Internat. J. Math. & Math. Sci.
VOL. II NO. 3 (1988) 507-516
507
MIXED FOLIATE CR-SUBMANIFOLDS IN A COMPLEX HYPERBOLIC SPACE ARE NON-PROPER
BANG-YEN CHEN
Department of Mathematics
Michigan State University
East Lansing, Michigan
USA
48824,
and
BAO-QIANG WU
Department of Mathematics
Xuzhou Teacher’s College
Jiangsu, People’s Republic of China
(Received June 4, 1987)
It
ABSTRACT.
1
in
conjectured
was
(also
II]
in
[2])
that
mixed
foliate
CR-submanifolds in a complex hyperbolic space are either complex submanffolds or
In this paper we give an affirmative solution to this
totally real submanifolds.
conjecture.
KEY WORDS AND PHRASES: CR-submanifolds, complex hyperbolic space, mixed foliate.
1980 AMS SUBJECT CLASSIFICATION CODE:
53B25, 53C40.
1.
INTRODUCTION.
A submanffold
M
of a Kaehler manifold
maximal complex subspace
x
k
is called a CR-submanifold ff (1) the
of the tangent space
,
defines a dffferentiable distribution
TxM
containing in
TxM, x
M,
(2)
called the holomorphic distribution, and
g of
in
TM is a totally real
structure of
complex
almost
the
denotes
J
the orthogonal complementary distribution
T
T,
Jgx
distribution, i.e.,
and
where
M
the normal space of
submanffolds of
at
x.
Complex submanifolds and totally real
are trivial examples of CR-submanffolds.
A CR-submanifold is
called proper if it is neither a complex submanffold nor a totally real submanffold.
The totally real distribution
integrable [1,3].
distribution
satisfies
of a CR-submanifold of a Kaehler manifold is always
A CR-submanifold
is integrable, and
M is called mixed ?oliate if (a) the holomorphic
(b) the second fundamental form o of M in M
#(ff) : {0}.
It is known that mixed foliate CR-submanifolds in
:m
[1 [] and mixed foliate CR-submanifolds in
pm
m
are exactly CR-products in
are non-proper [4].
[1 II] (also in [2]) that mixed foliate CR-submanifolds in
hyperbolic space H m are non-proper too.
conje.tl]red in
It was
coracle-’
508
B.Y. CHEN and B.Q. WU
In this paper, we solve this conjecture completely to give the following
M
Let
THEOREM 1.
be a mixed foliate CR-submanifold of
Hm.
Then
M
is
either a complex submanifold or a totally real submanifold.
PRELIMINARIES.
For simplicity, we assume that
is the (complex) m-dimensional complex
Hm
Let M be a
hyperbolic space with constant holomorphic sectional curvature -4.
mixed foliate CH-submanifold of Hm. Then, by definition, the holomorphic distribution
of M is integrable and the second fundamental form o of M in Hm satisfies
o(ff) {0}. We denote by < ) the metric tensor of Hm as well as the induced
one on M. Let D
denote the normal connection and the Weingarten map
and A
N is a complex
then
of
M in H m, respectively. If N is a leaf of
2.
Hm.
submanffold of
, D, A
Denote by
,
and
v
the second fundamental form, the
,
normal connection, the Weingarten map and the Levi-Civita connection of
N
(in Hm),
M. Then
D’, A the corresponding quantities for N in
’(X,Y) + (X,Y) for X,Y tangent to N. Since 0(,) _- {0},
we also have AjZ
AZ, on TN, for Z in f. Since N is a complex submanffold
of Hm, the almost complex structure J satisfies (JX,Y)
(X,JY), Aj(
J(X,Y)
normal to N.
JA, JA --AJ, for X,Y tangent to N and
For any vector X tangent to M, we put JX = PX + FX where PX and FX
are the tangential and the normal components of JX, respectively. For a vector
normal to M, we put J
t + f, where t( and f are the tangential and the
normal components of J, respectively.
Since Hm is of constant holomorphic sectional curvature -4, the curvature tensor
R of Hm is given by
respectively, and by
(X,Y)
we have
- -
(X,Y)Z
<X,Z>Y- <Y,Z>X + <JX,Z>JY
<JY,Z>JX 2<X,JY>JZ
for X, Y, Z tangent to Hm.
(2.1)
We need the following result of [1 II] for later use.
)ixed
CR-subBanifold
1.
be a
foliate
let
H
of
BTM.
Then
DXZ DZ
AZA AWA
DxJZ
-tD Z
Fv
D Z
(c) I=
(b)
N and
tangent to
0
for
+
and
X
0(2h)
(e)
W
Z
ortbonotwa] vectors Z ad W i a’.
