Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
40 (2007), pp.15–32

SUBMANIFOLD THEORYFROM THE VIEWPOINT OF CIRCLES

Sadahiro Maeda(Received: April 13, 2006 )Abstract. We study submanifolds by using the notion of circles in Riemannian
geometry. This expository paper consists of two parts. One is to improve Nomizu
and Yano’s result ([11]) which characterizes extrinsic spheres in a Riemannian manifold. The other is to characterize totally geodesic immersions and some parallel
immersions in terms of Frenet curves of proper order 2 (which are generalizationsof circles) on submanifolds.

1. Introduction
A smooth curve γ = γ(s) in a Riemannian manifold M n parametrized by its
arclength s is called a Frenet curve of proper order 2 if there exist a field of orthonormal frames {V1 = γ̇, V2 } along γ and a positive smooth function κ(s) satisfying the following system of ordinary differential equations: ∇γ̇ V1 (s) = κ(s)V2 (s)and ∇γ̇ V2 (s) = −κ(s)V1 (s), where ∇γ̇ denotes the covariant differentiation along
γ with respect to the Riemannian connection ∇ of M n . The function κ(s) andthe field of orthonormal frames {V1 , V2 } are called the curvature and the Frenetframe of γ, respectively. Note that we do not allow the curvature κ(s) to vanish at some point. Therefore curves with inflection points, such as y = x3 on a
Euclidean xy-plane, are not Frenet curves of proper order 2. A curve is called a
Frenet curve of order 2 if it is either a geodesic or a Frenet curve of proper order
2. A Frenet curve of order 2 with nonnegative constant curvature k is called acircle of curvature k. Needless to say a circle of null curvature is nothing but ageodesic.
Study of circles produces new knowledge in submanifold theory. For example
we here recall the following two surfaces. Let f1 be a totally umbilic embeddingof a 2-dimensional standard sphere S 2 (c) of curvature c into a Euclidean space2000 Mathematics Subject Classification. Primary 53B25, Secondary 53C40.
Key words and phrases. totally umbilic submanifolds, circles, constant isotropic immersions,Kähler submanifolds, Veronese embeddings, Frenet curves of proper order 2, Kähler Frenet
curves.
15

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S. MAEDA

R5 and f2 = ι ◦ f an isometric parallel immersion of S 2 (c) into R5 . Here f is thesecond standard minimal immersion of S 2 (c) into S 4 (3c) and ι is a totally umbilic
embedding of S 4 (3c) into R5 . It is known that for each great circle γ of S 2 (c),both of the curves f1 ◦ γ and f2 ◦ γ are circles in the ambient space R5 . This tells
us that we cannot distinguish f1 from f2 by the extrinsic shape of geodesics of
S 2 (c) in R5 . However we emphasize that we can distinguish these two isometricimmersions f1 and f2 by the extrinsic shape of small circles of S 2 (c) in R5 . In
fact, for each small circle γ on S 2 (c), the curve f1 ◦ γ is also a circle in R5 but
the curve f2 ◦ γ is a helix of proper order 4 in the ambient space R5 , namely inR5 the curve f2 ◦ γ has constant positive three curvatures κ1 , κ2 and κ3 along
this curve in the sense of Frenet formula. It is hence interesting to investigate
the extrinsic shape of circles of the submanifold.
The first half of this paper is motivated by the follwing fact due to Nomizu
and Yano [11]: Let M n be a Riemannian submanifold of an ambient Riemannian
fn+p (with Riemannian connection ∇)e through an isometric immersion
manifold Mnn+pff . Then (M , f ) is an extrinsic sphere of M
(that is, M n is a totally umbilic
fn+p ) if and only if, for somesubmanifold with parallel mean curvature vector of M
positive constant k and for every circle γ = γ(s) of curvature k on M n , the curve
fn+p . In this fact we pay particular attention to a conditionf ◦ γ is a circle in M
fn+p ”. It is natural to pose the following two“the curve f ◦ γ is a circle in M
problems.
Problem 1. If we replace this condition by a condition that the curve f ◦ γ hase γ̇ γ̇k is constant along f ◦ γ, what can weconstant the first curvature, that is, k∇
say about the isometric immersion f ?
Problem 2. If we replace this condition by a condition that the curve f ◦ γ is a
Frenet curve of order 2, what can we say about the isometric immersion f ?
The main purpose of the first half of this paper is to give complete answers
to these problems (Theorems 1 and 2). As an application of Theorem 1 we
obtain a characterization of Veronese embeddings of complex projective spaces
into complex projective spaces which are typical examples of Kähler immersions(Theorem 3).
In the latter half, using the notion of Frenet curves of order 2, we establish someresults. We first provide a characterization of every totally geodesic submanifold
fn+p by observing the extrinsic shape of a FrenetM n in a Riemannian manifold M
curve of proper order 2 on the submanifold M n (Theorem 4).
We consider a Frenet curve γ = γ(s) of proper order 2 in a Kähler manifold
M (with complex structure J). We put τγ = |hV1 , JV2 i|, which is well-defined.
Since we havedhV1 , JV2 i = h∇γ̇ V1 , JV2 i + hV1 , J∇γ̇ V2 i = κhV2 , JV2 i − κhV1 , JV1 i = 0,
ds

