Advances in Pure Mathematics, 2011, 1, 67-72
doi:10.4236/apm.2011.13015 Published Online May 2011 (http://www.scirp.org/journal/apm)
Real Hypersurfaces in Complex Two-Plane Grassmannians
whose Jacobi Operators Corresponding to D-Directions
are of Codazzi Ty pe
Carlos J. G. Machado1, Juan de Dios Pérez1, Young Jin Suh2
1Department of Geom etry and Topology, University of Granada, Granada, Spain
2Department of Mat hem at ic s, Kyungpook National U niversi t y , Taegu, Republic of Kore a
Received January 11, 2011; revised March 14, 2011; accepted March 20, 2011
Abstract
We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Jacobi
operators corresponding to the directions in the distribution D

are of Codazzi type if they satisfy a further
condition. We obtain that that they must be either of type (A) or of type (B) (see [1]), but no one of these sat-
isfies our condition. As a consequence, we obtain the non-existence of Hopf real hypersurfaces in such am-
bient spaces whose Jacobi operators corresponding to D

-directions are parallel with the same further con-
dition.
Keywords: Real Hypersurfaces, Complex Two-Plane Grassmannians, Jacobi Operators, Codazzi Type
1. Introduction
The geometry of real hypersurfaces in complex space
forms or in quaternionic space forms is one of interesting
parts in the field of differential geometry. Now let us
consider consider real hypersurfaces in complex two-
plane Grassmannian

2
2
m
G , which consists of all
complex 2-dimensional linear subspaces in 2m

. It is
known to be the unique compact irreducible Riemannian
symmetric space equipped with both a Kähler structure
J
and a quaternionic Kähler structure

(see Berndt
and Suh [2]). Let
M
be a real hypersurface in

2
2
m
G  and N a local normal unit vector field. We
can define the structure vector field of
M
by
=
J
N

. Moreover, if

123
,,
J
JJ is a local basis of

, we define =
ii
J
N

, =1,2,3i. Thus we can con-
sider two natural geometric conditions: that both



=Span

 and

123
=,,D Span


 are invari-
ant under the shape operator
A
corresponding to N.
Berndt and Suh, [1] proved the following:
Theorem A Let
M
be a connected real hypersur-
face in

2
2
m
G , 3m. Then both



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


1
2
m
G  in

2
2
m
G , or
(B) m is even, say =2mn, and
M
is an open
part of a tube around a totally geodesic n P in


2
2
m
G .
The structure vector field

of a real hypersurface
M
in


2
2
m
G  is said to be a Reeb vector field. If
the Reeb vector field

of a real hypersurface
M
in


2
2
m
G  is invariant by the shape operator,
M
is
said to be a Hopf hypersurface. In such a case the inte-
gral curves of the Reeb vector field

are geodesics
(see Berndt and Suh [2]). Moreover, the flow generated
by the integral curves of the structure vector field

for
Hopf hypersurfaces in


2
2
m
G  is said to be geodesic
Reeb flow. Moreover, if the corresponding principal cur-
vature

corresponding to

is non-vanishing we say
M
is with non-vanishing geodesic Reeb flow.
Jacobi fields along geodesics of a given Riemannian
manifold


,
M
g
 satisfy a very well-known differential
equation. This classical differential equation naturally
inspires the so-called Jacobi operators. That is, if R
 is
the curvature operator of
M
, the Jacobi operator (with
respect to
X
) at pM,

Xp
REndTM

, is defined
as











=,
X
RY pRYXXp
 , for all p
YTM,

C. J. G. MACHADO ET AL. 68
being a self-adjoint endomorphism of the tangent bundle
TM
 of
M
. Clearly each tangent vector field
X
to
M
 provides a Jacobi operator with respect to
X
.
Let R denote the Riemannian curvature tensor of the
complex two-plane Grassmannian

2
2
m
G . Now if
M
is a real hypersurface in

2
2
m
G  with normal
vector field N we can consider the normal Jacobi op-
erator N
R on

2
2
m
G . Moreover, it is clear that

=0
N
RN , so we can consider N R as a self adjoint
endomorphism of the tangent bundle TM of
M
. We
will call it the normal Jacobi operator on
M
. The Ja-
cobi operator associated to the Reeb vector field R

is
called the structure Jacobi operator on
M
, where R
denotes the curvature tensor of
M
.
Recently, Jeong, Pérez and Suh, (see [3]) have proved
the non-existence of real hypersurfaces
M
in

2
2
m
G  with parallel structure Jacobi operator when
a further condition is satisfied. Also, Jeong, Kim and Suh,
(see [3]) have proved the non-existence of real hypersur-
faces
M
in

2
2
m
G  with parallel normal Jacobi
operator. Further results can also be seen in [4].
In this paper we will consider the Jacobi operators as-
sociated to a basis of the distribution D

