Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
35 (2002), pp. 23–28

CIRCLES AND TOTALLY GEODESIC KÄHLER
SUBMANIFOLDS
Kaoru Suizu(Received: January 31, 2002)
Communicated by S. Maeda
Abstract. The purpose of this paper is to characterize all totally geodesic
Kähler submanifolds by some circles.

1. Introduction.
This paper is a part of a joint work with Professors T. Adachi and S. Maeda [SMA].
We first review the definition of circles. Let γ = γ(s) be a regular curveparametrized by its arclength s in a Riemannian manifold M . Then the curve
γ is called a circle if there exist a field V = V (s) of unit vectors along γ and aconstant k (≥ 0) satisfying(∇γ̇ γ̇ = kV,(1)∇γ̇ V = −k γ̇,where ∇γ̇ denotes the covariant differentiation along γ with respect to the Riemannian connection ∇ of M . The constant k is called the curvature of the circle.
A circle of curvature zero is nothing but a geodesic. For each point x ∈ M , eachorthonormal pair (u, v) of vectors at x and each positive constant k, there exists
locally a unique circle γ = γ(s) on M with initial condition that γ(0) = x, γ̇(0) = uand ∇γ̇ γ̇(0) = kv. For details, see [NY].
We here recall the following two parallel surfaces in Euclidean space. Let f1 be
a totally umbilic imbedding of a 2-dimensional standard sphere S 2 (c) of curvaturec into Euclidean space R5 , and let f2 = ι ◦ f be an isometric parallel immersion
of S 2 (c) into R5 . Here f is the second standard minimal immersion of S 2 (c) into
S 4 (3c) and ι is a totally umbilic imbedding of S 4 (3c) into R5 . We know that foreach great circle γ on S 2 (c), both of the curves f1 ◦ γ and f2 ◦ γ are circles in
the ambient space R5 . This implies that we cannot distinguish f1 from f2 by theextrinsic shape of geodesics of S 2 (c) in R5 . However we emphasize that we candistinguish these two isometric immersions f1 and f2 by the extrinsic shape ofThe author talked this paper in the international conference of Geometry of Riemannian Submanifolds, Tokyo Metropolitan University, December 17-19, 2001.
23

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KAORU SUIZU(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 theambient R5 (for details, see [S]).
In this context we are interested in the extrinsic shape of circles of the submanifold.
We recall a Kähler circle in a Kähler manifold (M, J) with complex structure J.
Let γ be a circle in a Kähler manifold M . Then we see from (1) that hγ̇, JV i isconstant along γ. Therefore it makes sense to define a Kähler circle as a circle γ
satisfying the condition that γ̇ and Y span a holomorphic plane, that is, Y = J γ̇or Y = −J γ̇. Note that if γ is a Kähler circle, then (1) reduces to ∇γ̇ γ̇ = kJ γ̇ or∇γ̇ γ̇ = −kJ γ̇.
Let M be a complex n-dimensional complex space form Mn (c), which is locally either a complex projective space CP n (c) of holomorphic sectional curvaturec (> 0), a complex Euclidean space Cn or a complex hyperbolic space CH n (c) ofholomorphic sectional curvature c (< 0). Kähler circles γ of curvature k on M are
the following plane curves:
γγγcircle⊂

Kähler

CP 1 (c) −−−−−−−−−→ CP n (c),circle⊂circle⊂C

1

totally geodesic
Kähler−−−−−−−−−→ Cn ,totally geodesic
Kähler

CH 1 (c) −−−−−−−−−→ CH n (c).
totally geodesic

In this paper we pay particular attention to Kähler circles. Nomizu characterized
totally geodesic complex projective spaces among Kähler submanifolds in a complex
projective space by an extrinsic property of geodesics in his paper [N]. He gives
the following : A Kähler submanifold Mn in a complex projective space CP n+p islocally congruent to CP n , which is a totally geodesic Kähler submanifold of CP n+p ,if every geodesic on Mn is locally contained in a complex projective line CP 1 in
CP n+p .
Motivated by this characterization, we shall characterize all totally geodesic
Kähler immersions into an arbitrary Kähler manifold by the extrinsic shape ofKähler circles on the submanifold.
2. Results.
For the characterization of totally geodesic Kähler immersions, we review thedefinition of a Frenet curve of order 2 on a Riemannian manifold.
A smooth curve γ = γ(s) parametrized by its arclength s on a Riemannianmanifold M is called a Frenet curve of order 2 if there exist a field V = V (s) ofunit vectors along γ and a positive function κ = κ(s) satisfying(∇γ̇ γ̇ = κV,(2)∇γ̇ V = −κγ̇.
The function κ is called the curvature of γ. Of course a circle of positive curvature
is a Frenet curve of order 2 in a trivial sense and a Frenet curve of order 1 is nothing