SUBMANIFOLD THEORY FROM THE VIEWPOINT OF CIRCLES

17

we see τγ does not depend on parameter s. A curve γ = γ(s) is called a Kähler
Frenet curve if it is either a geodesic or a Frenet curve of proper order 2 with
τγ = 1. In other words, a Kähler Frenet curve γ is either a geodesic or a curve
satisfying ∇γ̇ γ̇ = κ(s)J γ̇ or ∇γ̇ γ̇ = −κ(s)J γ̇ for some positive smooth functionκ = κ(s). We call γ a Kähler circle of curvature k if the curvature function κ ofa Frenet curve γ of order 2 is a nonnegative constant k.
We next give a characterization of all totally geodesic Kähler immersions ofKähler manifolds into Kähler manifolds by using the notion of Kähler Frenet
curves (Theorem 5). We finally characterize all parallel isometric immersions of
an n-dimensional complex space form Mn (c) (which is locally congruent to either
CP n (c), Cn or CH n (c)) of constant holomorphic sectional curvature c into a (2n+f2n+p (c̃) (which is locally congruent to eitherp)-dimensional real space form M
2n+p
2n+p
2n+p
R
,S
(c̃) or H(c̃)) of constant sectional curvature c̃ by observing theextrinsic shape of Kähler Frenet curves on Mn (c) (Theorem 6).
2. Fundamental equations and isotropic immersions
fm be
We review fundamental equations in submanifold theory. Let M n , M
nmf an isometric immersion. Throughout
Riemannian manifolds and f : M → M
fm . The
this paper we will identify a vector X of M n with a vector f∗ (X) of M
fm are denoted by the same notation h , i.
Riemannian metrics on M n , M
e the covariant differentiations of M n and M
fm , reWe denote by ∇ and ∇spectively. Then the second fundamental form σ of the immersion f is defined
by
(2.1)

e X Y − ∇X Y,
σ(X, Y ) = ∇where X and Y are vector fields tangent to M n . For a vector field ξ normal to
M n , we write(2.2)

e X ξ = −Aξ X + DX ξ,∇

where −Aξ X (resp. DX ξ) denotes the tangential (resp. the normal) componente X ξ. We define the covariant differentiation ∇¯ of the second fundamentalof ∇form σ with respect to the connection in (tangent bundle)⊕(normal bundle) as
follows:(2.3)¯ X σ)(Y, Z) = DX (σ(Y, Z)) − σ(∇X Y, Z) − σ(Y, ∇X Z).
(∇

We next review the notion of isotropic immersions. An isometric immersion
f is said to be isotropic at x ∈ M if kσ(X, X)k/kXk2 (= λ(x)) doesf :M →M
not depend on the choice of X(6= 0) ∈ Tx M . If the immersion is isotropic at every
point, then the immersion is said to be isotropic. When the function λ = λ(x)is constant on M , we call M a constant (λ-)isotropic submanifold. Note that a
totally umbilic immersion is isotropic, but not vice versa. The following is wellknown ([12]).

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S. MAEDA

f, h , i). Then f isLemma 1. Let f be an isometric immersion of M into (Misotropic at x ∈ M if and only if the second fundamental form σ of f satisfies
hσ(u, u), σ(u, v)i = 0 for an arbitrary orthogonal pair u, v ∈ Tx M .
3. Characterization of constant isotropic immersions
The following gives a geometric meaning of constant isotropic immersions interms of circles on submanifolds.
Theorem 1 ([5]). Let M n be an n-dimensional connected Riemannian submanfn+p through an isometricifold of an (n + p)-dimensional Riemannian manifold M
immersion f . Then the following are equivalent.
fn+p .
(i) M n is a constant (λ-) isotropic submanifold of M(ii) There exists a positive constant k satisfying that for each circle γ of curfn+p has constantvature k on the submanifold M n the curve f ◦ γ in M
the first curvature κ1 along this curve.
fn+p be a constant λ-isotropic immersion. We
Proof. (i) ⇒ (ii): Let f : M n → Mtake a circle of curvature k, which satisfies the following equations:(3.1)∇γ̇ γ̇ = kYs

and ∇γ̇ Ys = −k γ̇.

In the following, for simplicity we also denote f ◦ γ by γ. It follows from (2.1)
thate γ̇(s) γ̇(s) = kYs + σ(γ̇(s), γ̇(s)).
∇

(3.2)

e γ̇ γ̇k of the curve f ◦ γThen from √(3.2) we can see that the first curvature κ1 = k∇22
is equal to k + λ , which is constant.
fn+p be an isometric immersion satisfying the(ii) ⇒ (i): Let f : M n → M
condition (ii). We take a point x ∈ M and choose an arbitrary orthonormal pairof vectors u, v ∈ Tx M . Let γ = γ(s) be a circle of curvature k on the submanifold
M n with initial condition that γ(0) = x, γ̇(0) = u and ∇γ̇ γ̇(0) = kv. By conditione γ̇ γ̇k of the curve f ◦ γ is constant, so that equation(ii) the first curvature κ1 = k∇(3.2) implies kσ(γ̇, γ̇)k is constant. Hence we obtain
0=
(3.3)

d
kσ(γ̇, γ̇)k2 = 2hDγ̇ (σ(γ̇, γ̇)), σ(γ̇, γ̇)ids
¯ γ̇ σ)(γ̇, γ̇) + 2σ(∇γ̇ γ̇, γ̇), σ(γ̇, γ̇)i
= 2h(∇¯ γ̇ σ)(γ̇, γ̇), σ(γ̇, γ̇)i + 4khσ(γ̇, γ̇), σ(γ̇, Y )i.
= 2h(∇

Evaluating equation (3.3) at s = 0, we get(3.4)¯ u σ)(u, u), σ(u, u)i + 2khσ(u, u), σ(u, v)i = 0.
h(∇SUBMANIFOLD THEORY FROM THE VIEWPOINT OF CIRCLES

19

On the other hand, for another circle ρ = ρ(s) of the same curvature k on thesubmanifold M n with initial condition that ρ(0) = x, ρ̇(0) = u and ∇ρ̇ ρ̇(0) =−kv, we have(3.5)

