Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
36 (2003), pp.1–9

CIRCLES AND HYPERSURFACES IN SPACE FORMS

SADAHIRO MAEDA AND TOSHIAKI ADACHI(Received: December 20, 2002 )Abstract. In this expository paper, we study hypersurfaces of space forms byinvestigating circles on their hypersurfaces.

1. Introduction
In this paper we discuss hypersurfaces in a real space form, and real andcomplex hypersurfaces in a complex space form by observing the extrinsic shapeof circles of these hypersurfaces. An n-dimensional real space form M n (c) isa Riemannian manifold of constant curvature c, which is locally isometric toeither a standard sphere S n (c), a Euclidean space Rn or a real hyperbolic space
H n (c), according as c is positive, zero or negative. A complex n-dimensional
complex space form Mn (c) is a Kähler manifold of constant holomorphic sectional
curvature c, which is locally complex analytically isometric to either a complex
projective space CP n (c), a complex Euclidean space Cn or a complex hyperbolic
space CH n (c), according as c is positive, zero or negative. In this paper we meanby space form either a real space form or a complex space form.
A smooth curve γ on a Riemannian manifold M parametrized by its arclengthis called a circle if it satisfies ∇γ̇ ∇γ̇ γ̇ = −κ2 γ̇ with some nonnegative constant κ,
where ∇γ̇ denotes the covariant differentiation along γ with respect to the Riemannian connection ∇ on M . This condition is equivalent to the condition that
there exist a nonnegative constant κ and a field of unit vectors Y along this curve2000 Mathematics Subject Classification. Primary 53B25, Secondary 53C40.
Key words and phrases. hypersurfaces, circles, first curvature, space forms, real space forms,
complex space forms, Kähler submanifolds, totally geodesic, Veronese embeddings.
The first author partially supported by Grant-in-Aid for Scientific Research (C)(No. 14540080), Ministry of Education, Science, Sports and Culture.
The second author partially supported by Grant-in-Aid for Scientific Research (C)(No. 14540075), Ministry of Education, Science, Sports and Culture .
The first author talked a part of this paper in the conference of Analysis, Manifolds andMechanics during February 5-7, 2003 to celebrate the 90th Birth Anniversary of Professor
Manindra Chandra Chaki.
12

S. MAEDA AND T. ADACHI

which satisfy the following differential equations: ∇γ̇ γ̇ = κY and ∇γ̇ Y = −κγ̇.
We call the constant κ curvature of γ. As we have κ = k∇γ̇ γ̇k, we treat geodesics
as circles of null curvature. For given a point x ∈ M , an orthonormal pair oftangent vectors u, v ∈ Tx M and a positive constant κ, by the existence anduniqueness theorem on solutions for ordinary differential equations we have locally a unique circle γ = γ(s) with initial condition that γ(0) = x, γ̇(0) = u and∇γ̇ γ̇(0) = κv. It is well-known that in Euclidean space a circle of positive curvature κ is nothing but a circle of radius 1/κ in the sense of Euclidean geometry.
Our study is motivated by the following fact on extrinsic spheres due to Nomizu
f withand Yano [NY]. An extrinsic sphere is a totally umbilic submanifold of M
f. Their result states that a Riemannianparallel mean curvature vector in Mnn+pfsubmanifold M of M
through an isometric immersion f is an extrinsic spherefof M if and only if there exists a positive constant κ such that for every circle γ
f.
of curvature κ on M n the curve f ◦ γ is a circle in M
The key word in this paper is the first curvature of a curve in the sense ofFrenet formula. We establish our result by paying attention to the first curvaturee γ̇ γ̇k of the curve f ◦γ for a circle γ on a hypersurface in the ambient space formk∇f(c). Here we denote by ∇e the Riemannian connection of M
f(c). We investigate
Mn−1nf (c) and real hypersurfaces M 2n−1 in a
hypersuraces M
in a real space form M
fn (c) (Theorems 1, 2, 3 and 4). We also investigate
nonflat complex space form M
fn (c) (Theorems 5 and 6).
complex hypersurfaces Mn−1 in a complex space form M
At the end of this paper, as an application of our discussion on hypersurfaces wecharacterize some Kähler embeddings of complex projective spaces into complex
projective spaces which are called Veronese embeddings (Theorem 7).
2. Hypersurfaces of real space forms
We consider parallel hypersurfaces in a real space form. By the results due toFerus[F] and Takeuchi[T] a hypersurface M n−1 with parallel second fundamental
fn (c) is either totally umbilic in M
fn (c) or locallyform in a real space form M
congruent to one of the following product spaces:i) M n−1 = Rk × S n−k−1 (c1 ), 1 5 k 5 n − 2 in Rn ,ii) M n−1 = S k (c1 ) × S n−k−1 (c2 ), 1 5 k 5 n − 2 with (1/c1 ) + (1/c2 ) = 1/c
in S n (c),iii) M n−1 = H k (c1 ) × S n−k−1 (c2 ), 1 5 k 5 n − 2 with (1/c1 ) + (1/c2 ) = 1/c
in H n (c).
By investigating the first curvature of each geodesic on these parallel hypersurfaces we obtain the following well-known theorem:
fn (c) has parallel secTheorem 1. A hypersurface M n−1 of a real space form M
ond fundamental form if and only if every geodesic on M being considered as a
curve in the ambient space has constant first curvature.

