Acta Math. Hungar., 122 (12) (2009), 173186.
DOI: 10.1007/s10474-008-8004-y
First published online July 13, 2008
REAL HYPERSURFACES IN COMPLEX
TWO-PLANE GRASSMANNIANS WITH
PARALLEL STRUCTURE JACOBI OPERATOR∗
I. JEONG1 , J. D. PÉREZ2 and Y. J. SUH3
1
2
National Institute for Mathematical Sciences, Daejeon 305-340, Korea
e-mail: [email protected]
Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada,
18071-Granada, Spain
e-mail: [email protected]
3
Kyungpook National University, Department of Mathematics, Taegu 702-701, Korea
e-mail: [email protected]
(Received January 15, 2008; revised March 17, 2008; accepted March 19, 2008)
Abstract. We give some non-existence theorems for Hopf real hypersurfaces
in complex two-plane Grassmannians G2 (Cm+2 ) with parallel structure Jacobi
operator Rξ .
0. Introduction
In the geometry of real hypersurfaces in complex space forms or in quaternionic space forms there have been many characterizations of homogeneous
hypersurfaces of type (A1 ), (A2 ), (B), (C), (D) and (E) in complex projective spaces Pn (C), of type (A0 ), (A1 ), (A2 ) and (B) in complex hyperbolic
spaces Hn (C) or of type (A1 ), (A2 ) and (B) in quaternionic projective spaces
QP m , which are completely classied by Cecil and Ryan [5], Kimura [7],
Berndt [2], Martinez and Pérez [9] respectively.
∗ The rst author was supported by KRCF, Grant No. C-RESEARCH-2006-11-NIMS, the second author by MEC-FEDER Grant MTM 2007-60371 and the third author by grant Proj. No.
R17-2008-001-01001-0 from Korea Science & Engineering Foundation.
Key words and phrases: real hypersurfaces, complex two-plane Grassmannians, parallel structure Jacobi operator, Hopf hypersurface.
2000 Mathematics Subject Classication: primary 53C40; secondary 53C15.
c 2008 Akadémiai Kiadó, Budapest
02365294/$ 20.00 °
174
I. JEONG, J. D. PÉREZ and Y. J. SUH
In particular, Kimura and Maeda [8] have considered a real hypersurface
M in complex projective spaces Pn (C) with Lie ξ -parallel Ricci tensor and in
quaternionic projective spaces QP m Pérez and Suh [10] have classied real hypersurfaces in QP m with D⊥ -parallel curvature tensor ∇ξi R = 0, i = 1, 2, 3,
where R denotes the curvature tensor of M in QP m , and D⊥ is a distribution dened by D⊥ = Span {ξ1 , ξ2 , ξ3 }. In such a case they are congruent to
a tube of radius π4 over a totally geodesic QP k in QP m , 2 5 k 5 m − 2.
Let us consider a complex two-plane Grassmannian G2 (Cm+2 ) which consists of all complex 2-dimensional linear subspaces in Cm+2 . The complex
two-plane Grassmannian G2 (Cm+2 ) is known to be the unique compact irreducible Riemannian symmetric space equipped with both a Kähler structure
J and a quaternionic Kähler structure J not containing J (see Berndt and
Suh [3], [4]). So, in G2 (Cm+2 ) we have two natural geometric conditions for
real hypersurfaces that [ξ] = Span {ξ} or D⊥ = Span {ξ1 , ξ2 , ξ3 } is invariant
under the shape operator. By using such conditions, Berndt and Suh [3] have
proved the following:
Theorem A. Let M be a connected real hypersurface in G2 (Cm+2 ),
m = 3. Then both [ξ] and D⊥ are invariant under the shape operator of
M if and only if
(A) M is an open part of a tube around a totally geodesic G2 (Cm+1 ) in
G2 (Cm+2 ), or
(B) m is even, say m = 2n, and M is an open part of a tube around a
totally geodesic QP n in G2 (Cm+2 ).