LEMMA 2. Under the hypothesis of Lemma 7, /.f M is proper, then (a) each
leaf
#
N
of
lies in a complex (h+p)-dimensional totally geodesic complex
p = dim
submanifold Hh+P of Hm and (b) h+l
p
2 where
2 and h
(a)
(d)
and
3.
AZ,AJZ
h
dim
.
MOI LEMMAS.
M be a mixed foliate CR-submanifold of Hm. If M is non-proper, there is
2.
nothing to prove. Thus we may assume that M is proper. By Lemma 2, p
Let
From Lemma 1, we have
o
+
for orthonormal vectors
We put
Z,W
in
g.
Let
Zl)...,Zp
be an orthonormal frame of
g.
CR-SUBMANIFOLDS IN A COMPLEX HYPERBOLIC SPACE
A= Aza
A=,
(3.2)
l,...,p
AjZ a
509
From property (d) of Lemma l, each A* has eiEenvalues I and -I with the same
Let
h.
X,,...,X h be h orthonormal eiEenvectors of A, with
eigenvalue 1. Then JX,,...,JXp are eienvectors of AZ with eigenvalue -1. With
respect to the basis {X,...,Xh, JXl,...,JXh], we have
multiplicity
0
0
lh
-lh
J
-I h
0
where
denotes the
Ih
(3.3)
Ih
h
h
0
Thus, by (2.1), we have
identity matrix.
(3.4)
-I h
0
In particular, if we choose
a
1,
we obtain
-Ih
0
(3.5)
A,
-I h
0
0
Fro (2.1) end (3.1) we have
A= Ap,
a#,
.
0,
#,
-I h
=,
1,...,p.
(3.6)
Using (3.1), (3.5) end (3.6) we ay get
B
0
0
B
B
0
(3.7)
A,,
-B
0
Since
A,
(Lma 1), we also have
0(2h)
B
tB
0(h),
(3.8)
B,
tB
denotes the transpose of B.
LEMMA 3. If M is a proper mixed foliate CR-ubmanifold of Hm, then p 3.
PROOF. Under the hypothesis, Lemma
shows that if p < 3, then p = 2. If
p = 2, we may choose an orthonormal frame XI,..,X h, JX,,...,JX h, Z,, Zz, JZ,, JZ
such that, with respect to this frame, Az, Az, A$ and Az$ take the forms of
where
-
(3.5), (3.7)and (3.8).
We put
V : Span(Xl,..,Xh}.
Then
TN
|Xz,...,Xh}
=
V
JV.
Since
Ir
we may further choose
0
(3.10)
0
soe
0(h) with tB = B,
B has the form:
such that with respect to it,
B
or
B
(3.9)
r,
0
-Ih-r
r
h.
510
B.Y. CHEN and B.Q. WU
CSI I:
In this case we have
h.
r
0
lh
Ih
Ih
0
W
(Z, + JZ,),
0
A
At,
(3.11)
0
-I h
So, if we put
AW
then
AJW
0,
(3.12)
which contradicts statement (c) of
CASE 2:
r : 0.
CASE 3:
r
I.
This case is impossible by applying an argument similar to Case
1.
>
0
and
h
>
In this case we can decompose
r.
V
and
JV
into
orthogonal decompositions:
V
V’
where
and-I,
V
V"
and
)
=
JV
Jr"
JV
(3.13)
B {defined by (3.10)) with eigenvalues
{3.10) and Lemma I we have
are eigenspaces of
By
respectively.
a(X,T)
V’,
{3.7),
{3.5),
<JX,T>(JZz-Z,) + <X,T>(JZ,+ Zz),
(3.14)
for
,
a(YT) =-<JY,T>(JZ/Z)
V" and
X V
By Lemma 1 we have
DZz
XZ,
for some 1-form
on
of
T
/
<Y,T>(JZI-Z)
TN.
DZ =-XZ,, DJZI
N.
N
Since
JZz,
DJZ
-XJZ,
is a complex submanffold of
(3.15)
Hm,
the equation
Codazzi gives
-
(vx)(Y,z)
(W)(x,z)
(3.16)
,
,
(Vx)(Y,Z) Dx(Y,Z) -(vxY,Z) -(Y,vxZ) for X,Y,Z tangent to N.
In particular, ff X
V
Y e V" and W e JV
then by applying (3.14), (3.15)
and (3.16), we see that the Z-components of both sides of (3.16) yield
where
Because
<X,W>
(Y)<JX,W>
0,
(3.17) implies
=
2<vyX,W>
Similarly, if
<W, vyX> + <x,vyw>.
