Real hypersurfaces in a complex space form
and the generalized Tanaka-Webster connection
The 13th International Workshop
on Differential Geometry and Related Fields
5 – 7 Nov., 2009
Kyungpook National University
Mayuko Kon
Department of Mathematics,
Hokkaido University
Osaka City University
Table of Contents



Preliminaries
g-Tanaka-Webster connection
Holomorphic distribution
・ shape operator
・ Ricci tensor
・ sectional curvature
1. Preliminalies.
M n (c) : complex space form,
4c : holomorphic sectional curvature,
J : complex structure,
M : a real hypersurface in M n (c),
N : unit normal vector field,
・ almost contact metric structure
JX  X  ( X ) N ,
g
： induced connection.
We put

(φ，ξ，η，g )
  JN
 2 X   X   ( X ) ,   0,
 (X )  0,  ( X )  g ( X ,  ),
g (X , Y )  g ( X , Y )   ( X ) (Y ).
( X  Tx M ).
・ Gauss and Weingarten formulas
~
 X Y   X Y  g ( AX , Y ) N ,
~
 X N   AX .
( A : shape operator)
・ Hopf hypersurface
M : real hypersurface of CP n .
M : Hopf hypersurface   : function , A   .
def
・ the equation of Gauss
R : the Riemannian curvature tensor field of M .
R( X , Y ) Z  c{g (Y , Z ) X  g ( X , Z )Y  g (Y , Z )X
 g (X , Z )Y  2 g (X , Y )Z }
 g ( AY , Z ) AX  g ( AX , Z ) AY .
S : the Ricci tensor of M .
S ( X , Y )  (2n  1)cg ( X , Y )  3cg ( X ,  ) g (Y ,  )
trAg ( AX , Y )  g ( AX , AY ).
・ the equation of Codazzi
( X A)Y  (Y A) X
 g ( X ,  )Y  g (Y ,  )X  2 g (X , Y ) .
M : totally umbilic
M : totally η-umbilic
M :Einstein

AX  aX
def
(X  Tx M )
 AX  aX  bg (X ,  )
def
 S ( X , Y )  ag ( X , Y )
def
(X , Y  Tx M )
M :pseudo-Einstein  S ( X , Y )  ag ( X , Y )  bg ( X ,  ) g (Y ,  )
def
⇒ Are there any other useful condition?
⇒ Or, any other useful connection?
→ section 3
→ section 2
■ Examples of real hypersurfaces.
Example 1.1.
S 2n1  {( z1,, zn1 )  C n1 | nj11| z j |2  1},
 : S 2n1  CP n .
M 0 ' (2n, t ) : {( z1 ,, zn1 )  S 2n1 | nj1| z j |2  t | zn1 |2 }
M 0 (2n  1, t ) :  ( M 0 ' (2n, t ))
(t  0).
geodesic hypersphere, type A 1
▲ compact totally η-umbilical real hypersurface in CP n
 cot 



A
cot 

 0




,

2 cot 2 
0
t  cot 2  ,
A  2 cot 2.
type A 2
M ' (2n, m, t ) : {( z1 ,, zn1 )  S 2 n1 | mj1| z j |2  t nj 1m 2 | z j |2 }
 2 p  1  2 q 1  2q  1 
 S 2 p 1 
 S


2n 

 2n 
M (2n  1, m, t ) :  (M ' (2n, m, t )).
t  0,



,
p

m

1
,
q

n

m

1


▲ a tube of radius  over CP p
 cot 




cot 

A


0



0
 tan 

 tan 
M (2n  1, m, t ) : pseudo-Einstein  t 





,



2 cot 2 
m 1
.
nm
t  cot 2  ,
A  2 cot 2 .
type B
M ' (2n, t ) : {( z1 ,, zn1 )  S 2 n1 || nj11 z j 2 |2  t}
(t  0).
M (2n  1, t ) :  (M ' (2n, t )).
▲ a tube of radius  over Q n 1
A
 cot(   / 4)




0




cot(   / 4)


 tan(   / 4)

,





0
 tan(   / 4)



