Mem. Gra. Sci. Eng. Shimane Univ.
Series B: Mathematics
51 (2018), pp. 1–5

GEOMETRY OF GEODESIC SPHERES IN A COMPLEXPROJECTIVE SPACE IN TERMS OF THEIR GEODESICSSADAHIRO MAEDA
Communicated by Toshihiro Nakanishi(Received: March 9, 2017)

Abstract. This paper is the survey of joint works with K. Ogiue ([7]) and
B.H. Kim, I.B. Kim ([5]). Geodesic spheres G(r) are fundamental examples of(real) hypersurfaces in a Riemannian manifold. In this paper, as an ambient
space we take an n−dimensional complex projective space CP n (c), n ≧ 2 ofconstant holomorphic sectional curvature c(> 0). By√ observing geodesics on
G(r) in CP n (c) we characterize all G(r) (0 < r < π/ c ) (Theorems 1 and 2)
and some G(r) which are called Berger spheres (Theorem 3).

1. Introduction
In diﬀerential geometry, there are two geometries. One is intrinsic geometry andthe other is extrinsic geometry (submanifold geometry). Among real hypersurfaces
M 2n−1 isometrically immersed into CP n (c) , n √≧ 2, we pay particular attention togeodesic spheres G(r) of radius r (0 < r < π/ c ). The class of geodesic spheres
G(r) in CP n (c) is an abundant class which gives fruitful results in both of intrinsic
geometry and extrinsic geometry.
In intrinsic geometry, we recall a well-known result due to W. Klingenberg : 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. Thenthe length ℓ of every closed geodesic on M satisfies ℓ ≧ 2π/ L (see [4]). However
the odd dimensional version of the above result does not hold (cf. [11]). Indeed,√
let G(r) bea (2n − 1)-dimensional geodesic sphere of radius r (0 < r < π/ c )√with tan2 ( c r/2) > 2 in CP n (c). Then in√this case there exists a closed geodesicon G(r) whose length is shorter than 2π/ L , where L is the maximal sectional
curvature of G(r). √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 ≦c + (c/4) cot2 ( c r/2)(= L) at its each point. In this paper, geodesic spheres of2010 Mathematics Subject Classification. 53B25, 53C40.
Key words and phrases. geodesic spheres, complex projective space, geodesics, circles, Berger
spheres, Sasakian space forms.
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S. MAEDA√√
radius r (0 < r < π/ c ) with tan2 ( c r/2) > 2 in CP n (c), n ≧ 2 are called
Berger spheres.
In extrinsic geometry, we can say that all G(r) are the simplest examples of real
hypersurfaces in CP n (c). It is known that CP n (c) does not have a totally umbilic
real hypersurface and T. Cecil and P. Ryan ([3]) show that every geodesic sphere is
the only example of real hypersurfaces in CP n (c) (n ≧ 3) with at most two distinct
principal curvatures at its each point. This means that every G(r) is the simplest
real hypersurface of CP n (c).
The purpose of this paper is to characterize geodesic spheres in CP n (c) in extrinsic geometry (see Theorems 1, 2 and 3) and clarify geometric properties of such
hypersrfaces from the viewpoint of intrinsic geometry (see the last section).
This paper is related to a talk of the author in 2017 (October 6 ∼ 8) International
Conference on Diﬀerential Geometry and Applications, Mykonos Island, Greece.
We adopt usual notations and terminologies in the theory of real hypersurfaces
in CP n (c) (for examples, see [8, 9]).
2. circles in Riemannian geometry
The key word in our paper is a circle. We recall the definition of circles in
Riemannian geometry.
Let γ = γ(s) be a smooth real curve parametrized by its arclength s on a
Riemannian manifold M with Riemmanian connection ∇. If the curve γ satisfies
the following ordinary diﬀerential equations with some constant k(≧ 0):∇γ̇ γ̇ = kYs

