Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
44 (2011), pp. 27–32

η-UMBILICAL HYPERSURFACES IN P2 C AND H2 C
JONG TAEK CHO AND MAKOTO KIMURA(Received: February 6, 2011)
Abstract. We characterize totally η-umbilical hypersurfaces in P2 C or H2 C byusing the structure Jacobi operator or the Ricci operator.

1. Introduction
f, J, ge) be an n-dimensional Kähler manifold with a Kähler structure (J, ge)
Let (Mf with a unit normal vector N on
and let M be an orientable real hypersurface in M
M . Then the Reeb vector field ξ = −JN plays a fundamental role in real hypersurfaces in a Kähler manifold. In particular for a complex projective space Pn C, Cecil
and Ryan [1] proved that Hopf hypersurfaces (with ξ a principal curvature vectorﬁeld) are realized as tubes over certain submanifolds in Pn C, provided the rankof their focal maps is constant. In the geometry of hypersurfaces, the structure
Jacobi operator Rξ = R(·, ξ)ξ (along the Reeb ﬂow) has many interesting implications (cf. [2], [3], [4]). Recently, Ivey and Ryan [6] showed that there are no real
hypersurfaces whose structure Jacobi operator vanishes in P2 C or H2 C. In higher
dimensions, it was proved by the present authors [5].
From Codazzi equation, we can show that there are no totally umbilical real
hypersurfaces in a non-ﬂat complex space form. In this context, some authors
studies the so called totally η-umbilical structure in a real hypersurface in Pn C or
Hn C, that is, its shape operator A is represented by
A = λI + µη ⊗ ξfor λ, µ ∈ R. Indeed, totally η-umbilical real hypersurfaces in Pn C or Hn C areclassiﬁed in [1], [14] or [10]. They are realized as a geodesic hypersphere in Pn C
and a horosphere, a geodesic hypersphere or a tube over a complex hyperbolic
hyperplane Hn−1 C in Hn C. By Gauss equation we ﬁnd that Rξ is proportional to
I (identity transformation) on the orthogonal complement space ξ ⊥ of ξ for such
spaces. In the present note, we characterize totally η-umbilical hypersurfaces in
P2 C or H2 C by using the structure Jacobi operator or the Ricci operator.
2000 Mathematics Subject Classification. 53B20, 53C15, 53C25.
Key words and phrases. real hypersufaces, complex space forms, Ricci operators, structure
Jacobi operators.
27

28

J. T. CHO AND M. KIMURA

2. Preliminaries
All manifolds are assumed to be connected and of class C ∞ and the real hypersurfaces are supposed to be oriented. At ﬁrst, we review the fundamental facts on
fn (c) with constanta real hypersurface of a n-dimensional complex space form M
holomorphic sectional curvature 4c. Let M be an orientable real hypersurface offn (c) and let N be a unit normal vector on M . We denote by ge and J a KählerM
metric tensor and its Hermitian structure tensor, respectively. For any vector ﬁeld
X tangent to M , we put(2.1)

JX = ϕX + η(X)N,JN = −ξ,where ϕ is a (1,1)-type tensor ﬁeld, η is a 1-form and ξ is a unit vector ﬁeld on M ,which is called Reeb vector field. The induced Riemannian metric on M is denoted
by g. Then by properties of (eg , J) we see that the structure (ϕ, ξ, η, g) is an almost
contact metric structure on M , that is, from (2.1) it follows that
(2.2)

ϕ2 X = −X + η(X)ξ, η(ξ) = 1
g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y )for any vector ﬁelds X and Y tangent to M . From (2.2), we haveϕξ = 0, η ◦ ϕ = 0, η(X) = g(X, ξ)
The Gauss and Weingarten formula for M are given ase X Y = ∇X Y + g(AX, Y )N,∇e X N = −AX∇e and ∇ denote the Levi-Civita connecfor any tangent vector ﬁelds X, Y , where ∇fn (c), ge) and (M, g), respectively, A is the shape operator. From (2.1)
tions of (Me = 0, we then obtainand ∇J(2.3)(∇X ϕ)Y = η(Y )AX − g(AX, Y )ξ,∇X ξ = ϕAX.

