551
Internat. J. Math. & Math. Sci.
Vol. I0, No. 3 (1987) 551-556
TOTALLY UMBILICAL CR-SUBMANIFOLDS
OF SEMI-RIEMANNIAN KAEHLER MANIFOLDS
K.L. DUGGAL and R. SHARMA
Department of Mathematics and Statistics
University of Windsor, Windsor
Ontario, Canada N9B 3P4
(Received August 18, 1986 and in revised form October 20, 1986)
ABSTRACT.
We study totally umbilical CR-submanifolds of a Kaehler manifold carrying
a semi-Riemannlan metric.
It is shown that for dimension of the totally real
distribution greater than one, these submanifolds are locally decomposable into a
complex and a totally real submanifold of the Kaehler manifold.
For dimension equal
to one, we show, in particular, that they are endowed with a normal contact metric
structure if and only if the
second
fundamental form is parallel.
KEY WORDS AND PHRASES.
CR-submanifolds, Totally umbilical, Semi-Riemannian metric,
Kaehler manifold, normal contact metric structure, second fundamental form.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
53C40.
INTRODUCTION.
The notion of a Cauchy-Riemann (CR)-submanlfold of a Kaehler manifold was
introduced by Bejancu [1,2] and developed later by several researchers.
detailed treatment, we refer to Yano and Kon [3].
far been confined to a positive definite metric.
For a
The study on CR-submanifolds has so
The objective of this paper is to
study totally umbilical CR-submanifolds of a Kaehler manifold carrying a semiRiemannian metric.
According to Flaherty’s theorem [4], the Hermitian metric on a
Kaehler manifold cannot have the Lorentzlan signature (which is essential for a spacetime manifold of relativity).
This motivates us to consider CR-submanlfolds (whose
metric can have the Lorentzlan signature) of a Kaehler manifold.
2.
PRELIMINARIES.
Let
M be a submanifold isometrically immersed in a Kaehler manifold
a complex structure
We denote by the same
Riemannian.
J, a Hermitlan metric
g
M
the metric of
We also assume that
TM
the Levi-Civita connection induced in
VxY VxY + B(X,Y)
VxV -A + DxV
g
and
M, which is assumed to be semi-
is non-degenerate with respect to
M
M with
and the Levi-Civlta connection
g.
If
V
is
then we have
(Gauss Formula)
(2.1)
(Weingarten Formula)
(2.2)
552
K.L. DUGGAL AND R. SHARMA
X,Y tangential to M and
for arbitrary vector fields
the second fundamental form,
g(A,Y)
g(X,Y),
B(X,Y)
particular, when B
umbilical if
g(B(X,Y),V).
V
D
and
M
is
the
is said to be totally
M
In
M.
being the mean curvature vector field of
0, we say that
B
M; where
normal to
the shape operator associated to
In fact,
M.
normal connection of
V
is totally geodesic.
[I] of the Kaehler manifold M if there exists
a dlfferentlable distribution D on M such that (I) D is invarlant by J, i.e.
JD
D
for each x in M and (II) the complementary orthogonal distribution D
x
x
is antl-varlant (totally real), i.e. JD
for each x in
is a subspace of T (M)
x
x
M
is said to be a CR-submanlfold
A CR-manlfold is called proper if
M.
PX + FX,
JX
p3 +
p
FP
0,
f3
O,
+ f
0,
fF
into their tangential and normal components
(2.4a)
0
Pt
O,
tf
(2.4b)
0
AX + yB(X,Y)
(VxP)Y
=-B(X,PY) + fB(X,Y)
(VxF)Y
(2.6)
(Vxt)V
AfvX PX
(2.7)
(Vxf)V
-FA
(2.8)
B(X,tV)
(VxP)Y
Vx(PY)
PVxY, (VxF)Y Dx(FY)
FVxY
(Vxt)V
Vx(tV
tDxV (Vxf)V Dx(fV
fDxV
PROPOSITION 2.1.
Let
PROOF.
D
M
Let
D
the distributions
D
JV
and
(2.3)
Following relations hold [3]:
respectively.
where
JX
and
be a CR-submanifold of a Kaehler manifold.
D1
0 for all
Y
Then there exists a non-zero vector field
As
D.
in
D
X
0
because
TM
X
This proves non-degenerateness of
Y
in
is non-zero.
D.
X
in
are complementary and
0 for all
But
is non-degenerate.
Dx
and
orthogonal to each other, it follows that g(X,Y)
a contradiction.