, i R

,
=1,2,3i. A type

1,1 tensor T on
M
is called of
Codazzi type if



=
XY
TY TX for any ,
X
Y
tangent to
M
, where  denotes the covariant deriva-
tive on
M
. In this paper we will study real hypersur-
faces
M
in

2
2
m
G  whose Jacobi operators i R

,
=1,2,3
i are of Codazzi type. Namely, we will prove
the following.
Theorem 1.1 There do not exist any connected Hopf
real hypersurfaces
M
in


2
2
m
G , 3m, such that

=
XY
ii
RY RX

, =1,2,3i, for any ,
X
YTM

if the distribution D or the D-component of the
Reeb vector field is invariant by the shape operator.
As a consequence of Theorem 1.1, we immediately
obtain the following.
Theorem 1.2 There do not exist any connected Hopf
real hypersurfaces
M
in

2
2
m
G , 3m, whose
Jacobi operators i R

, =1,2,3i, are parallel if the dis-
tribution D or the D-component of the Reeb vector
field is invariant by the shape operator.
2. Preliminaries
For the study of Riemannian geometry of


2
2
m
GC

see [1]. All the notations we will use since now are the
ones in [1] and [2]. We will suppose that the metric g
of

2
2
m
GC
 is normalized for the maximal sectional
curvature of the manifold to be eight. Then the Rieman-
nian curvature tensor R of

2
2
m
GC
 is locally given
by




 
 

 
 

3
=1
3
=1
,=, ,,
,2,
,,
2,
,,,
RXYZgYZX gXZY gJYZJX
gJXZJY gJXYJZ
gJYZJXgJXZJY
gJXYJZ
gJ JY ZJ JXgJ JXZJ JY
 


 








(2.1)
where 123 ,,
J
JJ is any canonical local basis of

.
Let
M
be a real hypersurface of

2
2
m
GC
, that is,
a submanifold of


2
2
m
GC
 with real codimension one.
The induced Riemannian metric on
M
will also be
denoted by
g
, and

denotes the Riemannian con-
nection of


,
M
g. Let N be a local unit normal field
of
M
and
A
the shape operator of
M
with respect
to N. The Kähler structure
J
of

2
2
m
GC
 induces
on
M
an almost contact metric structure


,,,
g
 .
Furthermore, let 123
,,
J
JJ be a canonical local basis of
J
. Then each
J

induces an almost contact metric
structure


,,,
g

 on
M
. Using the above ex-
pression for the curvature tensor R, the Gauss and Co-
dazzi equations are respectively given by




 
  
  
  

    
 
3
=1
3
=1
3
=1
3
=1
,=, ,
,,2,
,,2,
,,
,,
,,
RXYZgYZXgXZY
gYZ XgXZYgXY Z
g
YZXgXZYgXYZ
gYZ XgXZY
YZXXZY
Xg YZYgXZ
gAYZAXgAXZAY
 

 

 




 
 
  












and








  

 

  

3
=1
3
=1
3
=1
=2,
2,
,
XY
A
YAXXYYXgXY
XYYX gXY
XY YX
XYYX
 






 



 






where R denotes the curvature tensor of
M
in


2
2
m
GC
.
In [2] the following Proposition is obtained.
Proposition 2.1 If
M
is a connected orientable real
hypersurface in


2
2
m
G  with geodesic Reeb flow,

C. J. G. MACHADO ET AL. 69
then


 





 
    
3
=1
,2,2 ,
=2 ,
22
gAAXY gAXY gXY
XYY XgXY
XY YX
 

 
 

 

 



for any ,
X
YTM where

=,gA

 .
Recently Lee and Suh (see [5]) have proved the fol-
lowing.
Proposition 2.2 Let
M
be a connected orientable
Hopf real hypersurface in


2
2
m
G , 3m. Then the
Reeb vector

belongs to the distribution D if and
only if
M
is locally co ng ru en t to an open part o f a tu b e
around a totally geodesic n
P in

2
2
m
G , where
=2mn.
3. Proof of Theorem 1.1
From the expression of the curvature tensor of


2
2
m
G  we get






  



  
33
=1 =1
3
=1
=3,
3, ,
,
,,
ii ii
i
ii ii
ii i
iiiiiii i
RX XXgX
gXgXX
gX
X
XgAAXgAXA


 



 

 
 

 

 

 


(3.1)
for any tangent vector field
X
. From (3.1) we have

 
        
          

     

3
1221
=1
1221
3
=1
=,
3,,
3,,
XXiiiXi
i
iiXiiiiiXi
iiii Xii
iiiiiXi
RY gYY
YAXgAX YgYYAXAX
qXYqXYY AXgAXYgY
YqXqXAX AX
g

  

 

 
 

 
  