CIRCLES AND HYPERSURFACES

3

Proof. In a neighborhood of each point of M we choose a unit normal vector field
fn (c). The Riemannian connections ∇e in Mfn (c) and ∇ in M are related
N in Mby the following formulas for arbitrary vector fields X, Y on M :(2.1)

e X Y = ∇X Y + hAX, Y iN ,
∇

(2.2)

e X N = −AX,∇fn (c). Suppose that every geodesic
where A is the shape operator of M in M
fn (c), constant first curvature.
γ : I → M on M has, considered as a curve in M
e γ̇ γ̇k is constant on the interval I.
This hypothesis means, by definition, that k∇e γ̇ γ̇k2 = hAγ̇, γ̇i2 , hence our hypothesis is
But from equation (2.1) it follows k∇equivalent to the constancy of hAγ̇, γ̇i on I. Therefore as ∇γ̇ γ̇ = 0, we haveh(∇γ̇ A)γ̇, γ̇i = ∇γ̇ hAγ̇, γ̇i = 0 on I. Thus we obtain h(∇X A)X, Xi = 0 for
every tangent vector X of M . By use of the Codazzi equation h(∇X A)Y, Zi =h(∇Y A)X, Zi this property shows that M has parallel second fundamental form.
By the same discussion as above we get the converse. ¤Next, we study the first curvature of each circle of positive curvature on parallel
hypersurfaces M n−1 in a real space form(cf. [M]):fn (c) through an isoTheorem 2. A hypersurface M n−1 of a real space form M
fn (c) if and only if there exists somemetric immersion f is totally umbilic in M
κ > 0 satisfying that every circle of curvature κ on M being considered as a curvein the ambient space has constant first curvature.
Proof. Suppose that for every circle γ : I → M of positive curvature κ on M thefn (c). We take acurve f ◦ γ has constant first curvature in the ambient space M
point x ∈ M and choose an arbitrary orthonormal pair of vectors u, v ∈ Tx M .
Let γ = γ(s), s ∈ I be a circle of curvature κ on the submanifold M n with initialcondition that γ(0) = x, γ̇(0) = u and ∇γ̇ γ̇(0) = κv. It follows fromp equation (2.1)
that the first curvature κe of the curve f ◦ γ is expressed as κ
e = κ2 + hAγ̇, γ̇i2 .
This equality tells us that hAγ̇, γ̇i is constant on I, so that we have(2.3) 0 =dhAγ̇, γ̇i = h(∇γ̇ A)γ̇, γ̇i + 2hAγ̇, ∇γ̇ γ̇i = h(∇γ̇ A)γ̇, γ̇i + 2κhAγ̇, Y i.
ds

Evaluating the equation (2.3) at s = 0, we get(2.4)

h(∇u A)u, ui + 2κhAu, vi = 0.