If the Reeb vector eld ξ of a real hypersurface M in G2 (Cm+2 ) is invariant by the shape operator, M is said to be a Hopf hypersurface. In such a case
the integral curves of the Reeb vector eld ξ are geodesics (see Berndt and
Suh [4]). Moreover, the ow generated by the integral curves of the structure
vector eld ξ for Hopf hypersurfaces in G2 (Cm+2 ) is said to be a geodesic
Reeb ow.
Let us introduce a structure Jacobi operator Rξ in such a way that
Rξ (X) = R(X, ξ)ξ
for the curvature tensor R(X, Y )Z of M in G2 (Cm+2 ), where ξ denotes the
Reeb vector; X , Y and Z are any tangent vector elds of M in G2 (Cm+2 ).
Then the structure Jacobi operator Rξ for the Reeb vector ξ is said to be parallel if the covariant derivative of the structure Jacobi operator Rξ vanishes,
that is, ∇X Rξ = 0 for any vector eld X on M .
Recently, some geometric properties for such a structure Jacobi operator
Rξ of real hypersurfaces in complex space forms Mn (c) have been studied by
many authors (see [6], [11] and [12]). Among them parallel properties of such
Acta Mathematica Hungarica 122, 2009
REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
175
a structure Jacobi operator was studied by Ki, Pérez, Santos and Suh [6].
Moreover, D-parallel or Lie ξ -parallel of the stucture Jacobi operator were
studied by Pérez, Santos and Suh (see [11] and [12]).
Related to such a structure Jacobi operator Rξ , in this paper we give a
non-existence theorem for real hypersurfaces M in G2 (Cm+2 ) with parallel
structure Jacobi operator, that is, ∇X Rξ = 0 along any vector elds X on
M as follows:
There do not exist any Hopf hypersurfaces in G2 (Cm+2 ),
m= 3, with parallel structure Jacobi operator if the distribution D or D⊥
component of the Reeb vector eld is invariant by the shape operator.
Theorem 1.
On the other hand, by Lemma 3.2 to be given in Section 3 we have proved
that the distribution D or D⊥ component of the Reeb vector eld is invariant
by the shape operator if the principal curvature α of the Reeb vector ξ for
Hopf hypersurface in G2 (Cm+2 ) is constant along the direction of ξ . So we
also assert the following
There do not exist any Hopf hypersurfaces in G2 (Cm+2 ),
m= 3, with parallel structure Jacobi operator if the principal curvature α is
constant along the direction of ξ .
Theorem 2.
1. Riemannian geometry of G2 (Cm+2 )
In this section we summarize basic material about G2 (Cm+2 ); for details we refer to [3] and [4]. By G2 (Cm+2 ) we denote the set of all complex two-dimensional linear subspaces in Cm+2 . The special unitary group
m+2 ) with stabilizer isomorphic to
G = SU
¡ (m + 2) acts ¢transitively on G2 (C
K = S U (2) × U (m) ⊂ G. Then G2 (Cm+2 ) can be identied with the homogeneous space G/K , which we equip with the unique analytic structure
for which the natural action of G on G2 (Cm+2 ) becomes analytic. Denote
by g and k the Lie algebra of G and K , respectively, and by m the orthogonal complement of k in g with respect to the CartanKilling form B of g.
Then g = k ⊕ m is an Ad (K)-invariant reductive decomposition of g. We put
o = eK and identify To G2 (Cm+2 ) with m in the usual manner. Since B is negative denite on g, its negative restricted to m × m yields a positive denite
inner product on m. By Ad (K)-invariance of B , this inner product can be
extended to a G-invariant Riemannian metric g on G2 (Cm+2 ). In this way
G2 (Cm+2 ) becomes a Riemannian homogeneous space, even a Riemannian
symmetric space.
