313
I nternat. J. tho & MaCho Sio
Vol. 6 No. 2 (983) 33-326
ON CONTACT CR-SUBMANIFOLDS
OF SASAKIAN MANIFOLDS
KOJI MATSUMOTO
Department of Mathematics
Faculty of Education
Yamagata University
Yamagata, Japan
(Received in June 9,1982 and is revised form in January 5, 1983)
ABSTRACT:
Recently, K.Yano and M.Kon [5] have introduced the notion of a contact
CR-submanifold of a Sasakian manifold which is closely similar to the one of a
CR-submanifold of a Kaehlerian manifold defined by A. Bejancu [i].
In this paper, we shall obtain some fundamental properties of contact CR-submanifolds of a Sasakian manifold. Next, we shall calculate the length of the second
fundamental form of a contact CR-product of a Sasakian space form (THEOREM 7.4). At
last, we shall prove that a totally umbilical contact CR-submanifold satisfying
certain conditions is totally geodesic in the ambient manifold (THEOREM 8.1).
KEY WARDS AND PHRASES
Kaehlerian manifold, Sasakian space form, contact CR-product.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE
i.
53B25.
INTRODUCTION.
This paper is directed to specialist readers with background
n
the area and
appreciative of its relation of this area of study.
Let M be a (2n + l)-dimensional Sasakian manifold with structure tensors
(,,,<,>) [4]
immersed in
and let M be an m-dimensional Riemannian manifold isometrically
and let
,
<,> be the
differentiations on M and
induced metric on
M. Let V and
be the covariant
respectively. Then the Gauss and Weingarten’s formulas
for M are respectively given by
K. MATS UMOTO
314
uv vuv
U
+ o (, v),
(1. l)
V
(1.2)
-A)U +
for any vector fields U,V tangent to M and any vector field % normal to
denotes the second fundamental form and V
fundamental tensor
A
is relted to
is the normal connection. The second
o by
(1.3)
<a(U,V)
<A)U,V>
M, where
The mean curvature vector H is defined by
(1.4)
trace o.
The submanifold M is called a minimal submanifold of
a totally geodesic submanifold of
o
if
if
H
0
and M is called
0.
For any vector field U tangent to M, we put
BU + FU,
U
(1.5)
where PU and FU are the tangential and the normal components of
Then P is an endomorphism of the tangent bundle TM of
U, respectively.
and F is a normal-bundle-
valued 1-form of TM.
For any vector field
normal to M, we put
t + f%,
where
(1.6)
t and f% are the tangential and the normal components of
is an endomorphism of the normal bundle
Then
valued 1-form of
TiM
,
respectively.
of M and t is a tangent-bundle-
TiM.
We put
where
1
and
are the tangential and the normal components of
,
(1.7)
respectively.
Then we can put
rl
where
D (U)
+1 2
r]
<$ ,>
any vector field
(I 8)
and
()
,>
<$
2
for any vector field U tangent to M and
normal to M.
By virtue of (1.5),(1.6),(1.7) and (1.8), we get
315
CONTACT CR-SUBMANIFOLDS OF BASAKIAN MANIFOLDS
P2U + tFU
-U +
FPU +
n (U) 2
0
2
JFU
(U)
(1.9)
(i.i0)
Pt + tj,
()
Ft% +
r)
f2%
(1.11)
-I + n ()
2
2
(1.12)
for any vector field U tangent to M and any vector field % normal to
M.
Let (k) be a Sasakian space form with constant -holomorphic sectional
curvature
k. Then the curvature tensor R of M(k) is given by
7< + 3
(X,Y) Z
{<Y,Z>%- <X,Z>Y} +
4
k,4
!-{D(X) <Y, Z>$
r(Y)<X,Z>
+ D(Y)](Z)X- D(X)N{Z)Y- <Y,Z>X+ <X,Z>Y + 2<%,Y>Z}
for any vector fields X,Y and Z in
(k) [3].
For the second fundamental form
with respect to the connection on
(1.13)
,
we define the covariant differentiation
TM @TiM by
(i. i4)
for any vector fields
associated with
V.
V and W tangent to
Then the equations of Gauss and Codazzi are respectively given by
(u,v;w,z)
R(s,v;w,z) + <o(u,,o(v,z)>- <o(s,z),o(v,3>,
((u, D D
for any vector fields
((U, V) W)
2.
We denote R the curvature tensor
denotes
(guo) (v, 0
U,V,W
(gv) (u,
and Z tangent to
(i.15)
(1.16)
M,
where
R(U,V;W,Z)
<R(U, V) W, Z> and
(U, V) W.
the normal component of
CONTACT CR-SUBMANIFOLDS OF A SASAKIAN MANIFOLD.