On the other hand, for another circle ρ = ρ(s) of the same curvature κ on thesubmanifold M n−1 with initial condition that ρ(0) = x, ρ̇(0) = u and ∇ρ̇ ρ̇(0) =−κv, we find(2.5)

h(∇u A)u, ui − 2κhAu, vi = 0,

4

S. MAEDA AND T. ADACHI

which corresponds to the equation (2.4). Thus, from (2.4) and (2.5) we can seethat hAu, vi = 0 for each orthonormal pair of vectors u, v at each point x of M ,fn (c).
so that the hypersurface M is totally umbilic in M
The converse is obvious from the fact that every circle on a totally umbilic subfn+p (c) is mapped to a circle in the ambient space M
fn+p (c). ¤manifold M n of M
3. Real hypersurfaces of nonflat complex space forms
It is well-known that there exist no real hypersurfaces with parallel second funfn (c), which is either a complex
damental form in a nonflat complex space form M
projective space or a complex hyperbolic space. We hence consider a bit weak
condition on real hypersurfaces in a complex space form.
fn (c) and N a unit normal
Let M 2n−1 be an orientable real hypersurface of M
fn (c). It is known that M admits an almost contact metricvector field on M in M
fn (c) which
structure (φ, ξ, η, h , i) induced from the Kähler structure J of M
satisfies
φ2 = −Id + η ⊗ ξ,

η(ξ) = 1,

andhφX, φY i = hX, Y i − η(X)η(Y ),where Id denotes the identity map of the tangent bundle T M of M . Then it is
known that the following equalities(∇X φ)Y = η(Y )AX − hAX, Y iξand∇X ξ = φAXhold. The condition that the structure vector ξ = −JN is principal is natural. It
is well-known that this condition is equivalent to the condition that every integral
curve of the vector field ξ is a geodesic on M 2n−1 . As was shown in [NR], for afn (c) (n = 2), if Aξ = αξ holds with some function
real hypersurface M 2n−1 in M
α on M then α is locally constant. In CP√n (c) each real hypersurface M lying
on a tube of constant radius r (0 < r < π/ c) around a complex submanifold of
CP n (c) satisfies this condition on ξ. In CH n (c) each real hypersurface M lying ona tube of radius r (0 < r < ∞) around a complex submanifold or around a totallyreal submanifold of CH n (c) satisfies this condition. Real hypersurfaces M 2n−1
fn (c) are calledwith the structure vector ξ as a principal curvature vector in MnHopf hypersurfaces. In CP (c) we have the following typical Hopf hypersurfaces:√(A1 ) A tube of radius r over hyperplane CP n−1 (c), where 0 < r < π/ c,(A2 ) a tube of radiusr over totally geodesic CP k (c) (1 5 k 5 n − 2), where√0 < r < π/ c.
n
In CH (c) we have the following typical Hopf hypersurfaces:(A0 ) A horosphere in CH n (c),(A1 ) a tube of radius r over CH k (c) (k = 0, n − 1), where 0 < r < ∞,(A2 ) a tube of radius r over CH k (c) (1 5 k 5 n − 2), where 0 < r < ∞.
These Hopf hypersurfaces are usually called hypersurfaces of type A. The following
theorem gives a characterization of such hypersurfaces([NR]):