0
X e V’,
Y e V"
2<vyX,W>
,x(Y)<3X,W>.
and
W
X(Y)<JX,W>
JV
(3.17)
(3.18)
the JZz-components yield
2<vxY,W>.
(3.19)
Combining (3.18) and (3.19) we find
<vxY,W>
which also implies
vV’V"
Since
J
<vxW,Y>
JV’
0
0.
(3.20)
Therefore
vV’JV’
.
V’.
(3.21)
JV"
(3.22)
is parallel, this also gives
vv’JV"
V’
Vv,V’
511
CR-SUBMANIFOLDS IN A COMPLEX HYPERBOLIC SPACE
Similrly, we nmy obtain
Vv,V’
V’,
vv,JV’
JV’,
(3.23)
vv,V"
V
vvoJV"
JV’.
(3.24)
U’ : V
Let
@
JV’
,
U"
and
vvoU’
:
V"
@
vv,U"
U’,
JV’. Then (3.21)
(3.24) show that
(3.25)
U’.
U’. Therefore, we
U" and vjV U"
In a similr way we nmy also obtain vjv,U’
and U
are both integrable and prallel distributions. Thus N is
see that U’
This is a contradiction
loclly the Riemannin product of two Kaehler nmnifolds.
admits no complex submanifold which is a product of two Kaehler
since
Hm
(Q.E.D.)
manifolds (cf. [1 I]).
Hm.
Let
be a proper mixed foliate CR-submanifold of
M
LBMMA 4.
3, then h = dime # = 2r is even and with respect o a suitable
p = diml
JXt,...,JX h, Z,...,Zp, JZ=,...,JZp,
orthonormaI frame Xt,...,X h,
-I h
0
-I h
As=
[B
-I h
Ir
0
0
(3.27)
B
As,
B
-B
As
4,
0
0
0
0
I p
Ih
At,
we have
0
0
-I r
0
Ir
As,
0
-C
then, for
=
C
Ir
0
0
e also hate
4,
0
Da
Ea
PR(X)F.
0(r)
such that
(3.28)
D=
A,
for some
Ea
tea =
tea
0
0
-E=.
Under the hypothesis, there is a suitable orthonormal frame X t,...,X h,
Zs,...,Zp, JXs,...,JXp such that Asp Asp As, and Az, take the desired
JXt,...,JX h,
forms (cf. (3.5), (3.7) and (3.10)).
Since
AaAs + AtAa
0
for
a
3,
(3.29)
.’,
where
Da
0(h)
Da
-Da
0
tD= = D=.
AAa = 0,
with
AaA= +
From this we see that each
0
Do=
where each
Ea
is a
0
From Lemma 1 we also have
AaA=,
A=$A=
=
0.
(3.30)
takes the following form:
Ea
a
tE
we also have
(3.31)
3,
0
(rx(h-r))-matrtx. Since Da
0(h), this implies
B.Y. CHEN and B.Q.
512
latEa
tlzaEa
and
Ir
-- - - -
Ea
It is clear that this is impossible unless
0,
r
h
instance, if
or
h
0,
then
A
that
2r
h
AsX,
Then
4,
As
AsA + AAs
and D
{Q.E.D.)
0
is in the desired form.
LEMMA 5.
(3.27)
addition to
e
Finally, for each
0{h),
we may conclude
p
4,
M be a proper mixed foliate CR-submanifold of Hm. If
Furthermore, we may choose the orthonormal frame such that, in
and (3.28), we also have
Let
2p-4.
h
then
AsX r.
,X h
by using the properties
Da
that
be chosen in such a way
are expressed in the forms given in (3.31).
As,
and
X,,...,X h
Now, let
which is even.
Xr+l
-
Therefore, we have
Similar argument works for the second case.
This contradicts to (c) of Lemma 1,
Consequently,
is a square matrix.
However, the first two cases cannot occur since, for
0 by virture of {3.30).
-A, which implies Ax
2r.
r,
r
(3.32)
lh_ r.
A,AsX,
AX
Xa-,
Yi
-Xr+_,,
1
i
AsXi,
Xr+i
p
(3.33)
4,
a
(3.34)
r.
As given in the proof of Lemma 3, we decompose the tangent bundle of
PROOF.
N
into orthogonal decomposition:
V’,
V’
V
JV,
V
TN
- -
Jr’ e JV’.
JV
Such a decomposition is given with respect to
vector in
that
V’.
Y
We put
AsY,...,ApY,
we conclude that
p-2
Xi
A,
Ai+zY,
for
AsXi,
and
as before.
i
i
2
2
Az. Now,
be a unit
implies
From this
(3.36)
,p-2
(3.37)
r.