and ∇γ̇ Ys = −k γ̇,where ∇γ̇ is the covariant diﬀerentiation along γ with respect to the Riemmanianconnection ∇ of M and Ys is the so-called unit principal normal vector of γ, wecall a circle of curvature k on M . We may regard a geodesic of null curvature. Byvirtue of the existence and uniqueness theorem on solutions to ordinary diﬀerential
equations we can see that at any point p of a Riemannian manifold M for eachpositive constant k and every pair {X, Y } of orthonormal vectors X, Y (∈ Tp M )there exists locally the unique circle γ of curvature k on M with initial condition
that γ(0) = p, γ̇(0) = X and Y0 = Y .
3. Main results
Since CP n (c) does not admit totally umbilic real hypersurfaces, there does notexist real hypersurfaces all of whose geodesics are mapped to circles in this ambient space (See [10] and Proposition 2 in [6]). So it is natural to investigate real
hypersurfaces some of whose geodesics are mapped to circles in CP n (c).
The following three theorems are established from this viewpoint. Theorem 3.3
is an immediate consequence of Theorem 3.2.
Theorem 3.1 ([7]). Let M 2n−1 be a real hypersurface of CP n (c), n ≧ 2 through an
isometric√ immersion. Then M is locally congruent to a geodesic sphere G(r) (0 <r < π/ c ) with respect to the full isometry group SU (n + 1) of the ambient
space CP n (c) if and only if there exist such orthonormal vectors v1 , v2 , . . . , v2n−2

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orthogonal to the characteristic vector ξp at each point p of M that all geodesics ofM through p in the direction vi + vj (1 ≦ i ≦ j ≦ 2n − 2) are circles of positive
curvature in CP n (c).
Theorem 3.2 ([7]). Let M 2n−1 be a real hypersurface of CP n (c),√n ≧ 2. Then M
is locally congruent to either
a geodesic sphere G(r) (0 < r < π/ c ) with n ≧ 2√
or a tube of radius π/(2 c ) around a totally geodesic CP ℓ (c) (1 ≦ ℓ ≦ n − 2) withn ≧ 3 if and only if there exist such orthonormal vectors v1 , v2 , . . . , v2n−2 orthogonal
to the characteristic vector ξp at each point p of M that all geodesics of M throughp in the direction vi (1 ≦ i ≦ 2n − 2) are circles of the same positive curvature k(p)in CP n (c). Here, the function k = k(p) on M is automaticallyconstant.
When√√
M is locally congruent to G(r), k is expressed as: k√ = ( c /2) cot( c r/2), andwhen M is locally congruent to a tube of radius√ π/(2 c ) around a totally geodesicℓCP (c) (1 ≦ ℓ ≦ n − 2), k is written as: k = c /2.
Theorem 3.3 ([5]). Let M 2n−1 be a real hypersurface of CP n (c), n ≧ 2. Then M islocally√congruent to a Berger sphere, namely a geodesic sphere G(r) of radius r withtan2 ( c r/2) > 2 if and only if at each point p of M there exist such orthonormal
vectors v1 , v2 , . . . , v2n−2 orthogonal to the characteristic vector ξp at each point p of
M that all geodesics of M through p in the direction vi √(1 ≦ i ≦ 2n − 2) are circles√of the same positive curvature k(p) with k(p) < c /(2 2 ) in CP n (c).
4. Several comments on Theorems 1, 2 and 3(1) In Theorems 3.1, 3.2 and 3.3, all geodesics on G(r) are mapped to circles
lying on a totally real totally geodesic RP n (c/4) of constant sectional curvature
c/4 in the ambient space CP n (c). Note that there exist many geodesics on G(r)which do not lie on RP n (c/4) (cf. Proposition 2.1 in [2]).
√(2) In Theorem 3.2, √to distinguish geodesic spheres G(r) (0 < r < π/ c ) and atube T of radius π/(2 c ) around a totally geodesic CP ℓ (c) (1 ≦ ℓ ≦ n − 2), we
recall the notion of extrinsic geodesics. We denote by M n a Riemannian submanifn+p through an isometric immersion f . A smoothfold of a Riemannian manifold M
real curve γ = γ(s) on the submanifold M n is an extrinsic geodesic if the curve
fn+p . In this case, as a matter of coursef ◦ γ is a geodesic in the ambient space M
the curve γ is also a geodesic on the submanifold M n .
The following√ fact implies that we can distinguish G(r) and a tube T , i.e., a tubeof radius π/(2 c ) around a totally geodesic CP ℓ (c) (1 ≦ ℓ ≦ n − 2), by counting
the number of congruence classes of extrinsic geodesics (see [9]).
√Fact. (i) A geodesic sphere G(r)√(0 < r < π/(2√ c )) has no extrinsic geodesics.
(ii) A geodesic sphere G(r) (π/(2 c ) ≦ r < π/ c ) has just one extrinsic geodesic
with respect to the full isometry group I(G(r)) of G(r).
(iii) The tube T has uncountably infinite many extrinsic geodesics with respect tothe full isometry group I(T ) of T .
(3) We comment on the length of an integral curve of the characteristic vector
field ξ on a geodesic sphere G(r) in CP n (c). We note that every integral curve