CIRCLES AND HYPERSURFACES

5

Theorem A. Let M 2n−1 be a real hypersurface in a nonflat complex space formfn (c). Then the following conditions are mutually equivalent:M1) M is locally congruent to a hypersurface of type A.
2) The structure tensor φ and the shape operator A of M are commutative:
φA = Aφ.
3) h(∇X A)X, Xi = 0 on M for each X ∈ T M 2n−1 .
It follows from Theorem A and the discussion in the proof of Theorem 1 yields
the following (cf. [NR]):fn (c) is
Theorem 3. A real hypersurface M 2n−1 of a nonflat complex form Mof type A if and only if every geodesic of M being considered as a curve in theambient space has constant first curvature.
Since we have no totally umbilic real hypersurfaces in a nonflat complex space
form, the proof of Theorem 2 implies
Theorem 4. There exist no real hypersurfaces M 2n−1 in a nonflat complex space
fn (c) satisfying that for some κ > 0 every circle of curvature κ on M being
form Mconsidered as a curve in the ambient space has constant first curvature.
4. Complex hypersurfaces of complex space forms
In this section we consider complex hypersurfaces with parallel second fundafn (c). It is known that they aremental form in a complex space form M
fn (c),
i) either totally geodesic in Mii) or locally congruent to a complex quadric Qn−1 (C) in the case c >0([NT]).
By investigating the first curvature of each geodesic on these parallel hypersurfaces we obtain the following theorem:
fn (c)
Theorem 5 [MO]. A complex hypersurface Mn−1 of a complex space form M
has parallel second fundamental form if and only if every geodesic on M being
considered as a curve in the ambient space has constant first curvature.
Proof. Suppose that every geodesic γ of Mn−1 , considered as a curve in thefn (c), has constant first curvature. Then by the Gauss formulaambient space M
e X Y = ∇X Y + σ(X, Y ) we see that kσ(γ̇, γ̇)k is constant along γ. This implies∇¯ γ̇ σ)(γ̇, γ̇), σ(γ̇, γ̇)i = 0 and hence we havethat h(∇(4.1)¯ X σ)(X, X), σ(X, X)i = 0h(∇¯for every vector field X tangent to M . Here, the covariant differentiation ∇of the second fundamental form σ with respect to the connection in (tangent
bundle)⊕(normal bundle) by¯ X σ)(Y, Z) = DX (σ(Y, Z)) − σ(∇X Y, Z) − σ(Y, ∇X Z),(∇

6

S. MAEDA AND T. ADACHI

fn (c).
where D is the normal connection of Mn−1 in M
fn (c). We denote by
Let J be the complex structure of the ambient space M
the same letter J the complex structure of the hypersurface Mn−1 . Then, by
replacing X by JX in (4.1), we obtain¯ X σ)(X, X), Jσ(X, X)i = 0
h(∇

(4.2)

¯ X σ)(X, X) = 0 for all X.
for all X. It follows from (4.1) and (4.2) that (∇¯ X σ)(Y, Z) = (∇¯ Y σ)(X, Z), we know
Then, thanks to the Codazzi equation (∇that M has parallel second fundamental form. The converse is obvious by thesame discussion as above. ¤We obtain the following by studying the first curvature of each circle of positive curvature on parallel complex hypersurfaces Mn−1 in a complex space formfn (c).
M
Theorem 6 [SMA]. A complex hypersurface Mn−1 of a complex space formfn (c) is totally geodesic in M
fn (c) if and only if there exists some κ > 0 satisfying
Mthat every circle of curvature κ on M being considered as a curve in the ambient
space has constant first curvature.
fn (c) be an isometric Kähler immersion satisfying theProof. Let f : Mn−1 → M
condition on the extrinsic shape of circles. For an arbitrary orthonormal pair(u, v) of vectors at a fixed point x of M we choose a circle γ = γ(s), s ∈ I of
curvature κ on the submanifold M with initial condition that γ(0) = x, γ̇(0) = u
and ∇γ̇ γ̇(0) = κv. By the same discussion as in the proof of Theorem 2 we havedhσ(γ̇, γ̇), σ(γ̇, γ̇)i = 2hDγ̇ (σ(γ̇, γ̇)), σ(γ̇, γ̇)ids
¯ γ̇ σ)(γ̇, γ̇), σ(γ̇, γ̇)i + 4κhσ(γ̇, Y ), σ(γ̇, γ̇)i.
= 2h(∇