The Lie algebra k has the direct sum decomposition k = su(m) ⊕ su(2)
⊕ R, where R is the center of k. Viewing k as the holonomy algebra of
Acta Mathematica Hungarica 122, 2009
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I. JEONG, J. D. PÉREZ and Y. J. SUH
G2 (Cm+2 ), the center R induces a Kähler structure J and the su(2)-part a
quaternionic Kähler structure J on G2 (Cm+2 ). If J1 is any almost Hermitian
structure in J, then JJ1 = J1 J , and JJ1 is a symmetric endomorphism with
(JJ1 )2 = I and tr (JJ1 ) = 0. This fact will be used frequently throughout
this paper.
A canonical local basis J1 , J2 , J3 of J consists of three local almost Hermitian structures Jν in J such that Jν Jν+1 = Jν+2 = −Jν+1 Jν , where the
index is taken modulo three. Since J is parallel with respect to the Rieman¯ of (G2 (Cm+2 ), g), for any canonical local basis J1 , J2 , J3
nian connection ∇
of J there exist three local one-forms q1 , q2 , q3 such that
¯ X Jν = qν+2 (X)Jν+1 − qν+1 (X)Jν+2
∇
(1.1)
for all vector elds X on G2 (Cm+2 ).
The Riemannian curvature tensor R̄ of G2 (Cm+2 ) is locally given by
(1.2)
R̄(X, Y )Z = g(Y, Z)X − g(X, Z)Y + g(JY, Z)JX
− g(JX, Z)JY − 2g(JX, Y )JZ
+
3
X
©
g(Jν Y, Z)Jν X − g(Jν X, Z)Jν Y − 2g(Jν X, Y )Jν Z
ª
ν=1
+
3
X
©
ª
g(Jν JY, Z)Jν JX − g(Jν JX, Z)Jν JY ,
ν=1
where J1 , J2 , J3 is any canonical local basis of J.
2. Some fundamental formulas for real hypersurfaces in G2 (Cm+2 )
Let M be a real hypersurface of G2 (Cm+2 ), that is, a hypersurface of
G2 (Cm+2 ) with real codimension one. The induced Riemannian metric on
M will also be denoted by g , and ∇ denotes the Riemannian connection of
(M, g). Let N be a local unit normal eld of M and A the shape operator of M with respect to N . The Kähler structure J of G2 (Cm+2 ) induces
an almost contact metric structure (φ, ξ, η, g) on M (see [3], [4], [15] and
[16]). Furthermore, let J1 , J2 , J3 be a canonical local basis of J. Then each
Jν induces an almost contact metric structure (φν , ξν , ην , g) on M . Using
the above expression for R̄, the Codazzi equation becomes (see [3], [4], [15]
and [16])
(∇X A)Y − (∇Y A)X = η(X)φY − η(Y )φX − 2g(φX, Y )ξ
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REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
+
3
X
©
ην (X)φν Y − ην (Y )φν X − 2g(φν X, Y )ξν
ª
ν=1
+
3
X
©
ην (φX)φν φY − ην (φY )φν φX
ª
ν=1
+
3
X
©
ª
η(X)ην (φY ) − η(Y )ην (φX) ξν .
ν=1
The following identities can be proved in a straightforward method and
will be used frequently in subsequent calculations:
(2.1)
(
φν+1 ξν = −ξν+2 ,
φν ξν+1 = ξν+2 ,
φν φν+1 X = φν+2 X + ην+1 (X)ξν ,
φξν = φν ξ,
ην (φX) = η(φν X),
φν+1 φν X = −φν+2 X + ην (X)ξν+1 .
Let us put
(2.2)
JX = φX + η(X)N,
Jν X = φν X + ην (X)N
for any tangent vector X of a real hypersurface M in G2 (Cm+2 ), where N
denotes a normal vector of M in G2 (Cm+2 ). Then from this and the formulas
(1.1) and (2.1) we have that
(2.3)
(∇X φ)Y = η(Y )AX − g(AX, Y )ξ,
(2.4)
∇X ξ = φAX,
∇X ξν = qν+2 (X)ξν+1 − qν+1 (X)ξν+2 + φν AX,
(2.5)
(∇X φν )Y = −qν+1 (X)φν+2 Y + qν+2 (X)φν+1 Y + ην (Y )AX
− g(AX, Y )ξν .
