordinary equations(1.1)∇γ̇ Vj (s) = −κj−1 (s)Vj−1 (s) + κj (s)Vj+1 (s),j = 1, · · · , d,where V0 ≡ Vd+1 ≡ 0 and ∇γ̇ denotes the covariant diﬀerentiation along γ. We
call Equation (1.1) the Frenet formula for the Frenet curve γ. The functions
κj (s)(j = 1, · · · , d − 1) and the orthonormal frame {V1 , · · · , Vd } are called thecurvatures and the Frenet frame of γ, respectively.
A Frenet curve is called a Frenet curve of order d if it is a Frenet curveof proper order r(5 d). For a Frenet curve of order d which is of proper orderr(5 d), we use the convention in (1.1) that κj ≡ 0 (r 5 j 5 d − 1) and Vj ≡0 (r +1 5 j 5 d). We call a smooth Frenet curve γ closed if there exists a nonzero
constant s0 with γ(s + s0 ) = γ(s) for every s. The minimum positive constantwith this property is called the length of γ and denoted by length(γ). For an

34

SADAHIRO MAEDA AND TOSHIAKI ADACHI

open Frenet curve, that is a Frenet curve which is not closed, we put its length
as length(γ) = ∞.
We call a smooth Fenet curve a helix when all its curvatures are constant. A
helix of order 1 is nothing but a geodesic and a helix of order 2, that is a curve
which satisﬁes ∇γ̇ V1 (s) = κV2 (s), ∇γ̇ V2 (s) = −κV1 (s), V1 (s) = γ̇(s), is called acircle of curvature κ.
Needless to say, a curve generated by a Killing vector ﬁeld is a helix. But ingeneral, the converse does not hold. From this point of view we are interested in
the following well-known result.
Proposition 1. In a complete and simply connected real space form, which isa Euclidean space Rn , a standard sphere S n or a hyperbolic space H n , a smooth
Frenet curve is a helix if and only if it is generated by a Killing vector ﬁeld inthis space.
In order to obtain the complex version of Proposition 1, we review the definition of complex torsions of Frenet curves in Kähler manifolds. Let M be ann-dimensional Kähler manifold with complex structure J and Riemannian metrich , i. For a Frenet curve γ = γ(s) in M of order d(5 2n) with the associatedFrenet frame {V1 , · · · , Vd }, we set τij (s) = hVi (s), JVj (s)i for 1 5 i < j 5 d andcall them its complex torsions. In the study of Frenet curves in a Kähler manifold their complex torsions play an important role. The following is the complex
version of Proposition 1 (see [15]).
Proposition 2. In a complete and simply connected complex space form, which
is a complex Euclidean space Cn , a complex projective space CP n or a complex hyperbolic space CH n , a smooth Frenet curve is generated by a holomorphic
Killing vector ﬁeld in this space if and only if all its curvatures and all its complex
torsions are constant functions.
In a Kähler manifold, we call a smooth curve γ a holomorphic helix whenboth of all curvatures and all complex torsions of γ are constant.
Let γ = γ(s) be a circle in a Kähler manifold M (with complex structure J)
satisfying the equations ∇γ̇ γ̇ = κV2 (s), ∇γ̇ V2 (s) = −κγ̇.
Then,∇γ̇ hγ̇, JV2 (s)i = h∇γ̇ γ̇, JV2 (s)i + hγ̇, J∇γ̇ V2 (s)i= κ · hV2 (s), JV2 (s)i − κ · hγ̇, J γ̇i = 0.
Hence the only complex torsion τ12 , say τ (−1 5 τ 5 1), of each circle in a Kähler
manifold M is constant. Circles of complex torsion ±1 are called holomorphic
circles and circles of null complex torsion are called totally real circles. As an
immediate consequence of Proposition 2 we can see that in a complete and simply
connected complex space form M , every circle of curvature κ(> 0) is generated bya Killing vector ﬁeld of M . In order to get rid of the inﬂuence of the action of thefull isometry group, we say that two smooth curves γ1 and γ2 are congruent each