¯ u σ)(u, u), σ(u, u)i − 2khσ(u, u), σ(u, v)i = 0h(∇which corresponds to equation (3.4). Thus, from (3.4) and (3.5) we can see thathσ(u, u), σ(u, v)i = 0 for any orthonormal pair of vectors u, v at each point x of
fn+p through the isometricM , so that the submanifold M n is (λ-)isotropic in M
immersion f by Lemma 1.
Next, we shall show that λ : M → R is constant. It follows from (3.4) and(3.5) that¯ u σ)(u, u), σ(u, u)i = 0 for every unit vector u at each point x of M n .
h(∇Then, for every geodesic τ = τ (s) on the submanifold M n we see that λ = λ(s)is constant along τ . Therefore we can conclude that λ is constant on M . ¤4. Characterization of extrinsic spheres
The following is an improvement of Nomizu and Yano’s result [11].
Theorem 2 ([4]). Let M n be an n-dimensional connected Riemannian submanifn ＋ p through an isometricfold of an (n+p)-dimensional Riemannian manifold M
fn ＋ p if and only if for someimmersion f . Then M n is an extrinsic sphere of M
positive constant k and for every circle γ = γ(s) of curvature k in M n , the curve
fn ＋ p .
f ◦ γ is a Frenet curve of order 2 in M
Proof. The “only if” part is obvious by virtue of the well-known result of Nomizu
and Yano ([11]). So it suffices to verify the “if” part.
Let x be an arbitrary point of M n and u, v an orthonormal pair of vectors in
Tx M n . Let γ = γ(s) be a circle of of positive curvature k in M n satisfying (3.1)
and the initial condition γ(0) = x, γ̇(0) = u and Y0 = v. By assumption there
exist a (nonnegative) smooth function κ̃ = κ̃(s) and a field of unit vectors Yes
fn+p satisfying thatalong f ◦ γ in M
(4.1)

e γ̇ γ̇ = κ̃Yes , ∇e γ̇ Yes = −κ̃γ̇.
∇It follows from (2.1) and the first equality of (4.1) that
(4.2)

κ̃Yes = kYs + σ(γ̇, γ̇),so that(4.3)

κ̃2 = k 2 + kσ(γ̇, γ̇)k2 .

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S. MAEDA

We here note that κ̃(s) > 0 for each s.
Now we shall compute the covariant differentiation of (4.2). For the left-handside, by using (4.1) and (4.2) we see
e γ̇ (κ̃Yes ) = κ̃˙ Yes + κ̃∇e γ̇ Yes∇κ̃˙= (kYs + σ(γ̇, γ̇)) − κ̃2 γ̇.
κ̃

(4.4)

For the right-hand side, using (2.1), (2.2), (2.3) and (3.1), we obtaine γ̇ (kYs + σ(γ̇, γ̇))∇(4.5)

e γ̇ Ys − Aσ(γ̇,γ̇) γ̇ + Dγ̇ (σ(γ̇, γ̇))= k∇¯ γ̇ σ)(γ̇, γ̇) + 2σ(∇γ̇ γ̇, γ̇)= k(∇γ̇ Ys + σ(γ̇, Ys )) − Aσ(γ̇,γ̇) γ̇ + (∇¯ γ̇ σ)(γ̇, γ̇).
= −k 2 γ̇ + 3kσ(γ̇, Ys ) − Aσ(γ̇,γ̇) γ̇ + (∇We compare the tangential components and the normal components for the submanifold M n in (4.4) and (4.5), respectively. Then we get the following:
˙ s − κ̃3 γ̇ = κ̃(−k 2 γ̇ − Aσ(γ̇,γ̇) γ̇),κ̃kY
¯ γ̇ σ)(γ̇, γ̇)).
˙ γ̇, γ̇) = κ̃(3kσ(γ̇, Ys ) + (∇
κ̃σ(

(4.6)
(4.7)

Equation (4.7) gives at s = 0¯ u σ)(u, u)),˙κ̃(0)σ(u,u) = κ̃(0)(3kσ(u, v) + (∇so that(4.8)¯ u σ)(u, u)).
˙κ̃(0)κ̃(0)σ(u,u) = κ̃(0)2 (3kσ(u, v) + (∇On the other hand, we see from (4.3) and (3.3) thatκ̃2 (0) = k 2 + kσ(u, u)k2 ,(4.9)
and that for each s¯ γ̇ σ)(γ̇, γ̇), σ(γ̇, γ̇)i + 4hσ(∇γ̇ γ̇, γ̇), σ(γ̇, γ̇)i.
˙ = 2h(∇2κ̃κ̃
So we have at s = 0(4.10)¯ u σ)(u, u), σ(u, u)i + 2khσ(v, u), σ(u, u)i.
˙κ̃(0)κ̃(0)= h(∇SUBMANIFOLD THEORY FROM THE VIEWPOINT OF CIRCLES

21

Substituting (4.9) and (4.10) into (4.8), we obtain(4.11)

3k(k 2 + kσ(u, u)k2 )σ(u, v) − 2khσ(u, u), σ(u, v)iσ(u, u)¯ u σ)(u, u), σ(u, u)iσ(u, u) − (k 2 + kσ(u, u)k2 )(∇¯ u σ)(u, u).
= h(∇Here we consider anther circle τ = τ (s) of the same curvature k in M n satisfying
the equations ∇τ̇ τ̇ = kZs and ∇τ̇ Zs = −k τ̇ , with initial condition τ (0) = x,τ̇ (0) = u and Z0 = −v. By assumption, the curve f ◦ τ is a Frenet curve of order
fn+p . So we can apply the above discussion to the curve τ . Then we can
2 in M
see that−3k(k 2 + kσ(u, u)k2 )σ(u, v) + 2khσ(u, u), σ(u, v)iσ(u, u)(4.11’)¯ u σ)(u, u), σ(u, u)iσ(u, u) − (k 2 + kσ(u, u)k2 )(∇¯ u σ)(u, u).
= h(∇Hence, from (4.11) and (4.11’) we obtain(4.12)

3(k 2 + kσ(u, u)k2 )σ(u, v) − 2hσ(u, u), σ(u, v)iσ(u, u) = 0.