Moreover, from JJν = Jν J , ν = 1, 2, 3, it follows that
(2.6)
φφν X = φν φX + ην (X)ξ − η(X)ξν .
Then from (1.2) and the above formulas, the equation of Gauss is given by
(2.7)
R(X, Y )Z = g(Y, Z)X − g(X, Z)Y
+ g(φY, Z)φX − g(φX, Z)φY − 2g(φX, Y )φZ
+
X3
ν=1
©
g(φν Y, Z)φν X − g(φν X, Z)φν Y − 2g(φν X, Y )φν Z
ª
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178
I. JEONG, J. D. PÉREZ and Y. J. SUH
+
X3
−
−
©
ν=1
X3
©
ν=1
X3
g(φν φY, Z)φν φX − g(φν φX, Z)φν φY
©
ν=1
ª
ª
η(Y )ην (Z)φν φX − η(X)ην (Z)φν φY
ª
η(X)g(φν φY, Z) − η(Y )g(φν φX, Z) ξν
+ g(AY, Z)AX − g(AX, Z)AY.
On the other hand, by Aξ = αξ we have
X3
ην (ξ)ην (φY ).
(2.8)
Y α = (ξα)η(Y ) − 4
ν=1
We recall a lemma due to Berndt and Suh [4] as follows:
Lemma 2.1. If M is a connected orientable real hypersurface in
G2 (Cm+2 ) with geodesic Reeb ow, then
¡
¢
αg (Aφ + φA)X, Y − 2g(AφAX, Y ) + 2g(φX, Y )
=2
X3
©
ν=1
ην (X)ην (φY ) − ην (Y )ην (φX) − g(φν X, Y )ην (ξ)
ª
− 2η(X)ην (φY )ην (ξ) + 2η(Y )ην (φX)ην (ξ) .
Putting X = ξ in (2.8) gives
X3
grad α = (ξα)ξ + 4
ν=1
ην (ξ)φξν .
From this, we have
(2.9)
Y (ξα) = ξ(ξα)η(Y ) − 4α
X3
ν=1
ην (ξ)ην (Y ) + 4
If we assume that ξα = 0, it follows that
X3
X3
(2.10)
ην (ξ)ην (AX) = α
ν=1
ν=1
X3
ν=1
ην (ξ)ην (AY ).
ην (ξ)ην (X).
Without loss of generality, we may put the structure vector ξ in such a
way that ξ = η(X0 )X0 + η(ξ1 )ξ1 for some units X0 ∈ D and ξ1 ∈ D⊥ . Then
we assert the following
Lemma 2.2. Let M be a Hopf real hypersurface in G2 (Cm+2 ). If the principal curvature α is constant along the direction of ξ , then the distribution
D or D⊥ component of the structure vector eld ξ is invariant by the shape
operator.
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REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
179
3. The parallel structure Jacobi operator
Let us denote by R(X, Y )Z the curvature tensor of M in G2 (Cm+2 ). The
the structure Jacobi operator Rξ of M in G2 (Cm+2 ) is dened by
(3.1)
Rξ X = R(X, ξ)ξ
for any vector eld X ∈ Tx M and x ∈ M .
Let us assume that the structure Jacobi operator Rξ on a Hopf hypersurface M in G2 (Cm+2 ) is parallel, that is (∇X Rξ )Y = 0. Then by using (2.3),
(2.4) and (2.5) in (3.1), we get
(3.2)
0 = (∇X Rξ )Y = −g(φAX, Y )ξ − η(Y )φAX
−
X3
ν=1
h
g(φν AX, Y )ξν − 2η(Y )ην (φAX)ξν
©
+ ην (Y )φν AX + 3 g(φν AX, φY )φν ξ + η(Y )ην (AX)φν ξ
ª
+ ην (φY )(φν φAX − αη(X)ξν )
i
©
ª
+ 4ην (ξ) ην (φY )AX − g(AX, Y )φν ξ + 2ην (φAX)φν φY
¡
¢
¡
¢
+ η (∇X A)ξ AY + α(∇X A)Y − αη (∇X A)Y ξ
− αg(AY, φAX)ξ − αη(Y )(∇X A)ξ − αη(Y )AφAX.