GEOMETRY OF A COMPLEX PROJECTIVE SPACE

35

other if there exist an isometry ϕ and a constant s0 with γ2 (s) = ϕ ◦ γ1 (s + s0 )for every s.
The congruence theorem for circles in a non-ﬂat complex space form M isstated as follows (see Theorem 5.1 in [15]):
Proposition 3. Two circles in a complete and simply connected non-ﬂat complex
space form M (c), that is M (c) = CP n (c) or CH n (c), are congruent if and only ifthey have the same curvatures and the same absolute values of complex torsion.
Therefore the moduli space of all congruency classes of circles on a non-ﬂat
complex space form M (c) is bijective to the set [0, ∞) × [0, 1]/ ∼, where (κ, τ )and (κ0 , τ 0 ) are equivalent if and only if (κ, τ ) = (κ0 , τ 0 ) in case of κκ0 6= 0 or
κ = κ0 = 0 in case of κκ0 = 0.
We are now in a position to study circles in CP n (c) (for details, see [2,3]).
We call a smooth curve simple if it does not have self-intersection points. Moreprecisely, an open curve σ is called simple if σ(s1 ) 6= σ(s2 ) for every s1 , s2 (s1 6=
s2 ), and a closed curve σ is called simple if σ(s1 ) 6= σ(s2 ) for every s1 , s2 (0 5s1 < s2 < length(σ)).
Theorem 1. Let γ be a circle of curvature κ and complex torsion τ in a complex
projective space CP n (c) of holomorphic sectional curvature c. Then the following
hold:4π(1) When τ = 0, γ is a simple closed curve whose length is √4k
.
2 +c
2π
(2) When τ = ±1, γ is a simple closed curve whose length is √k2 +c .
(3) When τ 6= 0, ±1, we denote by a, b and d(a < b < d) the nonzero solutionsfor√cλ3 − (4k 2 + c)λ + 2 cκτ = 0.
Then we ﬁnd the following:(i) If one of the three ratios a/b, b/d and d/a is rational, γ is a simple closed
4π
4π
curve. Its length is the least common multiple of √c(b−a)
and √c(d−a).
(ii) If each of the three ratios a/b, b/d and d/a is irrational, γ is a simple
open curve.
Theorem 1 tells us that there exist inﬁnitely many congruency classes of open
circles in CP n (c). In the following we pay attention to closed circles in CP n (c).
Theorem 2.
(1) Let γ1 and γ2 be closed circles with common length
l in CP n (c). Supposeqthat l satisﬁes the inequalities √2c π < l 5 43ncongruent by an isometryq (c).
³ of CPi(2) For each positive l 6∈

2
√
π, 43
c

5
cπ

5
c π.

Then γ1 and γ2 are, there exist at least two congruency

classes of closed circles in CP n (c) whose length is l.
We consider lengths of closed circles of the ﬁxed curvature κ.