0=

Evaluating this equation at s = 0, we find(4.3)

¯ u σ)(u, u), σ(u, u)i + 2κhσ(u, v), σ(u, u)i = 0.
h(∇Starting with the circle γ with initial condition that γ(0) = x, γ̇(0) = u and∇γ̇ γ̇(0) = −κv, we find the equation (4.3) turns to(4.4)¯ u σ)(u, u), σ(u, u)i − 2κhσ(u, v), σ(u, u)i = 0.
h(∇Hence these two equations (4.3) and (4.4) guarantee that
(4.5)

¯ u σ)(u, u), σ(u, u)i = 0,
h(∇

(4.6)

hσ(u, u), σ(u, v)i = 0

CIRCLES AND HYPERSURFACES

7

for each orthonormal pair of vectors u, v at every point x.
The equality (4.5) tells us that our hypersurface M has parallel second fundamental form (see the proof of Theorem 5), hence that the hypersurface M is
totally geodesic or locally congruent to a complex quadric Qn−1 (C). Next weshall show that M is not a complex quadric Qn−1 (C). It is well-known that theequality (4.6) implies that our immersion is λ-isotropic, namely at each pointof M the normal curvature vector σ(v, v) determined by a unit vector v has thesame length λ for every v (cf.[O]). Moreover, (4.5) shows this function λ : M → R
is locally constant. Indeed, for every geodesic τ = τ (s) on the hypersurface Mn−1
we see by differentiating kσ(τ̇ (s), τ̇ (s))k2 along τ that λ = λ(s) is constant alongτ.
e the curvature tensors of M andOn the other hand we denote by R and R
fn (c), respectively. Since M is a Kähler submanifold in M
fn (c), by substituting
M
c
e
R(X,
Y )Z = (hY, ZiX − hX, ZiX + hJY, ZiJX − hJX, ZiJY + 2hX, JY iJZ),
4
to the Gauss equationehR(X,Y )Z, W i = hR(X, Y )Z, W i + hσ(X, Z), σ(Y, W )i − hσ(X, W ), σ(Y, Z)i,
we find that the holomorphic sectional curvature K(X, JX) of Mn−1 determinedby a unit vector X is given by
K(X, JX) = hR(X, JX)JX, Xi = c − 2kσ(X, X)k2 = c − 2λ2holds for an arbitrary unit vector X. This implies that the hypersurface M isa complex space form, so that M is not a complex quadric Qn−1 (C). Therefore
we get the desirable result. Needless to say our totally geodesic hypersurface iszero-isotropic. ¤5. Appendix
As a generalization of Theorem 6 we shall provide a characterization of aKähler isometric full immersion of a complex projective space CP n (c) of constant holomorphic sectional curvature c into a complex projective space CP N (c̃)
of constant holomorphic sectional curvature c̃. By virtue of the classification
theorem ([C, NO]) this Kähler immersion is nothing but a Kähler embedding
fk : CP n (c/k) → CP N (c) given by
hr
i
k!
k0
kn
[zi ]05i5n 7→
z · · · zn
k0 ! · · · kn ! 0
k0 +···+kn =k,
where [∗] means the point of the projective space with the homogeneous coordinates ∗ and N = (n + k)!/(n!k!) − 1. We usually call fk the k-th Veronese
embedding. The embedding fk has various geometric properties.