Taking the inner product of both sides of equation (4.12) with σ(u, u), we get(3k 2 + kσ(u, u)k2 )hσ(u, u), σ(u, v)i = 0.
Hencehσ(u, u), σ(u, v)i = 0.
Since x is arbitrary, thanks to Lemma 1, we find that our immersion f is (say,λ-)isotropic. So, again by using Lemma 1, we get Aσ(γ̇,γ̇) γ̇ = λ2 γ̇. Hence, from
(4.6) we have˙ s − κ̃3 γ̇ = −κ̃(k 2 + λ2 )γ̇.
κ̃kY
Taking the inner product of bith sides of this equation with unit vectors Ys , weknow that κ̃ = κ̃(s) is constant along the curve f ◦ γ. Therefore we can see thatfn+p . Thus we get the statement
the curve f ◦ γ is a circle in the ambient space M
of Theorem 2. ¤5. Characterization of Veronese embeddings
As an application of Theorem 1 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 ([1, 10]) this Kähler immersion is equivalent to a Kähler embeddingfν : CP n (c/ν) → CP N (c) given by
hr
i
ν!
ν0
νn
[zi ]05i5n 7→
z · · · zn
ν0 ! · · · νn ! 0ν0 +···+νn =ν,where [∗] means the point of the projective space with homogeneous coordinates∗ and N = (n + ν)!/(n!ν!) − 1. We usually call fν the ν-th Veronese embedding.
The embedding fν has various geometric properties. We recall the work of [1,10] for the later use.

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S. MAEDA

Theorem A. Let f : Mn (c) → MN (c̃) be a Kähler isometric immersion of acomplex space form of constant holomorphic sectional curvature c into anothercomplex space form of constant holomorphic sectional curvature c̃. If c̃ > 0 and
f is full, then c̃ = νc, N = (n + ν)!/(n!ν!) − 1 and f is locally equivalent to theν-th Veronese embedding fν for some positive integer ν.
We are now in a position to prove the following:
Theorem 3 ([5]). Let f : Mn → MN (c) be a Kähler isometric full immersion of
an n-dimensional Kähler manifold Mn into an N -dimensional complex space form
MN (c) of constant holomorphic sectional curvature c > 0. Then the following
conditions are equivalent.
(i) For some positive integer ν, the submanifold Mn is locally congruent toCP n (c/ν), N = (n + ν)!/(n!ν!) − 1 and f is locally equivalent to the ν-thVeronese embedding fν .
(ii) There exists a positive constant k satisfying that for each circle γ of curvature k on the submanifold Mn the curve f ◦ γ in MN (c) has constantthe first curvature κ1 along this curve.
Proof. (i) ⇒ (ii): For each Veronese embedding fν : CP n (c/ν) → CP N (c) we seethat kσ(X, X)k2 = c(ν − 1)/2ν for any unit vector X at each point x ∈ CP n (c/ν)(see [13]). Then we find that for each circle γ of curvature
k on CP n (c/ν) theqcurve fν ◦ γ has constant the first curvature κ1 = k 2 + c(ν−1)
in the ambient2ν
Nmanifold CP (c).
(ii) ⇒ (i): Let f : Mn → MN (c) be a Kähler isometric full immersion satisfying the condition (ii). Then by virtue of Theorem 1 Mn is constant (λ-)isotropice the curvature tensor ofin MN (c). On the other hand we denote by R (resp. R)
Mn (resp. MN (c)). We recall 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.

Since M is a Kähler submanifold in MN (c), from this equation andceR(X,Y )Z = (hY, ZiX − hX, ZiX + hJY, ZiJX − hJX, ZiJY + 2hX, JY iJZ),
4
we find that the holomorphic sectional curvature K(X, JX) of Mn determinedby a unit vector X is given by
K(X, JX) = hR(X, JX)JX, Xi = c − 2kσ(X, X)k2 .
Thus we can see that our submanifold Mn is a complex space form. Therefore
from Theorem A we obtain the statement (i). ¤We here make mention of other curvatures in Theorem 3. It is known that foreach circle γ on CP n (c/ν) the curve fν ◦ γ is an integral curve of some Killing
vector field of the ambient space CP N (c), so that all curvatures of fν ◦ γ areconstant (cf. [7]).

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23

6. Characterization of totally geodesic immersions
In this section we consider Frenet curves of proper order 2 on submanifolds.
Our aim here is to prove the following:
Theorem 4 ([17]). Let M n be a connected Riemannian submanifold of a Riefn+p through an isometric immersion f . Asumme that there
mannian manifold M
exists a nonconstant positive smooth function κ = κ(s) satisfying that for every
Frenet curve γ = γ(s) of proper order 2 with curvature κ in M n , the curve f ◦ γ
fn+p . Then M n is totally
is a Frenet curve of order 2 in the ambient space M
fn+p .
geodesic in M
Proof. Let x be an arbitrary point of M n and u, v any orthonormal pair of vectors
in Tx M n . Without loss of generality we suppose that the function κ = κ(s) isdefined on some open interval −² < s < ². Let γ = γ(s) (|s| < ²) be a Frenet
curve of proper order 2 with curvature κ in M n satisfying the equations(6.1)∇γ̇ γ̇ = κ(s)Ys ,∇γ̇ Ys = −κ(s)γ̇and the initial condition γ(0) = x, γ̇(0) = u and Y0 = v. By assumption the
fn+p . So there exist a positive
curve f ◦ γ is a Frenet curve of proper order 2 in M
fn+psmooth function κ̃ = κ̃(s) and a field of unit vectors Yes along f ◦ γ in M
satisfying equation (4.1).
Then by the same discussion as in the proof of Theorem 2 we have the following
equations.
(6.2)

κ̃Yes = κYs + σ(γ̇, γ̇),(6.3)

κ̃2 = κ2 + kσ(γ̇, γ̇)k2 .