Then we have the following Gauss and Codazzi equations:
R(X, Y )Z = c{g(Y, Z)X − g(X, Z)Y(2.4)+ g(ϕY, Z)ϕX − g(ϕX, Z)ϕY − 2g(ϕX, Y )ϕZ}
+ g(AY, Z)AX − g(AX, Z)AY.

(2.5)(∇X A)Y − (∇Y A)X = c{η(X)ϕY − η(Y )ϕX − 2g(ϕX, Y )ξ}.

From (2.4) together with (2.2) the Ricci operator S is given by(2.6)

SX = c{(2n + 1)X − 3η(X)ξ} + HAX − A2 X,

where H = trace of A. Also, from (2.4) the structure Jacobi operator Rξ = R(·, ξ)ξ,which is a self-adjoint operator, is given by(2.7)

Rξ (X) = c(X − η(X)ξ) + g(Aξ, ξ)AX − η(AX)Aξ.

η-UMBILICAL HYPERSURFACES IN P2 C AND H2 C

29

Now we consider the vector ﬁeld U = ∇ξ ξ and denote αm = η(Am ξ). Then from
(2.2) and (2.3) we easily observe thatg(U, ξ) = 0, g(U, Aξ) = 0,
∥U ∥2 = g(U, U ) = α2 − α12 .
Then we see at once that ξ is a principal curvature vector ﬁeld if and only if
α2 − α12 = 0. Moreover, at that time, α1 is constant (cf. [8], [12]).
3. Real hypersurfaces satisfying Sϕ = ϕS and Rξ ϕ = ϕRξ
In [2] the ﬁrst author studies a real hypersurface M in a non-ﬂat complex space
fn (c), c ̸= 0, which satisﬁes Sϕ = ϕS and at the same time Rξ ϕ = ϕRξ .
form M
Unfortunately, there contain some incorrect arguments. So, in this section we add
the correction of it.
From the condition Sϕ = ϕS, we have
(3.1)

H(Aϕ − ϕA)X − (A2 ϕ − ϕA2 )X = 0.

Put X = ξ in (3.1) to get ϕA2 ξ = HU . Applying ϕ, then we haveA2 ξ = HAξ + (α2 − α1 H)ξ.

(3.2)

Using (2.7) the commutativity Rξ ϕ = ϕRξ implies that
(3.3)

α1 (Aϕ − ϕA)X = −g(U, X)Aξ − η(AX)U.

Put X = Aξ in (3.3) to get α1 AU = α1 ϕA2 ξ − α2 U . Using (3.2) we get
α1 AU = (α1 H − α2 )U.

(3.4)

Now, we shall prove that M is a Hopf hypersurface. Put Ω = {p ∈ M : (α2 −α12 )(p) ̸= 0}. Suppose that Ω is non-empty and proceed our arguments in Ω. Thenfrom (3.4) we can see that α1 ̸= 0 in Ω. Use the relation:(A2 ϕ − ϕA2 )X = A(Aϕ − ϕA)X + (Aϕ − ϕA)AX.
Then, from (3.1) and (3.3) we have()(3.5) α1 H(Aϕ−ϕA)X = − g(U, X)A2 ξ +η(AX)AU +g(U, AX)Aξ +η(A2 X)U .
Using (3.2), (3.3) and (3.4) in (3.5), then we obtain
(
)
(3.6)
(α2 − α1 H) α1 g(U, X)ξ − η(AX)U − g(U, X)Aξ + α1 η(X)U = 0.
Put X = U in (3.6) to get(3.7)(α2 − α1 H)(α2 − α12 )Aξ = (α2 − α1 H)(α2 − α12 )α1 ξ,which yields that α2 − α1 H = 0 in Ω. Hence, from (3.2) and (3.4) we obtain

30

J. T. CHO AND M. KIMURA

Lemma 3.1. In Ω,(3.8)

AU = 0andA2 ξ = HAξ.

(3.9)

Diﬀerentiating α1 = g(Aξ, ξ) covariantly, using 2nd equation of (2.3) and (3.8)
we easily get(3.10)g((∇X A)ξ, ξ) = dα1 (X),

where d denotes the exterior diﬀerentiation. Since U = ϕAξ, by using 1st equationof (2.3), (2.5) and (3.8), we have∇ξ U = α1 Aξ − α2 ξ + ϕ grad(α1 ),(3.11)

where grad(α1 ) denotes the gradient vector ﬁeld of α1 .
Diﬀerentiating (3.8) covariantly along Ω, then by using (2.5) and (3.11) we have
(3.12)(∇U A)ξ = −cϕU − α1 A2 ξ + α2 Aξ − Aϕ grad(α1 ).