If we put Y = ξ in (3.2), then we have
(3.3)
−
0 = (∇X Rξ )ξ = −φAX − αAφAX
X3
ν=1
©
ª
− ην (φAX)ξν + ην (ξ)φν AX + 3ην (AX)φν ξ − 4αην (ξ)η(X)φν ξ .
We want to verify the following lemma which will be used in the proof of
our main Theorems 1 and 2 stated in the introduction.
Lemma 3.1. Let M be a Hopf real hypersurface in a complex two-plane
Grassmannian G2 (Cm+2 ) with parallel structure Jacobi operator. If D and
D⊥ component of the Reeb vector ξ is invariant by the shape operator, then
the Reeb vector ξ belongs to either the distribution D or the distribution D⊥ .
Proof. When the function α = g(Aξ, ξ) identically vanishes, the lemma
can be veried directly by Berndt and Suh [3].
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180
I. JEONG, J. D. PÉREZ and Y. J. SUH
In this proof we consider only the case that the function α is nonvanishing. Let us put ξ = η(X0 )X0 + η(ξ1 )ξ1 for some unit X0 ∈ D. Then it
suces to show that η(X0 ) = 0 or η(ξ1 ) = 0.
Let us put X = ξ and Y = X0 in (3.2) and use that M is a Hopf. Then
it follows that
(3.4)
0 = 4αη1 (ξ)η(X0 )φ1 X0 + (ξα)AX0 + α(∇ξ A)X0 − 2α(ξα)η(X0 )ξ.
On the other hand, by virtue of Codazzi equation and the paper due to
Berndt and Suh [3] we have
(3.5)
(∇ξ A)X0 = (ξα)η(X0 )ξ + αφAX0 − AφAX0 + φX0 + η1 (ξ)φ1 X0 .
Then substituting (3.5) into (3.4) and using the expression ξ = η(X0 )X0 +
η(ξ1 )ξ1 to the obtained equation, we have the following:
(3.6)
0 = 4αη1 (ξ)η 2 (X0 )φ1 X0 + (ξα)AX0 − α(ξα)η 2 (X0 )X0
− α(ξα)η(X0 )η(ξ1 )ξ1 + α2 φAX0 − αAφAX0
+ αφX0 + αη1 (ξ)φ1 X0 .
From this, taking an inner product with ξ1 , we have
0 = (ξα)g(AX0 , ξ1 ) − α(ξα)η(X0 )η(ξ1 ) + α2 η(X0 )g(φ1 AX0 , X0 ).
Hence, together with the assumption in our Lemma, it follows that ξα = 0 or
η(X0 )η(ξ1 ) = 0, because the function α is non-vanishing. If the second case
holds, we get our result. So let us consider the rst case. Then AX0 = αX0
and (3.6) gives
(3.7) 0 = 4αη1 (ξ)η 2 (X0 )φ1 X0 + α3 φX0 − α2 AφX0 + αφX0 + αη1 (ξ)φ1 X0 .
On the other hand, we know the following
(3.8)
φX0 = −η1 (ξ)φ1 X0 ,
where we have used 0 = φξ = η(X0 )(φX0 + η(ξ1 )φ1 X0 ). Then from this, together with (3.7) it follows that
(3.9)
0 = 4αη1 (ξ)η 2 (X0 )φ1 X0 + α3 φX0 − α2 AφX0 .
On the other hand, if we put X = X0 in Lemma 2.1 and use AX0 = αX0 ,
we have
αAφX0 + α2 φX0 − 2αAφX0 + 2φX0 = −2η1 (ξ)φ1 X0 + 4η 2 (X0 )η1 (ξ)φ1 X0 .