DEFINITION 2.1:
A submanifold M of a Sasakian manifold M with structure tensors
-suman.lfoid if there is
(,$,N,<,>) is called a
bution
> D c TM on M satisfying the following conditions:
D:x
D,
(i)
(ii)
(iii)
satisfies
a differentiable distri-
D
C
T M for each z
in
M,
the complementary orthogonal distribution
#D c= TIMx
D :x--> D c T M
for each point x in M.
Let M be a contact CR-submanifold of a Sasakian manifold M. Then
e D C TM,
316
K. MATSUMOTO
so, the equations (1.9),(i.i0),(i.ii) and (1.12) can be written as
PU +
tFU
-U + n(U)$,
(2.1)
FU +
.FU
O,
(2.2)
Ptl +
t
Ftl +
fl
O,
(2.3)
(2.4)
for any vector field U tangent to M and any vector field
normal to
M, respectively.
By virtue of (1.5), we have
PROPOSITION 2.1:
In a contact CR-submanifold M of a Sasakian manifold
,
in
order to a vector field U tangent to M belong to D it is necessary and
sufficient
that
FU
0.
Taking account of (2.1) and PROPOSITION 2.1, we have
PZX
-X + D(X)$
(2.5)
for any % in D and we have from (1.6)
P
0.
(2.6)
Furthermore, we obtain
<,x,Pr>
<qbX,qbi,’>
<X,i,’>
n (x)n (i,’)
(2.7)
for any X and Y in D. Thus we have
PROPOSITION 2.2:
In a contact CR-submanifold M of a Sasakian manifold M, the
distribution D has an almost contact metric structure (P,,D,<,>) and hence
dim
D
odd.
We denote by H the complementary orthogonal subbundle of
in
TM.
Then we
have
TM
D
,
Dlj_.
(2.8)
Thus we have
PROPOSITION 2.3:
subbundle
3.
For a contact CR-submanifold M of a Sasakian manifold
has an almost complex structure
f
and hence
dim
E
even.
BASIC PROPERTIES.
Let M be a contact CR-submanifold of a Sasakian manifold M. Then we have
,
the
CONTACT CR-SUBMANIFOLDS OF SASAKIAN MANIFOLDS
(v,
+ (u,z))
-zU + Vu(Z
<u,z>
for any vector field U tangent to M and Z
in D
1.
<V,@X>
317
(3.1)
From (3.1), we get
<AczU, X> + q(X)<U,Z>
(3.2)
for any vector field U tangent to M, X in D
and Z in D
I.
In (3.2), if we put X
CX,
then (3.2) means
<vuz,x> <zU, X’> + n(x)<vuz,>
for any vector tleld U tangent to M, X in D
and Z in
obtain
<AzX F_,>
.
(or equivalently <o (x,{) ,z>
0
(3.3)
By virtue of (3.2), we
0)
(3.4)
for any X in D and Z in Di.
On the
other
hand, we have
.
<Vx
<((x,),>,>
for any X in D and
in
Vx,;>
<(hx,>,>
o
Thus we have from (3.4) and the above equation
o(X,)
for any X in D. So, we have from (1.7) and the last equation
x
Px
(3.5)
for any X in D. Thus we have
PROPOSITION 3.1:
In a contact CR-submanlfold M of a Sasaklan manifold M, the
distribution D has a K-contact metric structure
In (3.1), if we put
U
W
wz + o(z,w)) -z
E
D
y
+
(P,,,<,>).
then the equation [3.1) can be written as
vz
<v,z>,
from which
([Z,W])
where
[Z,W]
LEMMA 3.2:
A@gW A$W
+
V(hW
VZW- VwZ.
In a contact CR-submanifold M of a Sasakian manifold
V@W e D
for any Z and W in D
I.
(3.6)
,
we have
(3.7)
0
318
K. TSIOTO
PROOF:
For any Z,W
in D
and
I
in
H,
we obtain
<v z
W) +
<VzW- vfi,>
<(v VzW),>
0.
On the other hand, we can easily have
A,ZW A,
for any g and W in
(3.8/
D.
In fact, for any vector field U tangent to
and
and W in
D1, we have from (3.1)
<$
(v
+ (u,z))
,
<$v, +
<$ (u,z)
,
-< (u,z) ,$
fro which, we have (3.8).
gy rtue o (3.6)
In a contact ffR-subntfold
PROPOSITION 3.3
distribution D
and (3.8) and L 3.2, we have
of a Sasakian ntfold
,
is tntegrable.