Since
xAsYs
DZ
P
]
1-forms
ep
-AsAY =-AsX- =-Ya-z,
epZp,
on
VJZa
6.
X
2,
(3.38)
(Q.E.D.)
we also have (3.33). Formulas (3.34) are nothing but (3.37).
From properties (a) and (b) of Lemma 1 we have
for some
let
Then (e) of Lemma
{cf. p. 500 of [4 II]).
h
2p-4. Now, we put
V
which is equivalent to
Ai+zAsX
Xr+i
Yi
AX
AsX,
are orthonormal vectors in
r
Then, {3.27) holds.
Xr+,
(3.35)
N.
ep
-e#,
,p
1,
(3.39)
p.
(3.39) gives
Z OapJZp.
nder the hypotbes and the notations of Lena 5,
(3.40)
we have
2<VTXj,JXk>
jke,(T),
(3.41)
2<VTYj,JVk>
#jkO, (T),
(3.42)
(3.43)
CR-SUBMANIFOLDS IN A COMPLEX HYPERBOLIC SPACE
513
jke3(T) +
2<VTXj,Yk>
(3.44)
a4
for T
taEet
PROOF.
Xz,.’.,Xr,
(3.45)
<AXi,Yk>O3(T)
<VTXi,Xk>
<VTYi,Yk>
to N.
The proof of this lemma is based mainly on the equation of Codazzi.
Yz,...,Yr
be an orthonormal frame of
V’
T tangent to N,
as before, then for any vector
V"
V with
Lemma 4 gives
$
Yi
Xr+i
Let
AsXi
<JXi,T>(JZ-Z) + <Xi,T>(Z+JZ)
r(Xi,T
(3.46)
F.
+ <Yi,T>Zs + <JYi,T>JZ3 +
r(Yi,T =-<JYi,T>(JZx+Z)
(<AcX i,T>Zz + <Ac$X i,T>JZa),
<Yi,T>(Z-JZ)
F.
+
+ <Xi,T>Z s + <JXi,T>JZ
(<AYi,T>Z{x + <A,Yi,T>JZa).
.From (3.46), (3.47), (2.3) and Las 4 and 5, we obtain
(vxi) (JYj,JYk)
DX i (kZa-jkJZz)
+
<Yk,vXiY>(Z2-JZ1)
+
(<AczYk,vxiY>Zz
<Xk,vxiYJ>Zs
+
<X,vXiYk>Z
+
+
<JXk,VxiYj>JZs
<A,Yk,VXiYJ>JZ{x)
(3.48)
<yj,vXiYk>(Z2-JZl)
<JYj,vXiYk>(JZ2+Zz)
+
<JYk, VxiYJ>(JZ2+Zz
<JXj,vXiYk>JZ
Moreover, from (3.46), (3.47) and Lcmmas 4 and 5, we also obtain
(vy/j)(Xi,Y/k)
+
Djyj(dikJZ
+ F. <AXi,Yk>JZz
<Yk,VjyjXi>(JZ,+Z,)
<Xk,VjyjJXi>Zs
]
a,4
+
<Xk,VjTjXi>JZs
(<AYk, vjyjJXi>Za + <A,Yk, vjyjJXi)JZ)
<Xi,vjyjYk>(JZ-Z)
<Yi,vJyjJYk>Zs
]
4
<Yk,VjyjJXi>(Z-JZz)
(3.49)
<Xi,vjyjJYk>(Z+JZ)
<Yi,vjyjYk>JZ3
(<AXi,vJyJYk>Z
+
<A,Xi,vjyJYk>JZ).
Since the equation of Codazzi gives
(vxi)(JYJ,JYk)
(vJYj)(Xi,JYk),
(3.50)
514
B.Y. CHEN and B.Q. WU
Z,-components of both sides of (3.50) yield
the
2<JYk,vxiYJ>
(3.51)
jke=i(Xi)
and
X
(3.39), {3.40} and the fact that
Yk are orthogonal.
JZ-, JZ=-, and JZs-components of (3.50), we may also
where we used
Similarly, by comparing the
obtain
2<Jk,VJyjXi>
+
(3.52)
=4
ike=(JYj) +
2<Yk,VjyjXi>
-jkei3(Xi)
iko,s(JYj) + Z <AXi,Yk>0=(JYj),
<JXk,vxiYJ>
+
<AXi,Yk>e==(JYj),
(3.54)
<jxj,vxiYk>
<&xxi,k>e=(JYj)
<Xk,VjyjX i)
(3.53)
<i,vjyjYk>,
where we used (3.51) to derive (3.53).
Since A=A + A3x
0 for
4, Lemma 5 implies
-<AXk, Zi >.