Also, if we diﬀerentiate (3.9) covariantly along Ω, then together with (2.3) we get(3.13)

g(Aξ, (∇X A)Y ) + g((∇X A)ξ, AY ) + g(ϕAX, A2 Y )= dH(X)g(Aξ, Y ) + Hg((∇X A)ξ, Y ) + Hg(ϕAX, AY ).

From (3.13), using Codazzi equation (2.5), then it follows that
(3.14)()
c η(X)g(Aξ, ϕY ) − η(Y )g(Aξ, ϕX) − 2α1 g(ϕX, Y )+ g((∇X A)ξ, AY ) − g((∇Y A)ξ, AX) + g(ϕAX, A2 Y ) − g(ϕAY, A2 X)= dH(X)g(Aξ, Y ) − dH(Y )g(Aξ, X) + Hg((∇X A)ξ, Y ) − Hg((∇Y A)ξ, X)+ 2Hg(ϕAX, AY )for any vector ﬁelds X and Y tangent to Ω. Putting X = U and making use of(2.5) and (3.8), then we have()
(3.15) g((∇U A)ξ, AY ) = c 2(α1 − H)g(ϕU, Y ) − η(Y )g(U, U ) + dH(U )g(Aξ, Y ).
Hence, from (3.12) and (3.15), we have
(3.16)− cg(ϕU, AY ) + dα1 (ϕA2 Y )()= c 2(α1 − H)g(ϕU, Y ) − η(Y )g(U, U ) + dH(U )g(Aξ, Y ),where we have used (α2 − α1 H) = 0. If we put Y = ξ in (3.16), then use (3.9) toobtainα1 dH(U ) − Hdα1 (U ) = 2c(α2 − α1 2 ).
Putting Y = Aξ in (3.16) and using (3.9) again, then we obtain
H(α1 dH(U ) − Hdα1 (U )) = c(3α1 − H)(α2 − α1 2 ).
From the above two equations, we have (H−α1 )(α2 −α12 ) = 0. But, since α2 = α1 H,
we have α2 − α12 = 0. Eventually, we have shown that M is a Hopf hypersurface.

η-UMBILICAL HYPERSURFACES IN P2 C AND H2 C

31

Moreover, from (3.3) we have α1 (Aϕ − ϕA) = 0. Therefore, due to Okumura [13]and Motiel-Romero [11] we have (cf. [2])Theorem 3.2. Let M be a real hypersurface of Pn C and Hn C. If M satisfies
ϕS = Sϕ and ϕRξ = Rξ ϕ at the same time, then Aξ = 0 or M is locally congruent
to one of the so-called real hypersurfaces of type (A).
4. 3-dimensional real hypersurfaces
Let M be a real hypersurface in P2 C and H2 C. Then, since the Weyl curvaturetensor vanishes in dimension 3, we have()(4.1) R(X, Y )Z = ρ(Y, Z)X − ρ(X, Z)Y + g(Y, Z)SX − g(X, Z)SY − r/2 g(Y, Z)X − g(X, Z)Y
for any smooth vector ﬁelds X, Y, Z on M , where ρ(X, Y ) = g(SX, Y ) and rdenotes the scalar curvature. From (4.1) we get(4.2)

Rξ (X) = ρ(ξ, ξ)X − ρ(X, ξ)ξ + SX − η(X)Sξ − r/2(X − η(X)ξ).