 
 


 
 



     


     

  

 
 


12
21
12 21
3
=1
,,
,,
,,
,,
iiiiXi iii
ii iiiiiXi
ii
iii iXi
AXAXgq X
qXgAX AXgY
qXYqX YYAX
gAX YYAXgAXY
gAXAX g

 

 


 
  
  
 
 


 


 

 







 

 


 
 
     

      



12
21
3
1221
=1
,,
,
,,()
,,,
,
iii
ii iiii Xii
ii X
iii
iii Xii
i
qX
qXgAX AXgY
gY AXY
qXgYqXgYYAX
gAXYYAXgAXYgY
gY
 


 



 
 

 
 
 


 



 




 



       




 

     

 


 


3
1221
=1
12 21
,
,,
,
Xiii X
iii i
iiXii
iiii
ii Xi
Xii Xi
AX
qXYqXYY AXgAXY
YAXgAXYgY
YqX qXAXgAX
AX AX







  

 
 
 
 
 
 
 
 
 
 


 
 
12 21,, ii iiiiiii
Y
qXYqX YYAXgAXYYAXgAXY   
 


C. J. G. MACHADO ET AL. 70
 



 
   
 

 






12 21
,
,
,
,
Xiiiiiiii
iiiXiiiiii Xi
iXiiXiii X
iXXi iiX
gYYq XqX
gAXAX AX
AgAAY AAY
AYgAYAAYA
 

 
 

 
 

 
 
(3.2)
for any ,
X
Y tangent to
M
.
We will write
 
00 11
=XX

 , for a unit
0
X
D, where we suppose
 
01
0X  . Then we
have

11
,=0g

 , =1,2,3

. Notice this is true
even if D

.Thus the covariant derivative of 1 R

is
given by Equation (3.3), for any ,
X
Y tangent to
M
.
From this expression we have:
Lemma 3.1 Let
M
be a Hopf real hypersurface in

2
2
m
G  such that D or D-component of the
Reeb vector field is
A
-invariant. If

=
Xi
RY



Yi
RX

, =1,2,3i, for any ,
X
YTM, then
D

 or D 

.
Proof: As we suppose =A

 and have written
 
00 11
=XX

  with

0
X
 and


1

 non
null, where 0
X
D

is unit, as =0

 we get


0110
=
X
X  . Moreover, 00 =
A
XX

.
Taking 0 =
X
X in Proposition 2.1 we have






 
2
110110110
2
0110
2
=4
A
XXX
XX


.
From this, if =0

we obtain
 
01
=0X  , giving
us the result. Thus we suppose 0

. Therefore


22
10010
1
=4
A
XXX 

.
We also have

11 010
==
X
X   .





 

 
 
     
 

 
111
11 11
11111111
3
112 2111111
=1
112121111
=
=,
3,,
3,,
XX X
XX
XX
X
RYRY RY
gY Y
YAXgAX YgYYAXAX
qXYqXYY AXgAXYgY
YqXqXAX AX

 

 
 
 
 
 
 

 
 
 
 
 



      

     
   

 



1
3
12121111
=1
111
111 1
3
112 211111
=1
,, ,
,,
,
() (),()
X
X
XX
qXgYqXgYYAXgAXY
YAXgAXY gY
gY AX
qXYqXYY AXgAXYAXY
g





 


  

 
 

 

 
 

 




 

  

 


 
 
  
 
111
11212111
11 11
11 1
12 33 21111
112 313 2
,,
,
,,
,
X
X
XX
X
AX YgY
YqX qXAX
gAXAX AX
Y
qXYqXYYAXgAXYYAXgAXY
gYYq Xq X
 

 

 
  


 
 

 


 
 
  
 

 






11111111111
11 1111
11111
,
,
,
XX
XX X
XX X
gAXAX AX
AgAAY AAY
AYgAYAAYA

 
 


 
 
(3.3)

C. J. G. MACHADO ET AL. 71
From (3.3) we get



 
  
2
1100 0
1
32
001
,=
7.
g
RXX X
XX

 



(3.4)
and





 
10 0
11
23
01 0
,=4
44.
g
RX X
XX

 
 


(3.5)
As we suppose that 1 R

is of Codazzi type (3.4)
and (3.5) must be equal. This yields

2
0=0X
 . As
we suppose 0

 the result follows. □
With the hypothesis in Lemma 3.1, we can prove:
Lemma 3.2 If D 

 then

,0gADD

Proof: In this case, we can take 1 =

 . Thus the
condition of 1
R

being of Codazzi type is equivalent
to R

also being. Taking =Y

and
X
D we get



32
231
=2
2.
XRAXA AXAX
AX AX


  


 (3.6)
On the other hand



=.RXAX AX
 
   (3.7)
Therefore we have




 
32231
22
=
A
XA AXAXAXAX
AXA X

  
 