(6.4)

e γ̇ (κ̃Yes ) = κ̃˙ Yes + κ̃∇e γ̇ Yes∇κ̃˙= (κYs + σ(γ̇, γ̇)) − κ̃2 γ̇.
κ̃e γ̇ (κYs + σ(γ̇, γ̇))∇e γ̇ Ys − Aσ(γ̇,γ̇) γ̇ + Dγ̇ (σ(γ̇, γ̇))= κ̇Ys + κ∇

(6.5)

= κ̇Ys + κ(∇γ̇ Ys + σ(γ̇, Ys )) − Aσ(γ̇,γ̇) γ̇¯ γ̇ σ)(γ̇, γ̇) + 2σ(∇γ̇ γ̇, γ̇)+ (∇¯ γ̇ σ)(γ̇, γ̇).
= κ̇Ys − κ2 γ̇ + 3κσ(γ̇, Ys ) − Aσ(γ̇,γ̇) γ̇ + (∇

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S. MAEDA

We compare the tangential components and the normal components for the submanifold M n in (6.4) and (6.5), respectively.
˙ s − κ̃3 γ̇ = κ̃(κ̇Ys − κ2 γ̇ − Aσ(γ̇,γ̇) γ̇),κ̃κY¯ γ̇ σ)(γ̇, γ̇)).
˙ γ̇, γ̇) = κ̃(3κσ(γ̇, Ys ) + (∇
κ̃σ(

(6.6)
(6.7)

Equation (6.7) gives at s = 0¯ u σ)(u, u)),˙κ̃(0)σ(u,u) = κ̃(0)(3κ(0)σ(u, v) + (∇so that¯ u σ)(u, u)).
˙κ̃(0)κ̃(0)σ(u,u) = κ̃(0)2 (3κ(0)σ(u, v) + (∇(6.8)

On the other hand, we see from (6.3) and (3.3) thatκ̃2 (0) = κ2 (0) + kσ(u, u)k2 ,(6.9)
and that¯ u σ)(u, u), σ(u, u)i˙κ̃(0)κ̃(0)=κ̇(0)κ(0) + h(∇(6.10)+ 2κ(0)hσ(v, u), σ(u, u)i.

Substituting (6.9) and (6.10) into (6.8), we obtain
3κ(0)(κ(0)2 + kσ(u, u)k2 )σ(u, v) − 2κ(0)hσ(u, u), σ(u, v)iσ(u, u)¯ u σ)(u, u), σ(u, u)i)σ(u, u)= (κ̇(0)κ(0) + h(∇(6.11)¯ u σ)(u, u).
− (κ(0)2 + kσ(u, u)k2 )(∇Then by the same discussion as in the proof of Theorem 2 we find that

(6.11’)−3κ(0)(κ(0)2 + kσ(u, u)k2 )σ(u, v) + 2κ(0)hσ(u, u), σ(u, v)iσ(u, u)¯ u σ)(u, u), σ(u, u)i)σ(u, u)= (κ̇(0)κ(0) + h(∇¯ u σ)(u, u).
− (κ(0)2 + kσ(u, u)k2 )(∇It follows from (6.11) and (6.11’) that
(6.12)

3κ(0)(κ(0)2 + kσ(u, u)k2 )σ(u, v) − 2κ(0)hσ(u, u), σ(u, v)iσ(u, u) = 0.

So we have(3κ(0)2 + kσ(u, u)k2 )hσ(u, u), σ(u, v)i = 0.

SUBMANIFOLD THEORY FROM THE VIEWPOINT OF CIRCLES

25

Note that κ(0) 6= 0. Hencehσ(u, u), σ(u, v)i = 0.
This, combined with (6.12), shows that σ(u, v) = 0 holds for each orthonormal
pair of vectors u, v in Tx M n . Since x is an arbitrary point, our immersion f is
totally umbilic. Then by taking the inner product of both sides of equation (6.6)
with Ys we see that
(6.13)˙κ̃(s)κ(s)= κ̃(s)κ̇(s)for each s.

Moreover, from (6.7) we obtain(6.14)˙κ̃(s)hγ(s) = κ̃(s)(Dγ̇(s) h)γ(s)for any s,

where h is the mean curvature vector of our immersion. Therefore equations(6.13) and (6.14) imply(6.15)(Dγ̇(s) h)γ(s) =

˙
κ̇(s)
κ̃(s)
hγ(s) =
hγ(s) .
κ̃(s)κ(s)

In particular, at s = 0, we get(Du h)x =

κ̇(0)
hx .
κ(0)

This equation shows that (Du h)x is independent of the choice of a unit vectoru ∈ Tx M n . Changing u into −u we see (Du h)x = 0. Since x is an arbitrary
point, we have shown that the mean curvature vector h of f is parallel.
Finally, by assumption, as the function κ(s) is not constant, there exists somes0 ∈ (−², ²) with κ̇(s0 ) 6= 0. This, together with equation (6.15), yields
hγ(s0 ) =κ(s0 )(Dγ̇ h)γ(s0 ) = 0.
κ̇(s0 )

Since khk is constant on M n , we have h = 0 on M n . Consequently the immersion
fn+p is totally geodesic. ¤f : Mn → M
The discussion in the proof of Theorem 4 gives the following, which is an
improvement of Theorem 2.
Theorem 4’. Let M n be a connected Riemannian submanifold of a Riemannian
fn+p through an isometric immersion f . Then M n is an extrinsic
manifold M
fn+p if and only if there exists a positive smooth function κ = κ(s)sphere of M
satisfying that for every Frenet curve γ = γ(s) of proper order 2 with curvature κ
fn+p .
in M n , the curve f ◦ γ is a Frenet curve of order 2 in the ambient space M