It follows from (4.2) that(Rξ ϕ − ϕRξ )(X) = (Sϕ − ϕS)(X) − ρ(ϕX, ξ)ξ + η(X)ϕSξ.
Then we can easily show the following result.
Proposition 4.1. For a 3-dimensional real hypersurface M of P2 C and H2 C, thefollowing four conditions are equivalent:• Sϕ = ϕS;• M is pseudo-Einstein (or η-Einstein), which means S = aI + bη ⊗ ξ for smoothfunctions a and b;• Rξ ϕ = ϕRξ and Sξ = σξ;• Rξ = f (I − η ⊗ ξ) and Sξ = σξ, where f, σ are smooth functions.
Then, using Theorem 2 we have
Theorem 4.2. Let M be a real hypersurface in P2 C and H2 C which satisfies oneof four in Proposition 4.1. Then M is locally congruent to a geodesic hyperspherein P2 C and a horosphere, a geodesic hypersphere, a tube over a complex hyperbolicline H1 C in H2 C, or a Hopf hypersurface with Aξ = 0 in P2 C and H2 C.
The Reeb section is deﬁned by the plane spanned by {ξ, X} for a unit vector X orthogonal to ξ and the Reeb sectional curvature is deﬁned by K(X, ξ) =g(R(X, ξ)ξ, X). Then, we have
Corollary 4.3. Let M be a real hypersurface of P2 C or H2 C whose Ricci operator
S satisfies Sξ = σξ for a function σ. If the Reeb sectional curvature is pointwise
constant, then Aξ = 0, or otherwise M is locally congruent to a geodesic hypersphere in P2 C and a horosphere, a geodesic hypersphere or a tube over a complex
hyperbolic line H1 C in H2 C.
We close this paper by the following remark.

32

J. T. CHO AND M. KIMURA

Remark 4.4. Using Cecil-Ryan’s fundamental idea of tube construction in Pn C ([1]),Kimura and Maeda [9] found real hypersurfaces in Pn C with Aξ = 0, provided therank of their focal maps is constant. Indeed, they are realized as tubes of certain
complex submanifolds in Pn C of radius π/4. Very recently, Ivey and Ryan [7]constructed such a real hypersurface in H2 C with Aξ = 0 by a pair of Legendre
curves in the unit 3-sphere.
Acknowledgment. J.T. Cho was supported by Basic Science Research Program
through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2009-0071643) and M. Kimura waspartially supported by Grants-in-Aid for Scientiﬁc Research, The Ministry of Education, Science, Sports and Culture, Japan, No. 21540083.
References[1] T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space,
Trans. Amer. Math. Soc. 269 (1982), 481–499.
[2] J. T. Cho, Ricci operators and structural Jacobi operators on real hypersurfaces in a complex
space form, Taiwanese J. Math. 14 (2010), 1325–1336.
[3] J. T. Cho and U-H. Ki, Real hypersurfaces of complex projective space in terms of the Jacobi
operators, Acta Math. Hungar. 80(1-2) (1998), 155–167.
[4] J. T. Cho and U-H. Ki, Real hypersurfaces in complex space forms with Reeb flow symmetricstructure Jacobi operator, Canad. Math. Bull. 51 (2008), no. 3, 359–371.
[5] J. T. Cho and M. Kimura, Curvature of Hopf hypersurfaces in a complex space form, Results
Math., to appear.
[6] T. A. Ivey and P. J. Ryan, The structure Jacobi operator for hypersurfaces in CP2 and CH 2 ,Results Math. 56 (2009), 473–488.
[7] T. A. Ivey and P. J. Ryan, Hopf hypersurfaces of small Hopf principal curvature in CH 2 ,Geom. Dedicata 141 (2009), 147–161.
[8] U-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama
32 (1990), 207–221.
[9] M. Kimura and S. Maeda, On real hypersurfaces of a complex projective space II, Tsukuba
J. Math. 15 (1991), 547–561.
[10] S. Montiel, Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan 37 (1985),
515–535.
[11] S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom.
Dedicata 20 (1986), 245–261.
[12] R. Niebergall and P. J. Ryan, Real hypersurfaces of complex space forms, Math. Sci. Res.
Inst. Publ. 32 (1997), 233–305.
[13] M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math.
Soc. 212 (1975), 355–364.
[14] R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 15 (1975), 43–53.
Jong Taek Cho: Department of Mathematics, Chonnam National University,
Gwangju, 500-757, Korea
E-mail address: jtcho@chonnam.ac.kr
Makoto Kimura: Department of Mathematics, Faculty of Science and Engineering, Shimane University, Matsue, Shimane, 690-8504, Japan
E-mail address: mkimura@riko.shimane-u.ac.jp