Then both
are non-degenerate.
be degenerate.
such that g(X,Y)
that
Let
tV + fV
JV
be the decompositions of
# 0 and dlmDl # O.
dlmD
That
Dx
TM.
This shows
Hence we arrive at
is non-degenerate
can be proved likewise.
PROPOSITION 2.2.
The mean curvature vector
to
PROOF.
JD1 in
Consider any
TMi.
f
in
D
g(J(VxX),JV)
g(VxJX,JV)
g(J(VxX),JV)
g(VxX,V)
0.
and
of a totally umbilical
CR-sub-
V
in the complementary orthogonal subbundle
Then we have
Thus we observe that
i.e.
X
U
JD1.
manifold of a Kaehler manifold belongs to
Hence
g(X,X)g(,V)
g(VxJX +
g(VxX +
0.
belongs to JI>L.
g(X,JX)p,JV)
g(X,X),V)
O.
g(X,X)g(,V).
By prop. 2.1 it follows that g(,V)
0;
TOTALLY UMBILICAL CR-SUBMANIFOLDS
AFX
LEMMA 2.1.
(Yano-Kon [3])
AFyX,
X,Y
Y
for any
in
Di.
Let us
The above lemma holds for both positive definite and indefinite metrics.
dim D
denote
by
q.
q > I.
TOTALLY UMBILICAL CR-SUBMANIFOLDS WITH
3.
Then
be a CR-manlfold of a Kaehler manifold.
M
Let
553
For a positive definite metric, the following is known:
(Bejancu [5]).
THEOREM 3.1.
M.
of a Kaehler manifold
Let M be a totally umbilical proper CR-submanifold
q > I, then M is totally geodesic in M and is
If
D
locally a product of the leaves of
Di, i.e.
and
a CR-product.
That the above theorem holds also for an indefinite metric, can be proved as follows.
XY AFyX for all X,Y
xt tX. Since M
By lemma 2.1, we have
for any X in I>i we have
PROOF.
Di,
in
g(X,Y)
B(S,Y)
we have
X
in
.
g(,V)X.
D
Since q >
Hence we obtain
Di, that
g(t,t)
0.
f
"A
according to Chen’s theorem [6],
M
REMARK I.
{i,2,3,4
M
0 on
M
for
with
0.
J
0 and hence
Obviously we have
CR-submanifold is a CR-product iff
dim D
dim D
If we take
Thus we get
0.
VP
0.
VP
0"
This completes the proof.
is a CR-product.
p3 +p
f-structure
0.
Consequently
therefore reduces to a totally geodesic submanifold.
it follows that
X
t
I, it follows, on contraction of (3.1) at
Now, by prop. 2.2 we also have
Thus
lles
t
(3.1)
respect to an orthonormal base of
M
As
is totally umbilical
g(t,t)X
g(t,X)t
for all
X
and
Di.
in
2
so that
dim M
4
and assume that the
is globally framed, then there exists a local basis
PI 2’ P2 -I’ P3 P4
such that
+ + +
0.
One can verify
that
g
is Lorentzian with signature
3.1,
M
serves as a model of a class of decomposable space-times in general relativ-
ity.
For a recent study on such space-times we refer to [7].
TOTALLY UMBILICAL CR-SUBMANIFOLDS WITH q
4.
Subject to the hypothesis of theorem
I.
Chen proved the following theorem:
(Chen [8]).
THEOREM 4.1.
M.
Kaehler manifold
(i)
M
(ii)
q
(iii)
M
Let
M
be a totally umbilical CR-submanifold of a
Then
is totally geodesic, or
I, or
is totally real.
Note that if
M were a proper CR-submanifold in the above theorem, then the possibil-
ity (iii) would be ruled out.
Also note that (i) and (ii) are not mutually exclusive.
The case (ii) has been investigated by Chen [8], in the context of a locally Hermitian
symmetric space
M
and
dim M
5.
In the following theorem we study the case
by relaxing these conditions and assuming
THEOREM 4.2.
manifold
M.
(I)
M with
Let
q
M
to be proper.
M be a proper totally umbilical CR-submanlfold of a Kaehler
I. Suppose the mean curvature vector
is non-vanishing over
Then the following statements are equivalent:
M
has a normal contact metric (Sasakian) structure [3].
(2)
has a non-zero constant norm.
(3)
is parallel in the normal bundle.
K.L. DUGGAL AND R. SHARMA
554
(4)
PROOF.
and lles in
t.