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REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
181
Hence and by (3.8) it follows that
(3.10)
AφX0 =
α2 + 4η 2 (X0 )
φX0 .
α
Now substituting (3.10) into (3.9) and also using (3.8), we have 0 =
−8αη 2 (X0 )φX0 . This means φX0 = 0, which is a contradiction.
¤
By Lemma 2.2 we also assert the following
Lemma 3.2. Let M be a Hopf real hypersurface in a complex two-plane
Grassmannian G2 (Cm+2 ) with parallel structure Jacobi operator. If the principal curvature α is constant along the direction of ξ , then the Reeb vector ξ
belongs either to the distribution D or D⊥ .
Before giving our proof of the main theorem in the introduction, let us
check that real hypersurfaces of type (A) or type (B) in Theorem A satisfy
parallel structure Jacobi operator or not.
First we want to check whether the structure Jacobi operator Rξ of real
hypersurfaces of type (A) is parallel or not. So in order to do this we want
to use Proposition 3 in Berndt and Suh [3].
Now let us consider a unit eigenvector X ∈ Tβ , where Tβ is an eigenspace
√
¡√ ¢
with eigenvalue β = 2 cot
2r in [3]. In other words, we can substitute
X = ξ2 into (3.3). Then it follows that
0 = (∇ξ2 Rξ )ξ = β(αβ + 2)ξ3 .
Consequently,
√
¡ √we¢get β = 0 or αβ¡ + 2 =√0.¢The case β = 0 cannot occur since
β = 2 cot
2r for some r ∈ 0, π/ 8 .
Next, we
consider
we calculate αβ + 2
√
¡ √ the
¢ case αβ√+ 2 =¡ 0√. Actually
¢
where α = 8 cot
8r and β = 2 cot
2r :
¡√ ¢
¡√ ¢
¡√ ¢
0 = αβ + 2 = 2{ cot
2r − tan
2r } cot
2r + 2.
¡√ ¢
Thus we have cot
2r = 0 and this case also cannot occur for some
√ ¢
¡
r ∈ 0, π/ 8 . So we know that the structure Jacobi operator Rξ of real
hypersurfaces of type A in G2 (Cm+2 ) can not be parallel.
Let us check whether the structure Jacobi operator of real hypersurfaces
of type (B) in Theorem A is parallel or not. In order to do this we use
Proposition 2 in Berndt and Suh [3]. Putting X = ξ2 in (3.3) and using
Proposition 2, we have
0 = (∇ξ2 Rξ )ξ = −4βφξ2 .
This case can not happen for some r belonging to (0, π/4), which is a contradiction. So we also know that the structure Jacobi operator of real hypersurfaces of type (B) mentioned in Theorem A cannot be parallel.
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I. JEONG, J. D. PÉREZ and Y. J. SUH
4. Proof of the main theorem
In this section we consider a real hypersurface M in G2 (Cm+2 ) with parallel structure Jacobi operator, that is ∇X Rξ = 0 for any vector eld X on M ,
such that its Reeb vector eld belongs to D. Then rst we consider the
following
Let M be a connected Hopf real hypersurface in G2 (Cm+2 )
with parallel Jacobi structure operator. If the Reeb vector eld ξ belongs to
the distribution D⊥ , then g(AD, D⊥ ) = 0.
Proof. We will show that g(AX, ξν ) = 0 for any ν = 1, 2, 3 and X ∈ D.
In order to do this, we may put ξ = ξ1 , because ξ ∈ D⊥ . Then it suces to
show that η2 (AX) = η3 (AX) = 0.
On the other hand, we know that η2 (φAX) = −g(AX, φξ2 ) = g(AX, ξ3 )
= η3 (AX) and η3 (φAX) = −g(AX, φξ3 ) = −g(AX, ξ2 ) = −η2 (AX) for any
X ∈ D.
From these formulas, together with (3.3) we get
Lemma 4.1.