<A=Xi, k>
(3.55)
Therefore, (3.52) and (3.53) yield
<JYkVj Xi> + <JYivjy Xk>,
<Yk,VjyjXi> + <i,vjyjXk >.
ike, s(JYj)
ike=(JYj)
(3.56)
(3.57)
Furthermore, from {3.55), we see that the left-hand side of {3.54) is symmetric with
respect to the indices j and k and the right-hand side is skew-symmetric with
respect to
j
and
k,
thus we obtain
jkS(Xi)
<vjyYi,Yk>
<JYj,vXiXk>
+
(3.58)
<JYk,VXiXj>,
<AXi,Yk>e(JYj).
<vjyXi,Xk>
(3.59)
4
From {3.51) {respectively {3.62), {3.53) and {3.59)), we obtain {3.42) for T in V’
{respectively {3.43}, {3.44), and {3.45) for T in JV’). By using the same method,
we may obtain {3.41)
{3.45) for all T in TN. {The computation is long, but
straig h t-forward ).
(Q.E.D.
In the foHowing we denote by R and I the Riemann curvature tensor and
the normal curvature tensor of the leaf N.
L 7. Under the hypothesis and the notations o Lemma 5, we have
2R(X,Y;Y,X) +
2<VxY,VyXl>
<Dy=Z:,DxZ=> <DxZ:,gy,Z=>-
2<vyY,VxlX>
R’(X,,ys;z,z=)
+
=
From Lemma 5, we have
<AaX=,Y=> <AaX=,AX=> = <X,AaAsA=> = 0 for
Thus Lemma 6 implies 2<VTY,X=>
8=(T) = <DTZ,Z=>, from which we obtain
PIK)OF.
4.
=
(Q.E.D.)
(3.60).
4.
(3.60)
PROOF OF TH] l.
Under the hypothesis of Theorem
dim
J
If
I
if
M
is non-proper, Lemma 3 implies
3.
p
4,
then Lemmas 5 and 6 imply that, for
2
we have
p
CR-SUBMANIFOLDS IN A COMPLEX HYPERBOLIC SPACE
515
2<VT,,Xi>
Thu
we
2<VTX,,Yi>
E <^,X,,Xi>e2(T)
I
i+,(T),
2<VTY,,Xi>
2,...,r.
Thus,
@i+(T),
by
<X-2,i>e(T).
2<Vy,Y,,vx,x,>
Simirly,
2,...,r.
have
applying
Lemma 6,
we
have
we y obin
2<Vx,Y,,Vy, X,>
p-2
i=2
i+(Y,)[<vX, X,,Xi>
<vx,Y,,Yi>
2
+
i=2
+
i+,(X,)[<vy,Y,,Yi>
2,(X,)<Vy, X,,,>
erefore, by applyin
2<vy, Y,,vx, X,>
<,X,,Xi>
+ 2e,,(Y,)<vX,
6 aain, e may find
2<vx,Y ,,vY,X,>
<Dx,Za,Dy, Z>
<Dy,Z,Dx,Z>.
(4.1)
oinin (4.1) ith (3.60) of La 7, e ge
2R(X,,Y,;Y,,X,)
(X,,Y,;Z,,Z,).
(4.2)
Fr (2.7), (3.46), (3.47), La 5 d he equation of
(4.3)
the oher hd, (2.7), he equation of Eicci, La I d La 5 ive
(X,,,;Z,,Z,) e<A,X,,X,> 2.
(4.4)
quations (4.2), (4.3) and (4.4)
ive a contradiction.
obn (3.41)
(3.45) were dippeared.
the equation of Codazzi, we
summation
rm
in (3.43)
ma
obn a contradiction in a simir
For a CR-submanold
is ed-fol is equivalent
AP
=
CHEN, B.Y.:
4.
By applyin these equations, we
(.E.D.)
wa.
.
Dferen
CR-subnolds of a ehler nold, I,
16 (1981), 305-322;
If, bd, 16 (1981), 493-509.
.,
mery,
3.
If p
3, then, by (3.27) and
{3.45) in such form that the
M of a ehler man,old, the ndition that
-PA.
REFERCES
2.
e may
-2.
R(X,,Y,;Y,,X,)
1.
as,
BECU, A.: meFy of CR-submanods, Reidel Publishing
Dordrecht, 1986.
BIR, D.E. and CHEN, B.Y." On CR-sumanolds of Hermtn manolds,
fsrae
Mah., 34 (1979), 353-363.
BECU, A., KON, M. and YO, K.: CR-submanolds of a complex
Dferena mery, 6 (981), 37-45.
sce form,
.
.
M