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S. MAEDAγξ of the characteristic vector field ξ is a closed geodesic on G(r) with length√√e ξ ξ = √c cot(√c r)N
2π sin( c r)/ c . In fact, the curve γξ satisfies ∇ξ ξ = 0, ∇e ξ N = −√c cot(√c r)ξ with γ̇ξ = ξ, where ∇ and ∇e are the Riemannianand ∇nconnections of G(r) and CP (c), respectively. These
mean
be
√
√ that the2curve γξ can1
regarded as a small circle of positive curvature c | cot( c r)| on S (c)(= CP (c)).
Hence, the length ℓ of γξ is represented as:√√√√ℓ = 2π/ c cot2 ( c r) + c = 2π sin( c r)/ c .
We next consider an inequality:√√√√
2π sin( c r)/ c < 2π/ c + (c/4) cot2 ( c r/2),√
where c + (c/4) cot2 ( c r/2)√is the maximal sectional curvature of G(r). Solving
this inequality, we get tan2 ( c r/2) > 2.
√
(4) We review the length spectrum of every geodesic sphere√G(r) (0 < r < π/ c )(for details, see [2]). Every geodesic sphere G(r) (0 < r < π/ c ) admits countably
infinite many congruence classes of closed geodesics with respect to the full isometrygroup I(G(r)) of G(r). All integral curves of the characteristic vector field ξ arecongruent each other with respect
to I(G(r)) and the shortest closed geodesics√√(with common length 2π sin( c r)/ c ) on G(r). Furthermore, the lengths of all
closed geodesics exceptof the characteristic vector field ξ on G(r)√ integral curves√
2
are longer than 2π/ c + (c/4) cot ( c r/2).
√(5) We consider all geodesic spheres G(r) (0 < r < π/ c ) from the viewpointof contact geometry (cf. [1]). G(r) is a Sasakian manifold (with respect to thealmost contact metric structure(ϕ,√ξ, η, g) induced from the Käler structure J of√
CP n (c)) if and only if ( c /2) cot( c r/2) = 1. This Sasakian manifold M has
automatically constant ϕ-sectional curvature c + 1, so that it is a Sasakian
space
√
√
form of constant ϕ-sectional curvature c + 1. Since an inequality 1 < c /(2 2 )leads to an inequality c > 8, we find that all Sasakian space forms of constantϕ-sectional curvature c̃ with c̃ > 9 are Berger spheres.
(6) We comment on the√ sectional curvature K of Berger spheres, i.e., geodesic
spheres G(r) with tan2 ( c r/2) > 2 in CP n (c). K satisfies√ sharp inequalities2δL ≦ K ≦ L for some δ ∈ (0, 1/9), where L = c + (c/4) cot ( c r/2).
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[2] 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.
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S. Maeda: Department of Mathematics, Saga University, 1 Honzyo, Saga, 8408502, Japan
E-mail address: sayaki@ms.saga-u.ac.jp