(4.1) 0 = (∇X Rξ )ξ = −φAX − αAφAX − 2η3 (AX)ξ2 + 2η2 (AX)ξ3 − φ1 AX
for any vector eld X ∈ D. Taking an inner product (4.1) with ξ2 , we get
(4.2)
αg(AφAX, ξ2 ) + 2η3 (AX) = 0
for any vector eld X ∈ D.
On the other hand, by our assumption we can use Lemma 2.1. For any
X ∈ D we have
(4.3)
2g(AφAX, ξ2 ) = αg(AφX, ξ2 ) + αη3 (AX).
Then (4.2) can be changed into
(4.4)
0 = α2 g(AφX, ξ2 ) + (α2 + 4)η3 (AX)
for any X ∈ D. If α is non-vanishing, then we have
(4.5)
g(AφX, ξ2 ) = −
α2 + 4
η3 (AX).
α2
In particular, we see that if X ∈ D and ξ ∈ D⊥ , then φX ∈ D. By replacing
X by φX in (4.4) we have
¡
¢
0 = α2 g (A − X + η(X)ξ , ξ2 ) + (α2 + 4)η3 (AφX)
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REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
183
for any X ∈ D. Consequently, we have
−α2 g(AX, ξ2 ) + (α2 + 4)η3 (AφX) = 0.
(4.6)
Similarly, by taking an inner product with ξ3 to (4.1), we get
(4.7)
αg(AφAX, ξ3 ) − 2η2 (AX) = 0
for any X ∈ D. Substituting Y = ξ3 into Lemma 2.1 and using η3 (φAX)
= −η2 (AX) gives
(4.8)
2g(AφAX, ξ3 ) = αg(AφX, ξ3 ) − αη2 (AX).
Using (4.7) and (4.8), we get
0 = α2 g(AφX, ξ3 ) − (α2 + 4)η2 (AX),
(4.9)
for any vector eld X ∈ D. Suppose α 6= 0, then we have
(4.10)
g(AφX, ξ3 ) =
α2 + 4
η2 (AX).
α2
Also let us replace X by φX in (4.9), because we know that φX ∈ D for any
X ∈ D. Then we get
α2 g(AX, ξ3 ) + (α2 + 4)η2 (AφX) = 0.
(4.11)
For the case α = 0 we have g(AX, ξν ) = 0 for ν = 2, 3 and X ∈ D from
(4.2) and (4.7), respectively.
Next we consider the case α 6= 0. In order to show g(AD, D⊥ ) = 0 let us
substitute (4.10) into (4.6). Then we have
(
)
2 + 4)2
(α
0 = −α2 +
η2 (AX).
α2
2
2
Since −α2 + (α α+4)
6= 0, we have η2 (AX) = g(AX, ξ2 ) = 0 for any X ∈ D.
2
Also, let us substitute (4.5) into (4.11):
(
)
2 + 4)2
(α
0 = α2 −
η3 (AX).
α2
2
2
Since α2 − (α α+4)
6= 0, we have η3 (AX) = g(AX, ξ3 ) = 0 for any X ∈ D.
2
Hence, g(AX, ξν ) = 0 for ν = 1, 2, 3 and any X ∈ D, that is, g(AD, D⊥ ) = 0.
¤
Acta Mathematica Hungarica 122, 2009
184
I. JEONG, J. D. PÉREZ and Y. J. SUH
Next we consider a real hypersurface M in G2 (Cm+2 ) with parallel structure Jacobi operator Rξ and its Reeb vector ξ ∈ D⊥ as follows:
Let M be a connected Hopf real hypersurface in G2 (Cm+2 )
with parallel structure Jacobi operator. If the Reeb vector eld ξ belongs to
the distribution D, then g(AD, D⊥ ) = 0.
Proof. Using the assumption that ξ ∈ D, we substitute into (3.3) and
get
Lemma 4.2.
(4.12)
0 = −φAX − αAφAX −
X3
©
ν=1
ª
− ην (φAX)ξν + 3ην (AX)φν ξ .