For any X In D and X in
,
we have
x -xx.
(3.91
Next, we asse that the distribution D is integrable.
D, [X,Y] is an element of D, that is, [X,Y] e TM.
( ) {vr
+Ix,r]
we get
O(X,Y)
the
o(X,Y). From
for any X and Y in
Since we have
vrx + (x)r- (z)x} + {o(x,r)
and
Z in D
o(x,r)},
which we obtain
<o(x,r),z>
<o(x,y),z>
for any X and Y in D
en
(3.01
I.
Conversely, if (3.10) is satisfied, we can easily show that the distribution D
is integrable.
us
we have
PROPOSITION 3.4:
In a contact CR-subnifold M of a Sasaklan nlfold M, the
distribution D is integrable if and only If the equation (3.10) is satisfied.
Next, we can prove
PROPOSITION 3.5:
vector field
PROOF:
In a contact CR-subnifold M of a Sasakian manifold M, the
is parallel along D
For any Z In D
we have
319
CONTACT CR-SUBMANIFOLDS OF SASAKIAN MANIFOLDS
Cz
Vz + (z,).
Since the vector field
above equation
VZ
Z
Z is an element of D
is in
0, that is, the vector field
TiM.
Thus we have from the
is parallel along any vector
field in D
4.
SOME COVARIANT DIFFERENTIATIONS.
DEFINITION 4.1:
In a contact CR-submanifold M of a Sasakian manifold M, we
define
(4.1)
(4.2)
(4.4)
normal to M [2].
for any vector fields Uand V tangent to M and any vector field
DEFINITION 4.2:
t) is parallel_ if
?P
The endomorphism P (resp. the endomorphism
0
(resp.
V
0,
?F
0
and
?t
f, the 1-forms F and
0).
By virtue of (1.5) and (1.6), we can prove
PROPOSITION 4.1:
For the covariant differentiations defined in DEFINITION 4.1,
we have
(Vu)V--
<u,v> + (v)u + t(u,v) +
(VuF)V--
fO(U,V)
(Vut)%
Af%U- PA%U,
(VU:)I -FAIUor any
vector fields Uand
AvU,
(4.5)
(4.6)
o(U,mV),
(4.7)
(4.8)
o(U, tl)
V tangent to M and any vector field
normal to M.
By virtue of (4.5), we get
(V)Y
-<X,Y> + n(Y)X + to(X,Y)
(4.9)
for any X and Y in D. Thus we have
PROPOSITION 4.2:
structure
m.
(P,,,<,>)
In a contact CR-submanifold M of a Sasakian manifold M, the
is Sasakian if and only if
o(X,Y)
is in
for any X and Y in
K. MATSUMOTO
320
In a contact CR-submanifold M of a Sasakian manifold M, if
COROLLARY 4.3:
dim Dm
0, then the submanifold M is a Sasakian submanifold.
Next, we assume that the endomorphism P
(4.9)
<x,y>- n(z)x
(x,i,)
for any
is parallel. Then we have from
(4.10)
X and Y in D. By virtue of
o(X,)
0
(4.10), we nave X
and
a
for
any X in D, where a is a certain scalar field on D. Thus we have
In a contact CR-submanifold M of a Sasakian manifold M, if the
PROPOSITION 4.4:
endomorphism P is parallel, then
dim D
i.
THE DISTRIBUTION D
5.
In a contact CR-submanifold M of a Sasakian manifold M, we assume that the leaf
M of D
is totally geodesic in
M, that is,
VZW is
D
in
for any Z
and W in D
This
me ans
<VzW,X>
(5.1)
0
for any X in D and Z and W in D
<o (x, z)
I.
n (x) Cz, >
for any X in D and Z and W in D
By virtue of PROPOSITION 3.4 and (5.1), we have
(5.2)
0
I.
.
Conversely, if the equation (5.2) is satisfied, then it is clear that the leaf
of D
is totally geodesic in
PROPOSITION 5.1:
leaf
MI
of D
Thus we have
In a contact CR-submanifold
is totally geodesic in
of a Sasakian manifold
,
the
if and only if the equation (5.2) is satisfied.
Next, let us prove
THEOREM 5.2:
that the leaf M
Nu)v
In a contact CR-submanifold M of a Sasakian manifold M, we assume
of D
is totally geodesic in
n )u
M. If the endomorphism p satisfies
<u,v>
for any vector fields U and V tangent to
(5.3)
M,
then
dim
Dim
0, that is, the submani-
fold M is a Sasakian one.