TOTALLY GEODESIC KÄHLER SUBMANIFOLDS

25

but a geodesic.
Theorem. Let f be a Kähler isometric immersion of a Kähler manifold M intof. Then the following conditions are equivalent:
an arbitrary Kähler manifold M(a) f is a totally geodesic immersion.
(b) There exists k > 0 satisfying that f maps every Kähler circle of curvature kf.
on M to a circle in M(c) There exists k > 0 satisfying that f maps every Kähler circle of curvature kf.
on M to a Frenet curve of order 2 in M
Proof. It suffices to show that (c) implies (a). Let γ be a Kähler circle of curvature
k on M which satisfies ∇γ̇ γ̇ = ±kJ γ̇. By the assumption it is mapped to a Frenetf satisfying the equations
curve f ◦ γ of order 2 in the ambient manifold M(e γ̇ γ̇ = κ∇eV,(3)e∇γ̇ V = −eκγ̇,e denotes the covariant differentiation on Mf and κe γ̇ γ̇k. Here forwhere ∇e = k∇simplicity we usually denote the curve f ◦ γ by γ. We get from (3)e γ̇ ∇e γ̇ γ̇ = κe γ̇ (κ̃V ) = κe γ̇ γ̇ − κκe∇e∇e0 ∇e3 γ̇,dwhere κ
e0 = dsκe and s is the arclength parameter of γ. So we obtain equatione γ̇ ∇e γ̇ γ̇ + κe γ̇ γ̇.
κe(∇e2 γ̇) = κe0 ∇(4)e by use of the formulae of Gauss andWe calculate the covariant differentiation ∇Weingarten :(5)e X Z = ∇X Z + σ(X, Z),∇e X ξ = D X ξ − Aξ X∇

where σ denotes the second fundamental form of f . Here, we define the covariant¯ of the second fundamental form σ with respect to the connection
differentiation ∇in (tangent bundle) + (normal bundle) as follows:¯ X σ)(Y, Z) = DX (σ(Y, Z)) − σ(∇X Y, Z) − σ(Y, ∇X Z).
(∇It follows from (5) that(6)e γ̇ ∇e γ̇ γ̇ = ∇γ̇ ∇γ̇ γ̇ + σ(∇γ̇ γ̇, γ̇) − Aσ(γ̇,γ̇) γ̇ + Dγ̇ (σ(γ̇, γ̇))∇¯ γ̇ σ)(γ̇, γ̇)= −k 2 γ̇ ± 3kJ(σ(γ̇, γ̇)) − Aσ(γ̇,γ̇) γ̇ + (∇and(7)e γ̇ γ̇, ∇e γ̇ γ̇i = k 2 + kσ(γ̇, γ̇)k2 .
κe2 = h∇We obtain from (4),(6) and (7)(8)¯ γ̇ σ)(γ̇, γ̇)}
κe{kσ(γ̇, γ̇)k2 γ̇ ± 3kJ(σ(γ̇, γ̇)) − Aσ(γ̇,γ̇) γ̇ + (∇=κe0 {±kJ γ̇ + σ(γ̇, γ̇)}.

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KAORU SUIZU

Taking the tangential and the normal components for the submanifold M in equation (8), we obtain the following:(9)(10)κe(−Aσ(γ̇,γ̇) γ̇ + kσ(γ̇, γ̇)k2 γ̇) = ±e
κ0 kJ γ̇,¯ γ̇ σ)(γ̇, γ̇)} = κκe{±3kJ(σ(γ̇, γ̇)) + (∇e0 σ(γ̇, γ̇).