8

S. MAEDA AND T. ADACHIfN (c̃) be a Kähler isometric immersion
Theorem B [C, NO]. Let f : Mn (c) → M
of a complex space form of constant holomorphic sectional curvature c into another complex space form of constant holomorphic sectional curvature c̃. If c̃ > 0
and f is full, then c̃ = kc and N = (n + k)!/(n!k!) − 1 for some positive integerk.
We now show the following characterization of Veronese embeddings:
fN (c) be a Kähler isometric full immerTheorem 7 [SMA,M]. Let f : Mn → Msion of an n-dimensional Kähler manifold Mn into an N -dimensional complex
fN (c) of constant holomorphic sectional curvature c > 0. Then thespace form M
following conditions are equivalent:(1) For some positive integer k, the submanifold Mn is locally congruent toCP n (c/k), N = (n + k)!/(n!k!) − 1 and f is locally equivalent to the k-th
Veronese embedding fk .
(2) There exists κ > 0 satisfying that for each circle γ of curvature κ on thefN (c) has constant first curvaturesubmanifold Mn the curve f ◦ γ in M
along this curve.
Proof. (1) ⇒ (2). For each Veronese embedding fk : CP n (c/k) → CP N (c) we seethat kσ(v, v)k2 = c(k − 1)/2k for any unit vector v at each point x ∈ CP n (c/k)(see [Og]). We then find for each circle γ of curvature
κ on CP n (c/k) thatq

the curve fk ◦ γ has constant first curvature κ
e = κ2 + c(k−1)
in the ambient
2k
manifold CP N (c).
fN (c) be a Kähler isometric full immersion satisfying(2) ⇒ (1). Let f : Mn → M
the condition (2). By virtue of the discussion in the proof of Theorem 6 wefind Mn is a complex space form. This, combined with Theorem B, yields thestatement (1). ¤Remark 1. Theorem 7 is not true when we set κ = 0 in the statement (2). Everygeodesic on a parallel Kähler submanifold being considered as a curve in theambient space has constant first curvature.
Remark 2. We make mention of other curvatures of the curve f ◦ γ for a circle γ
on submanifolds in a space form through an isometric immersion f in all theorems
in this paper. All curvatures of every curve f ◦ γ are constant, since this curve isan orbit of one-parameter subgroup of the isometry group of the ambient space
form.
References[C] E. Calabi, Isometric imbedding of complex manifolds, Ann. Math. 269 (1982), 481–499.
[F] D. Ferus, Immersions with parallel second fundamental form, Math. Z. 140 (1974),
87–92.

CIRCLES AND HYPERSURFACES

9[M] S. Maeda, A characterization of constant isotropic immersions by circles, to appear inArchiv der Math..
[MO] S. Maeda and K. Ogiue, Geometry of submanifolds in terms of behavior of geodesics,
Tokyo J. Math. 17 (1994), 347–354.
[NO] H. Nakagawa and K. Ogiue, Complex space forms immersed in complex space forms,
Trans. Amer. Math. Soc. 219 (1976), 289–297.
[NR] R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, Tight andTaut Submanifolds, T. E. Cecil and S. S. Chern, eds., Cambridge University Press, 1998,
pp. 233-305.
[NT] H. Nakagawa and R. Takagi, On locally symmetric Kaehler submanifolds in a complex
projective space, J. Math. Soc. Japan 28 (1976), 638–667.
[NY] K. Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann.
210 (1974), 163-170.
[O] B. O’Neill, Isotropic and Kaehler immersions, Canadian J. Math. (1965), 905–915.
[Og] K. Ogiue, Differential geometry of Kaehler submanifolds, Advances in Math. 13 (1974),
73–114.
[SMA] K. Suizu, S. Maeda T. Adachi, A characterization of Veronese imbeddings into complex
projective spaces,, Math. Rep. Acad. Sci. Royal Soc. Canada 24 (2002), 61–66.
[T] M. Takeuchi, Parallel submanifolds of space forms, in honor of Y. Matsushima (1981),
Birkhäuser, Boston, 429–447.
Sadahiro MAEDA: Department of Mathematics Shimane University Matsue, Shimane, 690-8504, JAPANE-mail address: smaeda@math.shimane-u.ac.jp
Toshiaki ADACHI: Department of Mathematics Nagoya Institute of Technology Gokiso, Nagoya, 466-8555, JAPANE-mail address: adachi@math.kyy.nitech.ac.jp