PROOF:
The equation (5.2) means
<Asw
n (x)w,z>
(5.4)
0
for any X in D and Z and W in D
I.
CONTACT CR-SUBMANIFOLDS OF SASAKIAN MANIFOLDS
321
On the other hand, we have from (5.3)
o
AvU
(u,v) +
for any vector fields U and V tangent to 24. From this, we obtain
tc;(U,X)
0
for
any vector field U tangent to 24 and X in D. Thus we have
<(
(5.4),
6.
we get
--( (U,X) ,W>
for any X in D and W in D
0
A0
that is,
<AwX,U>
(U.X) ,W>
O,
Substituting this equation into
O.
dim D m
A CONTACT fiR-PRODUCT OF A SASI NIFOLD I.
In this section, we shall define a contact CR-product and give a necessary and
sufficient condition that a contact CR-submanifold is a contact CR-product.
a contact
CR-product
manifol is
241, where 24T denotes
A contact CR-submanifold 24 of a Sasakian
DEFINITION 6.1:
if it is locally product of
241
and
called
the
leaf of the distribution D.
THEOREM 6.1:
A contact CR-submanlfold 24 of a Sasakian manifold 24 is a contact
CR-product if and only if
for any
in D and W in D
.
(6.1)
Slnce (6. i) means
PROOF:
<o (X,Z)
n (X)Z,W>
for any X in D and Z and W in D
(6.2)
0
I,
the
leafM of D
is totally geodesic in
M.
Further-
more, we have
n (x)<z ,>
< c(x,y) ,z>
for any
0
So, by virtue of PROPOSITION 3.4, the distribution
X and Y in D and Z in D
D is integrable.
Let M
be the leaf of the distribution
<ACZX,Y>
D,
then we have from (6.1)
0
for any X and Y in D and
Z in D
I,
that is, the leaf M
M. Thus the submanlfold M is a contact CRproduct.
of D is totally geodesic in
K. MATS UMOTO
322
Conversely,
if the submanlfold M of a Sasakian manifold M is a contact
CR-product, then we have from (5.2)
(6.3)
for any X in D and W in D
AwX- (X)W
So, it is sufficient to prove the following:
(6.4)
D
for any I" in D and N in D
I.
In fact, since the dtstrtbutton D is totally geodesic in
M, we have
<Acw
for any X
(X)W,Z>
< (X,Y) ,W>
-<7Y, <vxy,w>
and Y in D and N in D I.
-< (X,Z) ,W> -< Y VY)
0
This means
(6.4). By virtue of (6.3) and (6.4), we
have (6.1).
7.
A CONTACT CR-PRODUCT OF A SASAKIAN MANIFOLD II.
In this section, we shall mainly study the second fundamental form of a contact
C-product.
Let M be a contact CR-product of a Sasakian manifold
late the
B(X,Z)
for any unit vectors X in D and Z in D
,
M. In M, we shall calcuwhere
B(X,Z)
is defined
by
-< (x, x) z, z>.
(x, z)
(7.1)
By virtue of (1.14) and (I.16), we get
<AzX,Vz> + <AZVX,Z> + <AzX,VxZ>.
Since the leaves
Vp
M and M are both totally geodesic in M, we have
e D
and
V
(7.2)
e D
for any vector field U tangent to M,Y tn D and g in D
1.
’I"hus we have from (6.1) and
(7.2)
<(X,4pX)Z,cZ>
<VIXo(dpX, Z),dpZ> <VIcXO(X,Z),4oZ> Q(VXX) + (V).
On the other hand, we obtain
(7.3)
ODNTACT CR-SUBMANIFOLDS OF SASAKIAN MANIFOLDS
n (vxOx)
<VxX, >
<xX, >
So, (7.3) can be written as
<R(x, ox)z,z>
+ n
-
<VxO%X,z),z>
+ n (x)
323
.
<v+xo(x,z),z> +
1- n(x)
(vxx).
(7.4)
Next, we have from (6,1)
<o (X, Y) qbZ>
<,X><Z,Y>,
(7.5)
<o(X,Z),Z>
<,X>,
(7.6)
from which
<(OX, Z),Z>
O.
(7.7)
Covariant differentiation of (7.6) and (7.7) along
<v
-<o(x,z),vxz> + <Vx,X>
xO(X,Z),z>
<v
(x, z
z>
-< (x, z)
X
and X respectively give us
+
(7.8)
VxZ>
(7.9)
Substituting (7.8) and (7.9) into (7.4), we get
-<(x,z),vCxCz> + <o(x,),v
z>
2(1- q(X)a).