36

SADAHIRO MAEDA AND TOSHIAKI ADACHI

Proposition 4.
(1) For each κ(> 0), the set of length(γ) of all closed circles of curvature κin CP n (c) is an unbounded discrete subset of the real positive line (0, ∞).
(2) The bottom of this unbounded subset is √κ2π
. The holomorphic circle of
2 +ccurvature κ is the only example of a closed circle of curvature κ whose
length is √κ2π
.
2 +c
4π
(3) The second lowest element of this unbounded subset is √4κ
. The totally
2 +creal circle of curvature κ is the only example of a closed circle of curvature
4π
κ whose length is √4κ
.
2 +c
Proposition 4 shows that holomorphic circles and totally real circles are niceexamples in the class of closed circles in CP n (c). These two closed circles areplane curves in CP n (c). Namely, they are lying on some real 2-dimensional
totally geodesic submanifolds of CP n (c). In fact, every holomorphic circle in
CP n (c) lies on CP 1 (c) which is a holomorphic totally geodesic submanifold of
CP n (c), and every totally real circle in CP n (c) lies on RP 2 ( 4c ) of curvature 4cwhich is a totally real totally geodesic submanifold of CP n (c). We remark thatevery circles in CP n (c) lies on a holomorphic totally geodesic CP 2 (c).
We shall provide characterizations of holomorphic circles and totally realcircles in CP n (c) from the viewpoint of submanifold theory. Let f0 be an isometricn+1
minimal imbedding of CP n (c) into a sphere S n(n+2)−1 ( n+12n c) of curvature 2n cand ι be a totally umbilic imbedding of S n(n+2)−1 ( n+12n c) into Euclidean spacen(n+2)R. Put f = ι ◦ f0 . Then f is an isometric parallel imbedding of CP n (c)n(n+2)
into R. It is well-known that the imbedding f is the only parallel full
imbedding of CP n (c) into a Euclidean space RN . This imbedding f has many
nice properties. For example, the imbedding f has a property that for eachgeodesic γ of CP n (c) the curve f ◦ γ is a circle in Rn(n+2) .
By using this imbedding f , we shall characterize holomorphic circles andtotally real circles in CP n (c) (cf. [7]).
Theorem 3. Let γ be a curve in CP n (c). Then γ is a geodesic or a holomorphic
circle if and only if the curve f ◦ γ is a circle in Rn(n+2) , where f is a parallel
imbedding of CP n (c) into Rn(n+2) .
Theorem 3 is a generalization of the fact that a curve γ on S 2 (c) is a circle,
that is γ is a great circle or a small circle of S 2 (c), if and only if γ is a circle in
R3 . By virtue of Theorem 3 we obtain a characterization of geodesics of CP n (c)from the viewpoint of submanifold theory:
Theorem 3’. Let γ be a curve in CP n (c). Then γ is a geodesic if and only if√
the curve f ◦ γ is a circle of curvature c in Rn(n+2) .
We establish a characterization of totally real circles of CP n (c).

GEOMETRY OF A COMPLEX PROJECTIVE SPACE

37

Theorem 4. Let γ be a curve in CP n (c). Then γ is a totally real circle if andonly if γ lies on totally real totally geodesic RP n ( 4c ) of CP n (c) and the curve f ◦γ
is a helix of proper order 4 in Rn(n+2) .
If we omit the condition that γ lies on RP n ( 4c ), Theorem 4 is not true
(for details, see [7]). The statement of a characterization of other circles γ withcomplex torsion τ (−1 < τ 6= 0 < 1) in CP n (c) is more complicated (see Theorem
6 in page 182 in [7]).
Finally we remark that in CP n (c) even if closed circles have the same curvature κ and the same length l,√they are not necessarily congruent. For example,√
and γ2 be alet γ1 be a circle of curvature 42c and complex torsion τ = 5595698
559√

circle of curvature

2c
4

12502
√
and complex torsion τ = 559. These two circles are
559
√

√

1118
closed circles of the same curvature 42c and the same length 4 3√π. But theyc
are not congruent each other (see [2]).

2. Characterizations of homogeneous real hypersurfaces in CP n .
Let M be a real hypersurface of CP n (of constant holomorphic sectional
curvature 4) and let N be a unit normal vector ﬁeld on M . The Riemanniane on CP n and ∇ on M are related byconnections ∇e X Y = ∇X Y + hAX, Y iN∇ande X N = −AX,∇

where h , i denotes the Riemmanian metric of M induced from the Fubini-Study
metric of CP n and A is the shape operator of M in CP n . Eigenvalues andeigenvectors of the shape operator A are called principal curvatures and principal
curvature vectors, respectively. It is known that M admits an almost contact
metric structure (φ, ξ, η, h , i) induced from the Kähler structure of CP n , which
satisﬁes
φ2 = −I + η ⊗ ξ, η(ξ) = 1
and
hφX, φY i = hX, Y i − η(X)η(Y ).
We recall the following ([17]).
Theorem A. Let M be a homogeneous real hypersurface of CP n . Then M is a
tube of radius r over the following Kähler submanifolds:(A1 ) hyperplane CP n−1 , where 0 < r < π2 ,(A2 ) totally geodesic CP k (1 5 k 5 n − 2), where 0 < r < π2(B) complex hyperquadric CQn−1 , where 0 < r < π4 ,n−1(C) CP 1 × CP 2 , where 0 < r < π4 and n(= 5) is odd,(D) complex Grassmann CG2,5 , where 0 < r < π4 and n = 9,
(E) Hermitian symmetric space SO(10)/U (5), where 0 < r < π4 , and n = 15.