26

S. MAEDA

7. Characterization of totally geodesic Kähler immersions
We consider the extrinsic shape f ◦ γ of a Kähler Frenet curve γ of a Kähler
f through a Kähler isometric immermanifold M in an ambient Kähler manifold M
sion f . We shall give a characterization of all totally geodesic Kähler immersions
from this point of view.
Theorem 5 ([8, 15]). Let f be a Kähler isometric immersion of a complex
n-dimensional Kähler manifold Mn into an arbitrary complex m-dimensional
fm . Then the following are equivalent.
Kähler manifold M(i) f is a totally geodesic immersion.
(ii) There exists a positive smooth function κ = κ(s) satisfying that f maps
every Kähler Frenet curve γ = γ(s) of curvature κ on Mn to a Frenet
fm .
curve of order 2 in M(iii) There exists a positive constant k satisfying that f maps every Kähler
fm .
circle γ = γ(s) of curvature k on Mn to a Frenet curve of order 2 in M
In order to prove Theorem 5 we prepare the following lemma:
Lemma 2. Let Mn be a Kähler manifold with complex structure J which isf through an isometric imimmersed into an arbitrary Riemannian manifold M
mersion f . If for a Kähler Frenet curve γ = γ(s) of positive curvature κ = κ(s)f, then following equalities
on Mn , the curve f ◦γ is a Frenet curve of order 2 in M
hold:
(7.1)
(7.2)

˙ γ̇ − κ̃3 γ̇ = κ̃{±κ̇J γ̇ − κ2 γ̇ − Aσ(γ̇,γ̇) γ̇},±κκ̃J¯ γ̇ σ)(γ̇, γ̇)}.
˙ γ̇, γ̇) = κ̃{±3κσ(γ̇, J γ̇) + (∇κ̃σ(In equalities (7.1) and (7.2), we adopt plus sign (resp. minus sign) for a Kähler
Frenet curve satisfying ∇γ̇ γ̇ = κJ γ̇ (resp. ∇γ̇ γ̇ = −κJ γ̇).
f by
Proof of Lemma 2. Since the curve f ◦ γ is a Frenet curve of order 2 in M
assumption, there exist a function κ̃ = κ̃(s) and a field of unit vectors Yes alongf satisfying (4.1). Then from (2.1) we havef ◦ γ in M
(7.3)

κ̃Yes = ±κJ γ̇ + σ(γ̇, γ̇).

The function κ̃ is positive because κ > 0.
For the left-hand side of (7.3), by using (4.1) and (7.3) again, we see(7.4)

e γ̇ (κ̃Yes ) = κ̃{κ̃˙ Yes + κ̃∇e γ̇ Yes } = κ̃κ̃˙ Yes − κ̃3 γ̇κ̃∇˙= κ̃{±κJγ̇ + σ(γ̇, γ̇)} − κ̃3 γ̇.

SUBMANIFOLD THEORY FROM THE VIEWPOINT OF CIRCLES

27

On the other hand, for the right-hand side of (7.3), it follows from (2.1), (2.2)
and (2.3) that
(7.5)
e γ̇ {±κJ γ̇ + σ(γ̇, γ̇)}
κ̃∇
n
o
e
= κ̃ ±κ̇J γ̇ ± κ∇γ̇ (J γ̇) − Aσ(γ̇,γ̇) γ̇ + Dγ̇ (σ(γ̇, γ̇))no¯ γ̇ σ)(γ̇, γ̇) + 2σ(∇γ̇ γ̇, γ̇)= κ̃ ±κ̇J γ̇ ± κ (∇γ̇ (J γ̇) + σ(γ̇, J γ̇)) − Aσ(γ̇,γ̇) γ̇ + (∇no¯ γ̇ σ)(γ̇, γ̇) ± 2κσ(J γ̇, γ̇)= κ̃ ±κ̇J γ̇ ± κ (J(±κJ γ̇) + σ(γ̇, J γ̇)) − Aσ(γ̇,γ̇) γ̇ + (∇no¯ γ̇ σ)(γ̇, γ̇) .
= κ̃ ±κ̇J γ̇ − κ2 γ̇ ± 3κσ(γ̇, J γ̇) − Aσ(γ̇,γ̇) γ̇ + (∇We compare the tangential components and the normal components of (7.4) and(7.5), respectively. Then we get equalities (7.1) and (7.2). ¤We shall prove Theorem 5. Needless to say, condition (i) implies both conditions (ii) and (iii). We have only to show that condition (ii) implies condition (i)for each given positive smooth function κ = κ(s). Let v ∈ T Mn be an arbitrary
unit vector tangent to Mn . We have Kähler Frenet curves γ1 , γ2 of cuvature κ on
Mn satisfying ∇γ˙1 γ˙1 = κJ γ˙1 , ∇γ˙2 γ˙2 = −κJ γ˙2 with condition γ˙1 (0) = γ˙2 (0) = v.
The curves γ1 , γ2 satisfy the following equalities by Lemma 2:±κκ̃˙ i J γ̇i − κ̃3i γ̇i = κ̃i {±κ̇J γ̇i − κ2 γ̇i − Aσ(γ̇i ,γ̇i ) γ̇i },¯ γ̇ σ)(γ̇i , γ̇i )},κ̃˙ i σ(γ̇i , γ̇i ) = κ̃i {±3κσ(γ̇i , J γ̇i ) + (∇
i

(7.6)
(7.7)
where κ̃i =p
κ2 + kσ(γ̇i , γ̇i )k2 (> 0). Since the immersion f is Kähler, we seehAσ(γ̇i ,γ̇i ) γ̇i , J γ̇i i = hσ(γ̇i , γ̇i ), σ(γ̇i , J γ̇i )i = hσ(γ̇i , γ̇i ), Jσ(γ̇i , γ̇i )i = 0.
Hence, by taking the inner product of both sides of equation (7.6) with J γ˙i , wefind κκ̃˙ i = κ̃i κ̇. Then equation (7.7) becomes¯ γ̇ σ)(γ̇i , γ̇i ) =±3κJσ(γ̇i , γ̇i ) + (∇
i

κ̃˙ i
κ̇
σ(γ̇i , γ̇i ) = σ(γ̇i , γ̇i ).
κ̃iκ

Evaluating this equation at s = 0, we get¯ v σ)(v, v) = κ̇(0) σ(v, v)
3κ(0)Jσ(v, v) + (∇κ(0)¯ v σ)(v, v) = κ̇(0) σ(v, v)−3κ(0)Jσ(v, v) + (∇κ(0)

for i = 1,
for i = 2.