(7.10)
By virtue of (3.9) and (6.1), we can calculate
<o(X,Z),V Z>-
<o(X,Z),VCXZ>
=2I](X,Z)
2
+ 2n(X) 2.
(7.11)
Thus we have
PROPOSITION 7.1:
B(x,z)
In a contact CR-product of a Sasakian manifold, we have
2(ll(x,z)II
1)
(7.12)
for any unit vectors X in D and Z in D
I.
Especially, if the ambient manifold
is a Sasakian space form
(k),
then we
have from (i. 15)
-(X, OX;qbZ,E)
k- i (1
2--
n(X) ).
(7.13)
Thus we have
PROPOSITION 7.2:
In a contact CR-product of a Sasakian space form M(k), we have
K. MATSUMOTO
324
(7. Z4)
for any unit vectors X in D and Z in D
I.
By virtue of PROPOSITION 3.4. and (7.14), we have
k + 3
Ilcr(X,Z 112
for
4
fi,
(7.15)
Thus- we have
In a Sasakian space form M(k) with constant -holomorphic
COROLLARY 7.3:
k < -3, there does
sectional curvature
not exist a contact CR-product of
M(k).
Next, we shall prove
THEORII 7.4:
Let M be a contact CR-submanifold of a Sasakian space form M(k).
Then we have
iiii 2
>
2(.h(< 2+
3)
+ i)
(7 16)
where
p
and bl
are both totally geodesic in
dimD
PROOF:
Let
onal basis of D
and
2h
dimD
(k).
A1,A2,...,Ah,@AI,A 2
and D
2h+l
i,j=l
qbAh,A2h+l
respectively. Then
IIo(Ai,Aj)112
(7.16) holds, then M
i. If the equality sign of
!!!1
2h+l
+ 2
i=i a=l
(= ) and
Bp
B1,B2
be orthog-
is given by
IIo(Ai,Ba)112
+
a,=l
IIo(Ba,S B)
2.
By virtue of (7.15), the above equation can be written as
p(h( 2+ 3))
iill
2h+l
+ ) +
[
i,j=l
a, 8=l
8
(7.17)
From the above equation, we have our theorem.
8.
TOTALLY UMBILICAL CONTACT CR-SUBMANIFOLDS.
Let M be a totally umbilical contact CR-submanifold of a Sasakian manifold M.
Then by definition we have
o(U, V)
<U, F’>H
(8. i)
for any vector fields U and V tangent to M. By virtue of
d#H
tH + .f?-l
Since the vector field tH is in
(1.6),
we can write
(8.2)
D
we have from (3.8)
CONTACT CR-SUBMANIFOLDS OF SASAKIAN MANIFOLDS
A
.
325
tW A wtS
for any W in
(8.3)
From this, we obtain
-<W,W>< tH, tH>
We assume that
dim D
(8.4)
< tH,W><H,@>
> 2. Then we can put W as the orthogonal vector field of tH.
The equation (8.4) means
0
tH
(8.5)
Next, let QI and Q2 be the projections of TM to D and D
respectively. Then for any
vector field U tangent to M we can put
QU + QU
U
(8.6)
and
QIU e D,
Q2U e
4#Dl c TM.
(8.7)
The equation (8.7) and the covariant differentiation of
QIVut%
-QA%U
H
teach us
(8.8)
Q,A2qU
for any vector field U tangent to M and any vector field
we put
t% +
%
normal to M. In (8.8), if
and taking account of (8.5), we have
QAHU
dpQAHU
(8.9)
for any vector field U tangent to M. For any % in D and any vector field U tangent
to M, we get
<QAsHU,%> <AsHU,%>
<Q
AHU %>
-<Q
<a(U,X),$H>
AHU %> -<AHU CX>
<U,X><H,H>
O,
-<o U CX), H>
-< U, CX><H ,H>
By virtue of (8.9) and the above two equations, we have
<H,H><U,X>
(8. I0)
0
for any vector field U tangent to M and any X in D. We assume that
if we take
(8.10)
H
U
X
such that the vector field % is orthogonal to
,
dimD
x
> 2
and
we have from
0. Thus we have
THEOREM 8.1:
Let M be a totally umbilical contact CR-submanifold of a Sasakian
326
manifold
.
K. MATSUMOTO
We assume that
dim D
x
> 2
and
dim D
x
> 2. Then the submanifold M is
totally geodesic in M.
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