2 cot 2 

M (2n  1, t ) : pseudo-Einstein  t 
1
.
n 1
t  cot 2  ,
A  2 cot 2 .
Theorem 1.2 (Cecil and Ryan, Takagi).
Let M be a totally η-umbilical real hypersurface of
CP n , n  2 , then M is locally congruent to a geodesic
hypersphere.
Theorem 1.3 (Cecil and Ryan, Kon).
Let M be a connected complete pseudo-Einstein
n
CP
, n  3 , then M is congruent to
real hypersurface of
M 0 (2n  1, t ), M (2n  1, m, (m  1/ n  m)) , or M (2n  1,1/(n  1)).
Example 1.4.
H12n1  {( z0 ,, zn )  C n1 |  | z0 |2  nj1| z j |2  1},  : H 2 n1  CH n .
M ' p ,q (t ) : {( z1 ,, zn1 )  H12 n1 | t ( | z0 |2   pj1| z j |2  nk  p 1| zn1 |2 }
 1  2q 1  t 
 H12 p 1 
 S


 t 1 
 1 t 
M p,q (t ) :  ( M ' p,q (t ))
(t  0).
n q 1
a tube of radius  over CH
M 0,n1 (t ) : geodesic hypersphere
 tanh 




tanh 

A


0



0
coth 

coth 





,



2 coth 2 
t  tanh 2  , A  2 coth 2.
holosphere
L(t ) : {( z1 ,, zn )  H12 n1 || z0  z1 |2  t} (t  0).
M n* (t ) :  ( L(t )).
▲ totally η-umbilical real hypersurface of CH n
0
1


 

A
.

1


2
0
Theorem 1.5 (Montiel).
Let M be a totally η-umbilical real hypersurface of
CH n , n  3, then M is locally congruent to one of the
following:
2
(a) a geodesic hypersphere M 0,n1 (tanh  ),
2
M
(tanh
 ),
(b)
n1,0
(c) holosphere M n* .
★ A connected, complete real hypersurface M in
CH n , n  3 is pseudo-Einstein
⇔
M is totally η-umbilic.
2. g-Tanaka-Webster connection
For a contact metric manifold (Tanno)
^
 X Y :  X Y  ( X )(Y )   (Y ) X    ( X )Y .
For a real hypersurface of a Kaehler manifold (Cho)
^ (k )
 X Y :  X Y  g (AX ,Y )   (Y )AX  k ( X )Y .
k : nonzero real number.
A  A  2k
 g-Tanaka-Webster connection coincides with
the Tanaka-Webster connection.
Then we have
^ (k )
^ (k )
   0, 
c. f .
 X   AX ,
^ (k )
^ (k )
  0,    0, 
g  0.
( X  )Y   (Y ) AX  g ( AX ,Y ) .
Def.
^
^ (k ) ^ (k )
^ (k ) ^ (k )
^ (k )
R( X ,Y ) Z :  X Y Z  Y  X Z  [ X ,Y ] Z ,
^
^
S (Y , Z ) : tr{ X  R( X ,Y ) Z }.
■ Problem
^
We want to study the shape operator A , the Ricci tensor S ,
^
n
the sectional curvature K of a hypersurface of CP
with respect to some conditions for g-Tanaka-Webster
connection.
(Early results)
M n (c) : complex space form,
4c : holomorphic sectional curvature.
Theorem 2.1 (Cho, 2008).
M : Hopf hypersurface of M n (c)
(c  0).
^
R0
⇒ M is locally congruent to one of the following:
(a)
n
CH
,
a horosphere in
1
2
2
(b) a homogeneous tube over Q and RP (CH )
in CP 2 (CH 2 ).
Theorem 2.2 (Cho, 2008).
M : Hopf hypersurface of M n (c)
X ,Y   ,
^
S ( X ,Y )  L( X ,Y ),
(c  0).
L : Levi form.
⇒ M is locally congruent to one of the following:
n
CH
,
a
horosphere
in
(a)
n
n
(b) a geodesic hypersphere in CP or CH ,
(c) a homogeneous tube over CH n 1 in CH n ,
1
2
2
Q
RP
(
RH
)
a
homogeneous
tube
over
and
(d)
in CP 2 (CH 2 ).
(Our results)
Theorem 2.3.
M : real hypersurface of CP n ( n  3), k 2  4,
^
X   , Y  Tx M ,
(  : function).
S ( X ,Y )  g ( X ,Y )
⇒
M is locally congruent to
(i) a geodesic hypersphere with k 2  (2n  2)(2n   ),
(ii) a tube over a totally geodesic CP p (1  p  n  2).
Theorem 2.4.
M : real hypersurface of CH n ( n  3),
^
X   , Y  Tx M ,
S ( X ,Y )  g ( X ,Y )
⇒
M is locally congruent to
(  : function).
(a) a geodesic hypersphere with k 2  (2n  2)(2n   ),
(b) a tube over a complex hyperbolic hyperplane with
k 2  ( 2n  2)(2n   ),
2
(c) a holosphere with k  (2n  2)(2n   ),
(d) a tube over a totally geodesic CH p (1  p  n  2).
Corollary 2.5.
M : real hypersurface of CP n (n  3),
^
S  0.
⇒
M is locally congruent to a geodesic hypersphere,
with k 2  4n( n  1).
M :Einstein