28

S. MAEDA

It follows that σ(v, v) = 0 for an arbitrary unit vector v ∈ T Mn . Therefore f isa totally geodesic immersion.
Remark. In the statement of Theorem 5, we can not omit the condition that
κ is positive. For example, we consider the second Veronese embedding f2 :2
CP n (c/2) −→ CP (n +3n)/2 (c) which is defined by√(z0 , . . . , zn ) 7→ (z02 , 2 z0 z1 , . . . , zn2 )in homogeneous coordinates. This (non totally geodesic) Kähler isometric embedding maps every geodesic on CP n (c/2) to a totally real circle of positive curvature√√2
c /2 in the ambient space CP (n +3n)/2 (c), that is a circle of curvature c /2 in
2
RP 2 (c/4) which is a totally real totally geodesic submanifold of CP (n +3n)/2 (c).
The following gives a characterization of the second Veronese embedding:
Proposition ([6, 9]). Let Mn be a non totally geodesic Kähler submanifold of
CP N (c) through a full Kähler isometric immersion f . Suppose that for each geodesic γ on Mn , the curve f ◦γ is a plane curve in the ambient space CP N (c). Thenthe submanifold Mn is locally congruent to CP n (c/2), the isometric immersion f
is locally equivalent to the second Veronese embedding f2 and N = (n2 + 3n)/2.
Here, a curve γ = γ(s) on a Riemannian manifold M is called a plane curve if
the curve γ is locally contained in some real 2-dimensional totally geodesic submanifold of M . As a matter of course, every plane curve with positive curvaturefunction is a Frenet curve of proper order 2. But in general, the converse does
not hold.
8. Characterization of some parallel isometric immersions
We shall provide a characterization of all parallel immersions of a complex
f2n+p (c̃) by observing the extrinsic
space form Mn (c) into a real space form M
f2n+p (c̃).
shape of a Kähler Frenet curve of Mn (c) in the ambient space M
Theorem 6 ([8]). Let f be an isometric immersion of a Kähler manifold Mn
f2n+p (c̃) of constant sectional curvature c̃. If there exists a
into a real space form M
positive smooth function κ satisfying that f maps every Kähler Frenet curve γ of
f2n+p (c̃), then f is a parallel immersion
curvature κ on Mn to a plane curve in M
and locally equivalent to one of the following.
f2n+p (c̃) = R2n+p ,(1) f is a totally geodesic immersion of Mn = Cn into M
where c̃ = 0.
f2n+p (c̃) = RH 2n+p (c̃),(2) f is a totally umbilic immersion of Mn = Cn into M
where c̃ < 0.
(3) f is a parallel immersion defined by
2
f1
f2
f2n+p (c̃),
f = f1 ◦ f2 : Mn = CP n (c) −→ S n +2n−1 ((n + 1)c/(2n)) −→ M

SUBMANIFOLD THEORY FROM THE VIEWPOINT OF CIRCLES

29

where f1 is the first standard minimal embedding, f2 is a totally umbilic
embedding and (n + 1)c/(2n) = c̃.
Proof. Let x be any point of Mn and v ∈ Tx Mn be any unit tangent vector atx ∈ Mn . Let γ1 = γ1 (s) and γ2 = γ2 (s) be Kähler Frenet curves of curvature
κ on Mn satisfying that γ1 (0) = γ2 (0) = x, γ̇1 (0) = γ̇2 (0) = v and ∇γ˙1 γ˙1 =κJ γ˙1 , ∇γ˙2 γ˙2 = −κJ γ˙2 . Then, by Lemma 2 equations (7.6) and (7.7) hold. Notethat κ̃i > 0. For the sake of brevity, we abbreviate the indices in the following.
It follows from (6.3) that
1 d 2
κ̃
2 ds
1 d
= κκ̇ +
hσ(γ̇, γ̇), σ(γ̇, γ̇)i
2 ds
¯ γ̇ σ)(γ̇, γ̇), σ(γ̇, γ̇)i ± 2κhσ(J γ̇, γ̇), σ(γ̇, γ̇)i.
= κκ̇ + h(∇κ̃κ̃˙ =
(8.1)

On the other hand, equation (7.2) implies that
(8.2)

¯ γ̇ σ)(γ̇, γ̇)}.
˙ γ̇, γ̇) = κ̃2 {±3κσ(γ̇, J γ̇) + (∇κ̃κ̃σ(

Substituting (6.3) and (8.1) into (8.2), at s = 0 we obtainno¯κ(0)κ̇(0) + h(∇v σ)(v, v), σ(v, v)i ± 2κ(0)hσ(v, v), σ(v, Jv)i σ(v, v)
n
on
o
(8.3)
¯ v σ)(v, v) .
= κ(0)2 + kσ(v, v)k2 ±3κ(0)σ(v, Jv) + (∇Then the same discussion as in the proof of Theorem 5 showsno2κ(0)hσ(v, v), σ(v, Jv)iσ(v, v) = 3κ(0) κ(0)2 + kσ(v, v)k2 σ(v, Jv),
so that(8.4)

no2hσ(v, v), σ(v, Jv)iσ(v, v) = 3 κ(0)2 + kσ(v, v)k2 σ(v, Jv).