38

SADAHIRO MAEDA AND TOSHIAKI ADACHIThe numbers of distinct principal curvatures of homogeneous real hypersurfaces are 2, 3, 3, 5, 5, 5, respectively.
In introduction we remark that there exist no real hypersurfaces all of whose
geodesics are circles in CP n . However, for each homogeneous real hypersurface
M by taking orthonormal vectors v1 , · · · , v2n−2 orthogonal to ξ at each point p of
M in such a way that v1 , · · · , v2n−2 are principal curvature vectors, all geodesics
γi = γi (s) on M with γi (0) = p and γ̇i (0) = vi (1 5 i 5 2n − 2) are circles in
CP n with positive curvature. Considering the converse of this fact, we establish
the following characterization of all homogeneous real hypersurfaces in CP n (cf.
[1]).
Theorem 5. Let M be a connected real hypersurface of CP n . Then M is locallycongruent to a homogeneous real hypersurface if and only if there exist orthonormal vectors v1 , · · · , v2n−2 orthogonal to ξ at each point p of M such that all
geodesics γi = γi (s) on M with γi (0) = p and γ̇i (0) = vi (1 5 i 5 2n − 2) arecircles in CP n with positive curvature.
In the hypothesis of Theorem 5 we do not need to suppose that we takethe vectors {v1 , · · · , v2n−2 } as a local ﬁeld of orthonormal frames in M . But,
for every homogeneous real hypersurface M in CP n , we can take a local ﬁeld of
orthonormal frames in M satisfying the hypothesis of Theorem 5. Every circle
in Theorem 5 is a totally real circle in CP n . We note that for any homogeneous
real hypersurface M , at each point p of M the geodesic γ = γ(s) with γ(0) = p
and γ̇(0) = ξ is a holomorphic circle of CP n .
In the following, we pay particular attention to geodesic spheres in CP n (4). It
is known that each geodesic sphere of radius r (0 < r < π2 ) in CP n (4) is congruent
to a tube of radius π2 − r around a totally geodesic complex hyperplane CP n−1 (4)in CP n (4), which is the simplest homogeneous real hypersurface of CP n . The
following is a direct consequence of Theorem 5 (see [14]).
Theorem 6. Let M be a connected real hypersurface of CP n . Then M is locallycongruent to a geodesic sphere of radius r (0 < r < π2 ) if and only if there exist
orthonormal vectors v1 , · · · , v2n−2 orthogonal to ξ at each point p of M such thatall geodesics of M through p in the direction vi + vj (1 5 i 5 j 5 2n − 2) arecircles in CP n with positive curvature.
In the next section we shall investigate the extrinsic shape of every geodesicof geodesic spheres in CP n (4) (for details, see [4]).
3. Extrinsic shape of geodesics of geodesic spheres.
Unless otherwise stated we here adopt the same terminology as that of thepreceding section. Let M be a geodesic sphere of radius r in CP n (4). Then the
shape operator A of M in CP n (4) is expressed as:
Aξ = (2 cot 2r)ξand Au = (cot r)uGEOMETRY OF A COMPLEX PROJECTIVE SPACE