def
^
S ( X ,Y )  g ( X ,Y )

^
S  0.
(X ,Y  Tx M )
Corollary 2.6.
n
CP
( n  3)
There are no real
hypersurfaces
of
^
which satisfies R  0.
Theorem 2.1 (Cho, 2008).
M : Hopf hypersurface of M n (c)
(c  0).
^
R0
⇒ M is locally congruent to one of the following:
(a)
n
CH
,
a horosphere in
1
2
2
(b) a homogeneous tube over Q and RP ( RH )
in CP 2 (CH 2 ).
3. Holomorphic distribution.
Hopf hypersurface
M : real hypersurface of M n (c)
M : Hopf hypersurface   : function , A   .
def
There are some classification theorems for Hopf hypersurfaces
of a complex space form.
⇒
We want to study
・ real hypersurfaces of a complex space form
without the assumption that they are Hopf hypersurfaces.
・ another condition that contains Hopf condition.
T0 : holomorphic distribution
T 0 ( x) : {X Tx (M ) | g ( X ,  )  0}
M : ruled real hypersurface

def
T 0 is integrable and its integral manifold is a totally
n1
geodesic submanifold M (c).
M : totally umbilic
M : totally η-umbilic
M :Einstein

AX  aX
def
(X  Tx M )
 AX  aX  bg (X ,  )
def
 S ( X , Y )  ag ( X , Y )
def
(X , Y  Tx M )
M :pseudo-Einstein  S ( X , Y )  ag ( X , Y )  bg ( X ,  ) g (Y ,  )
def
⇒ Are there any other useful condition?
■ Problem
T0 ( x)  { X  Tx ( M ) | g ( X ,  )  0}.
We want to study the shape operator A , the Ricci tensor S ,
n
the sectional curvature K of a hypersurface of M (c)
with respect to some conditions for T0 .
g ( AX , Y )  ag ( X , Y )
S ( X , Y )  ag ( X , Y )
(X , Y  T0 ( x))
(X , Y  T0 ( x))
g (( R( X , Y ) S ) Z ,W )  0
K ( X , )  c


(X , Y , Z ,W  T0 ( x))
( X  T0 ( x),| X | 1, c : constant )
totally η-umbilic
pseudo-Einstein

RS=0
□ Shape operator
M : real hypersurface of M n (c)(n  3),
We study the following condition:
X , Y   ,
i.e.
g ( AX , Y )  ag ( X , Y )
e1 ,, e2 n2 ,   : o.n.b,
0
a



A
0
a

 h1  h2 n 2
h1 

 
h2 n2 

b 
 Aei  aei  hi ,

2n2
 A  k 1 hk ek  b .
c. f .
totally η-umbilic
0 0
a





A
0
a 0


 0  0 b
Theorem 3.1 (Ortega 2002).
M : real hypersurface of M n (c)(n  3),
X , Y   ,
g ( AX , Y )  ag ( X , Y )
⇒ M is totallyη-umbilical or a ruled real hypersurface.
▲ Hopf hypersurface
▲ not a Hopf hypersurface
Corollary 3.2.
M : real hypersurface of CP n (n  3),
X , Y   ,
⇒
g ( AX , Y )  ag ( X , Y )
M is locally congruent to a geodesic hypersphere,
or a ruled real hypersurface.
Corollary 3.3.
M : real hypersurface of CH n (n  3),
X , Y   ,
g ( AX , Y )  ag ( X , Y )
⇒ M is congruent to one of the following:
(a)
(b)
(c)
(d)
a ruled real hypersurface,
a geodesic hypersphere M 0,n1 (tanh 2  ),
M n1,0 (tanh 2  ),
a holosphere M n* .
Theorem 3.4.
M : real hypersurface of M n (c)(n  3),
X , Y , Z ,W   ,
g ( R( X , Y ) AZ ,W )  0