Taking the inner product of both sides of this with σ(v, v), we getno2hσ(v, v), σ(v, Jv)ikσ(v, v)k2 = 3 κ(0)2 + kσ(v, v)k2 hσ(v, v), σ(v, Jv)i
hence

n
o
3κ(0)2 + kσ(v, v)k2 hσ(v, v), σ(v, Jv)i = 0.

Consequently hσ(v, v), σ(v, Jv)i = 0, because 3κ(0)2 + kσ(v, v)k2 > 0. Again
from (8.4) we find
(8.5)

σ(v, Jv) = 0

for any x ∈ Mn and any v ∈ Tx Mn .

30

S. MAEDA

Replacing v by Jv + u, we have σ(u, v) = σ(Ju, Jv) for each u, v ∈ Tx Mn . This,
together with Codazzi’s equation in a space of constant curvature, implies that¯ = 0 (see [3]). Next, owing to (8.5), we see
the immersion f is parallel: ∇σ
hAσ(γ̇,γ̇) γ̇, J γ̇i = hσ(γ̇, γ̇), σ(γ̇, J γ̇)i = 0. Therefore, taking the inner product ofboth sides of (7.1) with J γ̇, we haveκκ̃˙ = κ̃κ̇,(8.6)
so that equation (7.1) reduces toAσ(γ̇,γ̇) γ̇ = (κ̃2 − κ2 )γ̇,
which implies thathσ(v, v), σ(v, u)i = hAσ(v,v) v, ui = 0
for any orthonormal pair of vectors u, v ∈ Tx Mn at each point x ∈ Mn . Thus, byvirtue of Lemma 1, the immersion f is isotropic. Hence our submanifold Mn isa complex space form immersed as an isotropic submanifold with parallel second
f2n+p (c̃), so that the immersion ffundamental form into the ambient space M
is rigid (see Proposition 1 in [18]). Therefore we can see that the submanifold(Mn , f ) is locally congruent to one of (1), (2) and (3) (cf. [2, 16]).
Now we shall show that the examples (1), (2) and (3) satisfy the hypothesisof our theorem. For each positive smooth function κ the totally geodesic case(1) satisfies our hypothesis. Next, for the other cases (2) and (3) we take thefunction κ as a positive constant, say k. In the case of (2), for each√ circle γ ofcurvature k(> 0) on Mn the curve f ◦ γ is a circle of curvature k 2 − c̃ (seef2n+p (c̃).
page 169 in [11]), so that it is a plane curve in the ambient space M
Finally we explain
the case (3) in detail. The isometric immersion f given by√(3) is (λ =) c − c̃ -isotropic and the parallel second fundamental form σ of fsatisfies σ(JX, JY ) = σ(X, Y ) for each vector X, Y ∈ T (CP n (c)). Let γ = γ(s)be a Kähler circle of curvature k(> 0) on CP n (c). Then the curve f ◦ γ satisfies
e γ̇ γ̇ = ±kJ γ̇ + σ(γ̇, γ̇). Note that∇e γ̇ γ̇k =k∇p

k 2 + λ2 =p

k 2 + c − c̃ .

Here we put±kJ γ̇ + σ(γ̇, γ̇).
Yes = √
k 2 + c − c̃

SUBMANIFOLD THEORY FROM THE VIEWPOINT OF CIRCLES

31

So we have³´1e γ̇ ± kJ γ̇ + σ(γ̇, γ̇)∇k 2 + c − c̃³´1e γ̇ (J γ̇) + ∇e γ̇ (σ(γ̇, γ̇))=√± k∇
k 2 + c − c̃³´1
± k∇γ̇ (J γ̇) ± k · σ(γ̇, J γ̇) − Aσ(γ̇,γ̇) γ̇ + Dγ̇ (σ(γ̇, γ̇))=√k 2 + c − c̃³´1¯ γ̇ σ)(γ̇, γ̇) ± 2k · σ(J γ̇, γ̇)=√− k 2 γ̇ − λ2 γ̇ + (∇k 2 + c − c̃p
= − k 2 + c − c̃ γ̇.

e γ̇ Yes = √∇nTherefore we can see that for each Kähler√ circle γ of curvature k(> 0) on CP (c)the curve f ◦ γ is a circle (of curvature k 2 + c − c̃ ), so that it is a plane curve
f2n+p (c̃). ¤in M

Remark. Theorem 6 also holds under the condition κ ≡ 0 (see [14]).
As a consequence of Theorem 6 we establish the following corollary ([8]):Corollary. Let f be an isometric immersion of a Kähler manifold Mn into af2n+p (c̃) of constant sectional curvature c̃. If there exists a nonreal space form M
constant positive smooth function κ satisfying that f maps every Kähler Frenet
f2n+p (c̃), then f is a totally
curve γ of curvature κ on Mn to a plane curve in M
f2n+p (c̃) = R2n+p , where c̃ = 0.
geodesic immersion of Mn = Cn into M
Proof. By assumption, as the curvature function κ is not constant, there exists
˙ 0 ) 6= 0 from (8.6). In thesome s0 with κ̇(s0 ) 6= 0. Since κ, κ̃ > 0, we find κ̃(s¯following, we will use the fact that ∇σ = 0 and equation (8.5). Then equation(7.2) yields σ(γ̇(s0 ), γ̇(s0 )) = 0. Now, we can see that kσ(γ̇, γ̇)k is constant along
the curve γ. In fact, we haved¯ γ̇ σ)(γ̇, γ̇), σ(γ̇, γ̇)i ± 4κhσ(J γ̇, γ̇), σ(γ̇, γ̇)i = 0.
kσ(γ̇, γ̇)k2 = 2h(∇ds
Hence we conclude σ(v, v) = 0 for an arbitrary unit vector v ∈ Tx Mn at eachf2n+p (c̃) is a totally geodesic immersion. ¤point x ∈ Mn . Thus f : Mn → M
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Department of Mathematics, Shimane University Matsue, Shimane, 690-8504,
JAPAN
E-mail address: smaeda@riko.shimane-u.ac.jp