39

for every tangent vector u ∈ T M orthogonal to ξ. Moreover, this real hypersurface M satisﬁes the following (cf. [16]).
1) The structure tensor φ and the shape operator A of M in CP n (4) arecommutative: φA = Aφ.
2) The covariant derivative of the shape operator A satisﬁes(∇X A)Y = −{hφX, Y iξ + η(Y )φX}.
Let ι be an isometric imbedding of a geodesic sphere M into CP n (4). We shallshow that for every geodesic γ on M the curve ι ◦ γ in CP n (4) is a helix of order
4. Note that hγ̇(s), ξi is constant along γ. Indeed,∇γ̇ hγ̇(s), ξi = hγ̇(s), φAγ̇i = hγ̇, Aφγ̇i = −hφAγ̇, γ̇i = 0.
We shall call this constant the structure torsion of γ and denote by sin θ with
0 5 |θ| 5 π2 . By direct computation we obtain the following:
Proposition 5. Let M be a geodesic sphere of radius r (0 < r < π2 ) in CP n (4).
We denote by ι an isometric imbedding of M into CP n (4). Then the extrinsic
shape ι ◦ γ of a geodesic γ on M is as follows:(1) Suppose the radius r satisﬁes π4 5 r < π2 . If the structure torsion of γ is
± cot r, the curve ι ◦ γ is a geodesic.
(2) When r 6= π4 , if the structure torsion of γ is ±1( i.e. γ̇ = ±ξ), thecurve ι ◦ γ is a circle of curvature 2| cot 2r| and of complex torsion ∓1 in
CP n (4). This circle lies on a totally geodesic CP 1 (4).
(3) If γ has null structure torsion ( i.e. γ̇ is orthogonal to ξ), the curve ι ◦ γ
is a circle of curvature cot r and null complex torsion in CP n (4). This
circle lies on a totally geodesic RP 2 (1).
(4) Generally, if the structure torsion of γ is of the form sin θ (0 < |θ| <π
2 , sin θ 6= ± cot r), then the curve ι ◦ γ is a holomorphic helix of properorder 4 whose curvatures are described as
κ1 = | cot r − tan r · sin2 θ|, κ2 = tan r · | sin θ| cos θ, κ3 = cot r.
Its complex torsions are described as½− sin θ,if cot r − tan r · sin2 θ > 0,
τ12 =sin θ,
if cot r − tan r · sin2 θ < 0,½
−sgn(sin θ) cos θ,
if cot r − tan r · sin2 θ > 0,
τ14 =sgn(sin θ) cos θ,
if cot r − tan r · sin2 θ < 0,
τ23 = sgn(sin θ) cos θ, τ34 = sin θ, τ13 = τ24 = 0,
where sgn(a) denotes the signature of a real number a. This helix ι ◦ γlies on a totally geodesic CP 2 (4).

40

SADAHIRO MAEDA AND TOSHIAKI ADACHI

It follows from Propositions 2 and 5 that every geodesic on a geodesic spherein CP n is generated by a Killing vector ﬁeld on CP n as a curve in CP n . Thus
we have
Corollary 1. Every geodesic on a geodesic sphere in a complex projective space
is a simple curve.
The fact that the isometric imbedding ι of a geodesic sphere M into CP n
is an equivariant map, together with Propositions 2 and 5, shows the following
congruence theorem about geodesics on M :Proposition 6. On a geodesic sphere M in a complex projective space, twogeodesics are congruent with respect the isometry group of M if and only if theabsolute values of their structure torsion coincide.
We are now in a position to study length of closed geodesics on a geodesic
sphere in a complex projective space. Let Π : S 2n+1 (1) → CP n (4) denote the
Hopf ﬁbration of a unit sphere. For a smooth curve σ on CP n a smooth curve σ̃˙˙is called a horizontal lift of σ if σ̃(s)is a horizontal vector and dΠ(σ̃(s))= σ̇(s)for all s. Our idea lies on considering a horizontal lift of a holomorphic helix ι ◦ γ
for every geodesic γ on a geodesic sphere. The following elementary lemma isuseful in our argument.
Lemma 1. Let σ be a smooth simple curve on CP n (4). Suppose a horizontal liftσ̃ of σ on S 2n+1 (1) is represented as
√