Take the inner product of both hand-sides of (9) with J γ̇. We then find±e
κ0 k = −eκhAσ(γ̇,γ̇) γ̇, J γ̇i = −eκhσ(γ̇, γ̇), σ(γ̇, J γ̇)i = −eκhσ(γ̇, γ̇), J(σ(γ̇, γ̇))i = 0.
This implies that the only curvature function κe of the curve f ◦ γ is constant, sof. Thisthat this curve is a circle of positive curvature in the ambient manifold M
shows (c) implies (b).
For an arbitrary unit tangent vector v ∈ T M we denote by γi (i = 1, 2) Kähler
circles of curvature k with γ̇i (0) = v ∈ T M :∇γ̇1 γ̇1 = kJ γ̇1 , ∇γ̇2 γ̇2 = −kJ γ̇2 .
Following (10) we have¯ γ̇1 σ)(γ̇1 , γ̇1 ) = 0,3kJ(σ(γ̇1 , γ̇1 )) + (∇¯ γ̇2 σ)(γ̇2 , γ̇2 ) = 0.
−3kJ(σ(γ̇2 , γ̇2 )) + (∇Evaluating these at s = 0, we obtain¯ v σ)(v, v) = 0 = −3kJ(σ(v, v)) + (∇¯ v σ)(v, v),3kJ(σ(v, v)) + (∇which lead us to σ(v, v) = 0. As v is an arbitrary unit vector we find that M isf.
totally geodesic in M
A curve is said to be a plane curve in a Reimannian manifold M if it is locallycontained on some real 2-dimensional totally geodesic submanifold of M .
As an immediate consequence of this Theorem, we obtain the following characterization of totally geodesic Kähler submanifolds in a complete simply connectedfm (c).
complex space form M
Corollary. Let f be a Kähler isometric immersion of a Kähler manifold Mn
fm (c). Then the following
into a complete simply connected complex space form M
conditions are equivalent:(a) f is a totally geodesic immersion.
(b) There exists k > 0 satisfying that f maps every Kähler circle of curvature k
fm (c).
on Mn to a circle in M(c) There exists k > 0 satisfying that f maps every Kähler circle of curvature k
fm (c).
on Mn to a Frenet curve of order 2 in M(d) There exists k > 0 satisfying that f maps every Kähler circle of curvature kfm (c).
on Mn to a plane curve in M

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27

3. Remarks.
We first claim following.
fm (c)(= CP m (c) or CH m (c))
Proposition. In a nonflat complex space form M
generally, a Frenet curve γ of order 2 is not a plane curve.
Proof. We take a Frenet curve γ of order 2 in a Kähler manifold M with complex
torsion τ which is defined by τ = hγ̇, JV i. This complex torsin τ (−1 ≤ τ ≤ 1) isconstant from (2). In fact by direct computation, we find that∇γ̇ hγ̇, JV i = h∇γ̇ γ̇, JV i + hγ̇, J∇γ̇ V i = κhV, JV i − κhγ̇, J γ̇i = 0.
Needless to say we can take a Frenet curve of order 2 with each complex torsionτ (−1 ≤ τ ≤ 1) in an arbitrary Kähler manifold. Indeed, for any unit vectorfm (c) and any constant τ (|τ | < 1, τ 6= 0) we can take a unit vector
X ∈ TMfm (c) satisfying hX, Y i = 0 and hX, JY i = τ as follows: For a unit vector
Y ∈ TMfm (c) which satisfies hX, Y1 i = hJX, Y1 i = 0 we put
Y1 ∈ T M√
Y := −τ JX + 1 − τ 2 Y1 .
Then the vector Y satisfies kY k = 1 and√hX, Y i = hX, −τ JX + 1 − τ 2 Y1 i = 0,√hX, JY i = hX, τ X + 1 − τ 2 JY1 i = τ.

fm (c), γ is a plane curve if and only
Note that for a Frenet curve γ of order 2 in Mif τ = 0, ±1.
When τ = 0, this plane curve γ is as follows:
totally realγ ⊂ RP 2 ( 4c ) −−−−−−−−−→ CP n (c),totally geodesic
totally realγ ⊂ RH 2 ( 4c ) −−−−−−−−−→ CH n (c).
totally geodesic

When τ = ±1, this plane curve γ is as follows:Kählerγ ⊂ CP 1 (c) −−−−−−−−−→ CP n (c),totally geodesic
Kählerγ ⊂ CH 1 (c) −−−−−−−−−→ CH n (c).
totally geodesic

In the condition (d) of Corollary, the plane curve f ◦ γ is nothing but a Frenet
fm (c).
curve of order 2 with complex torsion τ = 1 or τ = −1 in M
Remark. If we put k = 0 in the statements of Theorem and Corollary, these results are no longer true. For example, we consider the second Veronese imbedding
2f : CP n (c/2) −→ CP (n +3n)/2 (c) which is defined by√(z0 , . . . , zn ) 7−→ (z02 , 2z0 z1 , . . . , zn2 ),28

KAORU SUIZU

where z0 , . . . , zn is the homogeneous coordinates of CP n . This non-totally geodesic
Kähler isometric
imbedding f maps every geodesic on CP n (c/2) to a circle of√curvature c/2 in a real projective plane RP 2 (c/4).
References
[N]
[NY]
[S]
[SMA]

K. Nomizu, A characterization of the Veronese varieties, Nagoya Math. J. 60(1976),
181–188.
K. Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann.
210(1974), 163–170.
K. Suizu, Characterizations of parallel imbeddings of projective spaces into space forms
by circles, Math. Japonicae 55(2002), 43–52.
K. Suizu, S. Maeda and T. Adachi, Characterization of totally geodesic Kähler immersions, to appear in Hokkaido Math. J..

Department of Mathematics, Shimane University, Matsue, Shimane, 690-8504
JAPAN
E-mail address: suizu@math.shimane-u.ac.jp