M
second fundamental form of
For any
# 0 and
As
Di.
Now since
X
tangent to
g(,)p2x
P
Operating
is parsllel.
JD
lies in
I,
q
t # 0
by prop. 2.2, it follows that
any vector field in
D
is a scalar multiple of
M we can show, using (2.7), that
g(t,t)X
g(t,X)t
on this gives
(4.1)
g(t,t)
g(,t)
Hence we get
g(t,t)(p2x
+ X)
(4.2)
g(t,X)t
In this case too, equation (3.1) holds, which shows (q
hence g(,) # 0. Equation (4.2) becomes
p2X
Di
assume
g(t,t)
-a-2t
a
n
and
p2X
2
is a real-valued function on
a
,
M.
Under the setting
equatln (4.3) takes the form
+ (X)
P
It follows that
(4.3)
is non-degenerate we can, without any loss of generality,
where
as the dual of
-X
g(t,t) # 0 and
[g(t,t)]-Ig(t,X)t
-X +
As the distribution
I) that
(4.4)
0, rank (P)
0, hOP
g(PX,PY)
dimM-
and
(4.5)
(X)(Y)
g(X,Y)
With the help of (2.5), (2.7) and (4.3) we derive
(VxP)Y
VX
{g(,Y)X- g(X,Y)}
a
(4.6)
(4.7)
a PX
Equations (4.4)-(4.7) show that M has a normal contact metric structure iff.
2
g(,) is a non-zero constant. This proves the equivalence of (I) to (2).
By
a
virtue of the equality
tD
2
X
X in a
t
tDx
the statement (2) is equivalent to
0.
(obtained earlier), using (2.8) and operating
f(Dx)
DX
Thus (2) is equivalent to
0.
By differenting the result
f2
f
0
on the derived equation we get
0, i.e. the statement (3).
The state-
ment (4) means
Dx(B(Y,Z))
B(X,Y)
Substitution
+
B(VxY,Z
g(X,Y)
B(Y,VxZ).
in the above shows that (4) is equivalent to (3).
This completes the proof.
Let us compare theorem 4.2 with the following theorem of Chen [8]:
REMARK 2.
"Let M be
M.
If dim M
(a)
(b)
a totally umbilical CR-submanifold of a locally Hermitlan symmetric space
5
and
q
I, then
is parallel
if
M
is not totally geodesic,
I/a and rank M
dim M
+ I.
M
Moreover, M
is locally isometric to a sphere of radius
is a totally umbilical hypersurface of a
TOTALLY UMBILICAL CR-SUBMANIFOLDS
flat, totally geodesic submanlfold of
M."
We observe that theorem 4.2 is consistent with the above mentioned theorem.
the conclusion (b) of the theorem shows that
(I) of theorem 4.2.
ACKNOWLEDGEMENT.
555
M
In fact,
is Sasaklan, which is the statement
Also the conclusion (a) is the statement (3) of theorem 4.2.
We thank the Natural Sciences and Engineering Research Council of
Canada for supporting with a research grant.
We are also thankful to the referee for
valuable suggestions for the improvement in the original version of this paper.
REFERENCES
I.
BEJANCU, A. CR submanlfolds of a Kaehler Manifold I, Proc. Amer. Math. Soc. 69
(1978), 135-142.
2.
BEJANCU, A. CR submanlfolds of a Kaehler Manifold II, Trans. Amer. Math. Soc.
250 (1979), 333-345.
3.
YANO, K. and KON, M. CR submanlfolds of Kaehlerian and Sasaklan Manifolds,
Birkhauser, Boston, 1983.
4.
FLAHERTY, E.J.
5.
BEJANCU, A. Umbilical CR submanlfolds of a Kaehler Manifold, Rendlcontl dl Mat.
(3) 15 Serle VI (1980), 431-446.
6.
CHEN, B.Y. CR-submanlfolds of a Kaehler Manifold. I, J. Diff. Geom. 16 (1981),
305-322.
7.
DUGGAL, K.L. and SHARMA, R. Lorentzlan Framed Structures in General Relativity,
Gen. Rel. Gray. 18 (1986), 71-77.
8.
CHEN, B.Y. Totally Umbilical Submanlfolds of Kaehler Manifolds, Arch. der Math.
36 (1981), 83-91.
Hermltlan and Kaehlerlan Geometry in Relativity, Lecture Notes in
Physics, Springer Verlan, Berlin, 1976.