√

√

√

σ̃(s) = Ae −1as + Be −1bs + Ce −1cs + De −1ds ,which is a curve on Cn+1 with non-zero vectors A, B, C, D ∈ Cn+1 and mutually
distinct real numbers a, b, c, d which satisfy a + b + c + d = 0 and a 6= 0. Thenσ is closed if and only if all the ratios b/a, c/a, d/a are rational. In this case, its
length isµ¶
1
11length(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, . . . }.
By virtue of this lemma we establish the following:
Theorem 7. Let γ be a geodesic on a geodesic sphere M of radius r (0 < r <π/2) in CP n of holomorphic sectional curvature 4.
(1) If the structure torsion of γ is ±1, then γ is closed and its length isπ sin 2r.
(2) If γ has null structure torsion, then γ is also closed and its length is
2π sin r.

GEOMETRY OF A COMPLEX PROJECTIVE SPACE

41(3) When the structure torsion of γ is of the form sin θ (0 < |θ| < π2 ), thenit is closed if and only if±qpsin θ =
sin r p2 tan2 r + q 2
with some relatively prime positive integers p and q with q < p tan2 r. In
this case, its length is( p
2π p2 sin2 r + q 2 cos2 r, if pq is evenplength(γ) =π p2 sin2 r + q 2 cos2 r,
if pq is odd.
Remark. When the structure torsion is not equal to ±1, every horizontal lift γ̂of ι ◦ γ for a geodesic γ on M is closed if and only if γ is closed.
When we study length spectrum of geodesics on a Riemannian manifold N , inorder to avoid the inﬂuence of the action of the isometry group of N , we consider
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 deﬁne the length spectrum LN : Geod(N ) →R ∪ {∞} of N by LN ([γ]) = length(γ), where [γ] denotes the congruency classcontaining 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 (1)) = {2π}.
As a direct consequence of Theorem 7, for a geodesic sphere M of radius r
in CP n (4), we can see that[
Lspec(M ) = {π sin 2r} {2π sin r}¯¯ p and q are relatively prime [½ q¯
2π p2 sin2 r + q 2 cos2 r ¯¯ positive integers which satisfy¯ pq is even and q < p tan2 r ¯¯ p and q are relatively prime [½ q¯
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 8. On a geodesic sphere M in CP n , there exist inﬁnitely many congruency classes of closed geodesics. Moreover the length spectrum Lspec(M ) of
M is an unbounded discrete subset in the real line R.
For a length spectrum λ ∈ Lspec(N ) we call the cardinality mN (λ) of theset L−1
N (λ) the multiplicity of λ. When the multiplicity of a length spectrum is
1 we say it is simple. Clearly for a geodesic sphere M in a complex projective
space, we see by the expression of Lspec(M ) that mM (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.

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SADAHIRO MAEDA AND TOSHIAKI ADACHI

Proposition 7. Let M be a geodesic sphere of radius r (0 < r < π/2) in CP nof holomorphic sectional curvature 4.
(1) The ﬁrst length spectrum of M is π sin 2r, which is the length of geodesics
with structure torsion ±1. It is simple.
(2) The second length spectrum of M is also simple. When 0 < r 5 π/4,
it is 2π sin r, which is the length of geodesics with null structure torsion.
When π/4 < r < π/2, it is π, which is the length of geodesics withstructure torsion ± cot r.
(3) The third length spectrum is also simple. When π/4 < r < π/2, it is
2π sin r,√which is the length √of geodesics with null structure torsion.
When 2m − 1 q
5 cot r < 2m + 1 (m = 1, 2, . . . ), in particular, 0 <r 5 π/4, it is π

4m(m + 1) sin2 r + 1, which is the length of geodesicspwith structure torsion ±1/(sin r (2m + 1)2 tan2 r + 1).