Taking an inner product with ξµ for any µ = 1, 2, 3,
(4.13)
αg(AφAX, ξµ ) = 0,
µ = 1, 2, 3
for any X ∈ Tx M and any point x ∈ M .
On the other hand, let us substitute X = ξµ and Y = X into (3.2). Then
for ξ ∈ D we have
(4.14)
0 = g(AφX, ξµ )ξ − η(X)φAξµ
−
X3
ν=1
[g(φν Aξµ , X)ξν − 2η(X)ην (φAξµ )ξν + ην (X)φν Aξµ
ª
©
+ 3 g(φν Aξµ , φX)φν ξ + η(X)ην (Aξµ )φν ξ + ην (φX)φν φAξµ
+ 2ην (φAξµ )φν φX ]
¡
¢
¡
¢
+ η (∇ξµ A)ξ AX + α(∇ξµ A)X − αη (∇ξµ A)X ξ
− αg(AX, φAξµ )ξ − αη(X)(∇ξµ A)ξ − αη(X)AφAξµ .
From this, taking an inner product with ξ and using (4.13) and Aξ = αξ ,
we get
(4.15)
0 = g(AφX, ξµ ) −
X3
ν=1
X3
ην (X)g(φν Aξµ , ξ) − 3
ν=1
X3
−2
ν=1
ην (φX)g(φν φAξµ , ξ)
ην (φAξµ )g(φν φX, ξ)
for any tangent vector eld X . From this, let us replace X by φX to get
(4.16) g(AX, ξµ ) =
X3
ν=1
X3
ην (φAξµ )g(φν X, ξ) + 3
Acta Mathematica Hungarica 122, 2009
ν=1
ην (X)g(φν φAξµ , ξ)
REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS
185
for any µ = 1, 2, 3 and any tangent vector eld X ∈ Tx M , x ∈ M .
Since ξ ∈ D, we see that φν ξ ∈ D for all ν = 1, 2, 3. Let D0 = {X ∈ D |
X⊥ξ , φ1 ξ , φ2 ξ , φ3 ξ} ⊂ D. Then the tangent vector space Tx M for any point
x ∈ M is
Tx M = D ⊕ D⊥ = [ξ] ⊕ [φ1 ξ, φ2 ξ, φ3 ξ] ⊕ D0 ⊕ D⊥ .
In order to show that g(AX, ξµ ) = 0 for any X ∈ D and µ = 1, 2, 3, we
consider that X = ξ . Then we have g(Aξ, ξµ ) = αg(ξ, ξµ ) = 0 for any µ =
1, 2, 3.
Next, we consider X ∈ [φ1 ξ, φ2 ξ, φ3 ξ]. Since η(ξν ) = 0 for any ν = 1, 2, 3,
we see that g(∇ξµ ξ, ξν ) = −g(ξ, ∇ξµ ξν ) for any µ = 1, 2, 3. Thus we have
g(Aφν ξ, ξµ ) = 0 for ν, µ = 1, 2, 3.
Next, we consider X ∈ D0 where D0 = {X ∈ D | X⊥ξ, φ1 ξ, φ2 ξ, φ3 ξ}.
Since X⊥φν ξ for all ν , we have from (4.16) g(AX, ξµ ) = 0.
From these facts we assert that g(AX, ξµ ) = 0 for any X ∈ D and any
µ = 1, 2, 3.
¤
Summing up Lemmas 3.1, 4.1 and 4.2, and using Theorem A in the introduction, we know that any Hopf hypersurface in G2 (Cm+2 ) with parallel
structure Jacobi operator is congruent to of type (A) or of type (B) if the distribution D or D⊥ -component of the Reeb vector ξ is invariant by the shape
operator. But, in Section 3 we have checked that the structure Jacobi operator Rξ of any hypersurfaces of type (A) or of type (B) in Theorem A cannot
be parallel. So we completed the proof of our Theorems 1 and 2.
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