 

Taking its scalar product with 2

it follows




  

23
22
,2
=,.
gA AXAX
AXgA X

 

 

 (3.8)
and the scalar product with 3

yields


 
 
32
33
,2
=,
gA AXAX
AXgA X

 






 (3.9)
Now the Codazzi equation gives
 



22
23
,= ,
=,.
X
gAXg A
g
AAX AX





and







 
33
32
,= ,
=, .
X
gAXg A
g
AAX AX





From (3.8) and (3.9) we get



 


2
32
2
23
2=0.
2=0.
AX AX
AX AX
 
 

  (3.10)
If


=0  we have finished. If


0  , from (3.10)
we obtain
  
2
23
2
=
A
XAX




 and


3
A
X 
 
2
2
2
=
A
X



. Clearly, this yields

2
A
X 


3
==0AX  , finishing the proof. □
From this Lemma and Proposition 2.2, in order to fin-
ish the proof of our Theorem, we only have to see if the
real hypersurfaces of either type


A
or type


B
satisfy our condition.
In the case of a real hypersurface of type


A
we get
from Proposition 3 in [1], considering 1 =

 and taking
2
=X

, =Y

, that if our condition is satisfied we
should have




2
2=RR
 

 . This yields


2
233
2=0.A

  As



2=A 






3
q

 we have



 
33
2=0.q     (3.11)
From (3.11) we have

30q
  
and


2=0  . If =0

, from the second equality we
also obtain =0

, but

=2cot 2r  for some
π
0, 8 r 

. Thus this is impossible.
If =

 , from the second equality we get
22=0
 , having a contradiction. Thus


3=0q
 .
From the second equality we get 2=0

 , with


=8cot 8r  and

=2cot 2r  for some
π
0, 8 r 

. Then

2
2 = 2cot2= 0r  , which is
impossible and we can conclude that type


A
real hy-
persurfaces do not satisfy our condition.
In the case of a real hypersurface of type


B let us
suppose it satisfies our condition. From Proposition 2 in
[1] it is easy to see that
 

11
1,=4gR


   and




2
11
1,=4gR


  . As both expressions
must be equal, we obtain 2
4
=4



, where now


=2tan2r   and

=2cot2r  , for some π 0, 4 r 

.
This yields


2
tan2= 2r

, which is impossible and the
proof concludes.
As a conclusion we have obtained that Jacobi opera-
tors corresponding to D

-directions have the same be-
haviour as the normal Jacobi operator and structure Ja-
cobi operator if we consider their covariant derivatives in
the direction of any tangent vector field are null. In order
to continue this research it is interesting to investigate

C. J. G. MACHADO ET AL. 72
what occurs if the covariant derivatives are taken in di-
rections corresponding to the two distributions appearing
on the real hypersurface, namely D and D. Also we
can consider as a future work what happens if we deal
with Lie derivatives of these Jacobi operators instead
covariant derivatives.
4. Acknowledgements
Second author is supported by MEC-FEDER Grant
MTM2010-18099. Third author is supported by Grant
BRSP-2010-001-0020931 from National Research Foun-
dation of Korea. The authors express their gratitude to
the referee who has contributed to improve the paper.
5. References
[1] J. Berndt and Y.-J. Suh, “Real Hypersurfaces in Complex
Two-Plane Grassmannians,” Monatshefte für Mathematik,
Vol. 127, No. 1, 1999, pp. 1-14.
doi:10.1007/s006050050018
[2] J. Berndt and Y.-J. Suh, “Real Hypersurfaces with Iso-
metric Reeb Flow on Real Hypersurfaces in Complex
Two-Plane Grassmannians,” Monatshefte für Mathematik,
Vol. 137, No. 2, 2002, pp. 87-98.
doi:10.1007/s00605-001-0494-4
[3] I. Jeong, J. D. Pérez and Y.-J. Suh, “Real Hypersurfaces
in Complex Two-Plane Grassmannians with Commuting
Normal Jacobi Operator,” Acta Mathematica Hungarica,
Vol. 117, No. 3, 2007, pp. 201-217.
doi:10.1007/s10474-007-6091-9
[4] I. Jeong, C. J. G. Machado, J. D. Pérez and Y.-J. Suh,
“Real Hypersurfaces in Complex Two-Plane Grassman-
nians with D

-Parallel Structure Jacobi Operator,” In-
ternational Journal of Mathematics, Vol. 22, No. 5, 2011,
pp. 655-673.
[5] H.-J. Lee and Y.-J. Suh, “Real Hypersurfaces of Type B
in Complex Two-Plane Grassmannians Related to the
Reeb Vector,” Bulletin of the Korean Mathematical Soci-
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doi:10.4134/BKMS.2010.47.3.551