Since the sectional curvature of a geodesic sphere M of radius r in CP n (4)22lies in the√ interval [cot r, 4 + cot r], the ﬁrst length spectrum of M is smaller
than 2π/ 4 + cot2 r if tan2 r > 2. Hence M is an example of a Bereger sphere, as
was pointed out in [18]. But for other length spectrum we ﬁnd that the following
Klingenberg’s lemma holds:
Corollary 2. Let M be a geodesic sphere of radius r in CP n of holomorphic
sectional curvature 4. Except geodesics
with structure torsion ±1, every geodesic√
γ on M satisﬁes length(γ) > 2π/ 4 + cot2 r.
Length spectrum is of course not necessarily simple. For example when M
is a geodesic sphere of radius π/4 in CP n , we have
n √
√
√
√
√
√
√
Lspec(M ) = π, 2π, 10π, 2 5π, 26π, 34π, 50π, 2 13π,
o
√
√
√
√
√
√
√
58π, 2 17π, 74π, 82π, 10π, 106π, 2 29π, 130π, . . .
√and the multiplicityofstructure√ 130π is two; it is the length of geodesics of √√torsions 3/ 65 and 7/ 65. Every spectrum which is smaller than 130π is
simple. Our aim here is to establish the following:
Theorem 9. Let M be a geodesic sphere of radius r (0 < r < π/2) in CP n ofholomorphic sectional curvature 4.
(1) If tan2 r is irrational, every length spectrum of M is simple.
(2) If tan2 r is rational, the multiplicity of each length spectrum of M is ﬁnite.
But it is not uniformly bounded; lim supλ→∞ mM (λ) = ∞. In this case,the growth order of mM is not so rapid. It satisﬁes limλ→∞ λ−δ mM (λ) =0 for arbitrary positive δ.

GEOMETRY OF A COMPLEX PROJECTIVE SPACE

43

This theorem guarantees that on a geodesic sphere of radius r with irrationaltan2 r in a complex projective space, two closed geodesics are congruent if andonly if they have the same length. On the other hand, if tan2 r is rational, thistheorem shows that we can not classify congruency classes of geodesics only by
their length.
Now we make mention of the growth of the number of congruency classes
of geodesics with respect to their length spectrum for a geodesic sphere in acomplex projective space. For a Riemannian manifold N we denote by nN (λ)
the cardinality of the set {[γ] ∈ Geod(N ) | LN ([γ]) 5 λ}.
Theorem 10. For a geodesic sphere M of radius r (0 < r < π/2) in CP n ofholomorphic sectional curvature 4 we havenM (λ)
3r
=
.
λ→∞
λ2
π 4 sin 2r
lim

4. Characterizations of space forms.
Our discussion is based on the expansion for the second fundamental formof geodesic spheres due to Chen and Vanheche ([8]). For the sake of reader’sconvenience, we here place their result.
Let M be a Riemannian manifold of dimension greater than 2 with Riemannian metric h , i. We denote by Gm (r) a geodesic sphere with center m andradius r in M , and by Am,r the shape operator of Gm (r) in M with respect to theoutward unit normal vector ﬁeld. The following is a key lemma in this section.
Lemma 2 (Theorem 3.1 in [8]). For non-zero tangent vectors v, w ∈ Tm M at apoint m ∈ M , we choose a unit tangent vector u ∈ Tm M orthogonal to both v andw. We respectively denote by vr , wr ∈ Texpm (ru) M the parallel displacements ofv and w along the geodesic segment expm (su), 0 5 s 5 r. Then for suﬃciently
small r we have
hAm,r vr , wr i =1
rhv, wi + hR(u, v)w, ui + O(r2 ),r3

where R is the curvature tensor of M .
In this section we characterize real space forms in terms of the extrinsic shapeof geodesics on geodesic spheres. It is well-known that for a real space form, every
geodesics on each geodesic sphere is a circle of positive curvature. We know theconverse holds.
Theorem 11. Let M be a Riemannian manifold with dim M = 3. Then thefollowing conditions are equivalent.
(1) M is a real space form.