M is a geodesic hypersphere or a ruled real hypersurface.
As an application of this theorem, we have
Theorem 3.5 (Maeda).
n
CP
(n  3),
There are no real hypersurface of
which satisfies RA  0
□ Ricci tensor
Theorem 3.6 (2007).
M : real hypersurface of M n (c)(n  3),
X , Y , Z ,W   ,
g ( R( X , Y ) SZ ,W )  0

M is a pseudo-Einstein real hypersurface.
As an application of this theorem, we have
Theorem 3.7 (Kimura and Maeda).
n
CP
(n  3)
There are no real hypersurface of
which satisfies RS  0.
Theorem 3.8 (Ki, Nakagawa, Suh).
M : real hypersurface of M n (c)(n  3),
X ,Y , Z  Tx M ,
R( X ,Y )SZ  R(Y , Z )SX  R( Z , X )SY  0

M : pseudo-Einstein
Proposition 3.9.
M : real hypersurface of M n (c)(n  3) with constant
principal curvatures.
X , Y  ξ ,
g ( SX , Y )  ag ( X , Y )
 M is a pseudo-Einstein real hypersurface.
★ NEXT PROBLEM
M : real hypersurface of M n (c)
M : Hopf hypersurface   : function , A   .
def
We study the condition
 : function , S   .
● with constant principal curvatures
● g ( A ,  ) : constant
⇒
Hopf hypersurface
□ Sectional curvature
Theorem 3.10.
M : real hypersurface of M n (c)(n  3),
X   , | X | 1,
⇒
K ( X , )  k
(k : constant )
M is congruent to one of the following:
(a) Hopf hypersurface which satisfies A  0 (k  c),
(b) totally η-umbilical real hypersurface (k  c).
■ Early results
M : real hypersurface of CP m (m  3),
H : sectional curvature on a holomorphic 2-plane,
H ( X )  g ( R( X ,X )X , X ),
X   , | X | 1.
Theorem 3.11 (Kimura）.
If H is constant, then M is one of the following:
(a) an open subset of a geodesic hypersphere ( H  4),
(b) a ruled hypersurface,
(c) a real hypersurface on which there is a foliation of
codimension two such that each leaf of the foliation
is contained in some CP n1 as a ruled hypersurface ( H  4).
Corollary 3.12.
M : real hypersurface of CP n (n  3),
The focal map  r has constant rank on M ,
X   , | X | 1,
K ( X , )  k
(k : constant )
⇒
M is congruent to one of the following:
(a) Hopf hypersurface which satisfies A  0 (k  1),
(a-1) a homogeneous real hypersurface which
s

/
4
CP
(1  s  n  1),
lies on the tube of radius
over
(a-2) a nonhomogeneous real hypersurface which
lies on the tube of radius  / 4 over a Kähler
submanifold N with nonzero principal curvatures  1,
(b) A geodesic hypersphere M (2n  1, t )of radius 
(t  cot 2  , k  t  1).
Corollary 3.13.
M : real hypersurface of CH n (n  3),
The focal map  r has constant rank on M ,
X   , | X | 1,
K ( X , )  k
(k : constant )
⇒
M is congruent to one of the following:
(a) a tube of radius  over an anti-holomorphic submanifold
 M which satisfies A  0 (k  1),
(b) a geodesic hypersphere M 0,n1 (t ) of radius  (k  1/ t ),
(c) a tube M n1,0 (t ) of radius  over a complex hyperbolic
hyperplane (k  t ),
*
(d) a holosphere M n .
Thank you for your attention !
Theorem 2.1 (Cho, 2008).
Theorem 2.2 (Cho, 2008).
Theorem 2.3.
Theorem 2.4.
Corollary 2.5.
Corollary 2.6.
Theorem 3.4.
Theorem 3.6.
Proposition 3.9.
Theorem 3.10.
^
R0
X ,Y   ,
X   , Y  Tx M ,
^
S ( X ,Y )  L( X ,Y ),
^
S ( X ,Y )  g ( X ,Y )
^
S 0
^
R0
X ,Y , Z ,W   ,
g ( R( X ,Y ) AZ ,W )  0
X ,Y , Z ,W   ,
g ( R( X ,Y )SZ ,W )  0
X ,Y   ,
g ( SX ,Y )  ag( X ,Y )
X   , | X | 1, K ( X , )  k (k : constant)