44

SADAHIRO MAEDA AND TOSHIAKI ADACHI(2) At any point m ∈ M , every geodesic on a geodesic sphere Gm (r) of M isa circle of positive curvature in M for each suﬃciently small r.
(3) At any point m ∈ M , every geodesic sphere Gm (r) of M is totally umbilic
in M for each suﬃciently small r.
The following is an improvement of Theorem 11. The same discussion as in[14], together with Lemma 2 and Theorem 11, yields
Theorem 11’. A Riemannian manifold M with dim M = n = 3 is a real space
form if and only if at any point m ∈ M for each suﬃciently small geodesic sphereGm (r) of M , there exist orthonormal vectors v1 , v2 , . . . , vn−1 at each point p ofGm (r) such that all geodesics of Gm (r) through p in the direction vi + vj (1 5i 5 j 5 n − 1) are circles of positive curvature in the ambient manifold M .
Finally we study geodesic spheres in a Kähler manifold M with complex
structure J. Let N be the outward unit normal vector ﬁeld on Gm (r). Since
Gm (r) is a real hypersurface in M , it admits an almost contact metric structure(φ, ξ, η, h , i) induced from the Kähler structure J of M .
Paying attention on the characteristic vector ﬁeld ξ of suﬃciently small geodesic spheres Gm (r), we obtain the following characterization of complex space
forms in the class of Kähler manifolds.
Theorem 12. Let M be a complex n(= 2)-dimensional Kähler manifold. Thenthe following conditions are equivalent each other.
(1) M is a complex space form.
(2) At any point m ∈ M , each suﬃciently small geodesic sphere Gm (r) of M
is a Hopf hypersurface of M , that is, the characteristic vector ξ of Gm (r)is a principal curvature vector in M at each point p ∈ Gm (r).
(3) At any point m ∈ M , for each suﬃciently small geodesic sphere Gm (r),every integral curve of the vector ﬁeld ξ is a geodesic on Gm (r).
(4) At any point m ∈ M , for each suﬃciently small geodesic sphere Gm (r),the geodesic on Gm (r) through p in the direction of the vector ξ is a circleof positive curvature in M at every point p ∈ Gm (r).
Next we give attention to the extrinsic shape of geodesics on Gm (r) orthogonal to the characteristic vector ξ. The following is a complex version for Theorem
11 in some sense.
Theorem 13. A complex n(= 2)-dimensional Kähler manifold M is a complex
space form if and only if at an arbitrary point m ∈ M , for any geodesic sphereGm (r) of suﬃciently small radius r, every geodesic through any ﬁxed point p ofGm (r), which is orthogonal to the vector ξ at the point p, is a circle of positive
curvature in the ambient manifold M .
As an immediate consequence of our argument we ﬁnd the following result
which corresponds to Theorem 11’.

GEOMETRY OF A COMPLEX PROJECTIVE SPACE

45

Theorem 13’. A complex n(= 2)-dimensional Kähler manifold M is a complex
space form if and only if at an arbitrary point m ∈ M , for any geodesic sphereGm (r) of suﬃciently small radius r, there exist orthonormal vectors v1 , v2 , . . . , v2n−2
orthogonal to the characteristic vector ξ at each point p of Gm (r) such that all
geodesics on Gm (r) through p in the direction vi + vj (1 5 i 5 j 5 2n − 2) arecircles of positive curvature in the ambient manifold M .

References[1] T. Adachi, M. Kimura and S. Maeda, A characterization of all homogeneous real hypersurfaces in a complex projective space by observing the extrinsic shape of geodesics, to appear
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Dedicata 30 (1989), 93–114.
[16] M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer.
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[18] A. Weinstein, Distance spheres in complex projective spaces, Proc. Amer. Math. Soc. 39(1973), 649–650.

Department of Mathematics, Shimane University, Matsue 690-8504, Japan

46

SADAHIRO MAEDA AND TOSHIAKI ADACHI

E-mail address: maeda@math.shimane-u.ac.jp
Department of Mathematics, Nagoya Institute of Technology, Gokiso, Nagoya
466-8555, Japan
E-mail address: adachi@math.kyy.nitech.ac.jp