ON GENERIC SUBMANIFOLDS OF A LOCALLY CONFORMAL KAHLER MANIFOLD WITH PARALLEL CANONICAL STRUCTURES

The study of CR-submanifolds of a Khler manifold was initiated by Bejancu (I). Since then many papers have appeared on CR-submanifolds of a Khler manifold. Also, it has been studied that generic submanifolds of KJihler manifolds (2) are generalisations of holomorphic submanifolds, totally real submanifolds and CR-submanifolds of Khler manifolds. On the other hand, many examples


2-form given byg{JX,Y)
I{X,Y), g being the Hermitian metric.Then M is called a locally conformal Khler manifold (1.c.k.) if there is a closed l-form , called the Lee-form on M, such that di9 A, where d and A denote exterior derivative operator and wedge product, respectively [4].Let M be a l.c.k, manifold.Then the vector field B Lee field of M is defined as g(X,B) (X).Now suppose that V be the Levi-Civita connection of g.Then, we have [4] VxY= VxY-(X)Y + oJ(Y)X g(X,Y)B (1.1) where V is a torsionless linear connection on M called the Weyl connection of g.
The following result is known [4].
THEOREM.The almost Hermitian manifold is a 1.c.k.manifold if and only if there is a closed l-form 0o on such that the Weyl connection be almost complex i.e.VJ=O.
DEFINITION.Let M be a submanifold of a l.c.k, manifold M. The holomorphic tangent spaces to M at xeM are defined as D T M JT M, x x x where D is the maximal complex subspace of T M. If the dimension of x x D is constant along M and it defines a differentiable distribution on X M, then M is called a generic submanifold of M. We call D the x holomorphic distribution and the orthogonal complementary distribution D is called purely real distribution [2].A generic submanifold M of x a l.c.k, manifold is a CR-submanifold [I] if the orthogonal

JD
T M, where T M is the normal space of M at x. x x Let M be a generic submanifold of a 1.c.k.manifold M and let V be covariant differentiation on M induced by P and V, respectively.
Then Gauss and Weingarten formulae w.r.t.V and P are given by .
VxY FxY+h(X,Y), VXN -ANX + VxN (1.2) VxY VxY+h{X,Y), VXN -ANX + VxN (1.3) for any vector fields X,Y tangent to M and N normal to M.Here h(h) is the second fundamental form of M with respect to (7) and 7 (7 is the normal connection.Moreover, g(ANX,Y) g(h(X,Y),N). (1.4) The transforms JX and JN of X and N by J are decomposed into tangential and normal parts as JX PX + FX, (1.5) JN tN + fN, (1.6) J_ where P and f are endomorphisms of TM and T M and F and t are normal bundle valued l-form on TH and tangent bundle valued l-form on T M, respectively.
For Lee vector field B tangent to M, we put B (B) + (B 2) x e M (1.7) x x x' and (B 2) are the tangential and normal components of B, where (BI)x x respectively.
The following relations hold for a generic submanifold [2].
We define the covariant differentiation of P,F,t and f as follows: ( for any vector fields X and Y tangent to M normal to M. We say that P (respectively f,F or (respectively f=O, VF=O or t=O).

M. Then and any vector t is parallel
Let M be a generic submanifold of a l.c.k.
(2.8) for any vector fields X,Y tangent to M and any vector field N to }4.
The following result is known [4].
LEMMA A. The holomorphic distribution D is integrable iff normal g(h(X,JY),FZ) g(h(JX,Y),FZ) +/for any vector fields X,YeD and ZD Using {2.6) in Lemma A, we have COROLLARY 2.2.Let M be a generic submanifold of a 1.c.k.manifold M. Then the holomorphic distribution D is integrable iff g(h(X,JY)-h(JX,Y)+(X,Y)B2,FZ) 0 J_ for ,any X,YeD and ZeD From (1.3),(1.5)and (1.6), on comparing the tangential and normal components, we obtain LEMMA 2.3.Let M be a generic submanifold of a 1.c.k.manifold .
GENERIC SUBMANIFOLDS WITH P=O.
We now assume that the canonical structure P generic submanifold of a l.c.k, manifold i.e., (2.5), we have for all vector fields X,Y tangent to M.
+/-Thus, for X,ZeD and YeD equation (3.2) gives us the following: PROPOSITION 3.1.Let M be a generic submanifold of a l.c.k.manifold such that BED.If P is parallel, then the holomorphic distribution D is integrable, that is, We now assume that the Lee field B is normal to M. Then +/- (2.6), (2.7), (3.1) and using the fact that t(T M)=D for X=_D and YeD Then, for ZD and using (1.6) in (3.3), we get whence h(X,Z) z g(X,Z) h(X,JZ) h(JX,Z) -g(X,JZ)B 2 which can be written as g(h(X,JZ)-h(JX,Z)+f)(X,Z)B2,FY) O.
manifold M with B normal to M. If P is parallel, then the holomorphic distribution D is integrable. 4.

GENERIC SUBMANIFOLDS WITH WF=O
In this section, we obtain some results assuming that F is parallel.
PROPOSITION 4.1.Let M be a generic submanifold of a l.c.k.
manifold .If F is parallel, then distribution D is integrable and leaf of D is totally geodesic in M.
PROOF.Since F is parallel, we have from (2.2) that for all X,Y D, FxY O. Consequently, xYD for any X,YD, which shows that the distribution D is integrable and leaf of D is totally geodesic in M.
PROPOSITION 4.2.Let M be a generic submanifold of a l.c.k.manifold M. If F is parallel, then AfNX+ANPX g(B2,N)X-g(B2,S)PX--(X)tN 337 for any vector field X tangent to M and N normal to M.
Hence, for any vector field N normal to M, we get g Thus, if F is parallel, we get the result.
From the above result, we immediately get the following COROLLARY 4.3.Let M be a generic submanifold of a l.c.k.manifold and let BeD.If P is parallel, then (a) A fN M, and X ANPX 2 -o3(X)tN, for any vector field X tangent to (b) g(Jh(X,Y),N) O, or g(Jh{D,D ),N) 0 for all XeD and YeD.
Using Lemma A in the above equation, we get TXt N -0(tN)X+ g(X tN)B +h(X tN)+ g(X tN)B AfNX +/-+/- (5.1) for any vector field X tangent to M and N normal to M.
Thus we have PROPOSITION 5.1.Let M be a generic submanifold of a l.c.k.
manifold .If t is parallel, then AfNX-PANX (N)PX-co(JN)X + g(X,tN)B for any vector field X tangent to M and N normal to M.
The following is immediate.
COROLLARY 5.2.Let M be a generic submanifold of a l.c.k.manifold such that BeD.If t is parallel, then (a) A fN x PANX g(X,tN)B for any X tangent to M, and 3_ (b) g(Jh(X,Y),N) O, or g(Jh(D,D ),N) 0 for any XD and YD PROPOSITION 5.3.Let M be a generic submanifold of manifold .Then F is parallel if and only if t is parallel.
PROOF.Let t be parallel.Then, from (5.1)we have a .c.k.
g(AfN x y)_g(PAN x y) + g(B,JN)g(X Y) i ,y)_ ! 2 g'X'tN'g(B1 z g(PX,Y)g(B2,N) 0 for any vector fields X,Y tangent to M and any vector field N to M. The above equation becomes -g(fh(X,Y),N)+g(h(X,PY),N) g(B2,N)g(X,Y and M be any purely real submanifold of dimension p in the complex number space C p. Then it can be verified that the Riemannian product M X M is a generic submanifold in C r+p satisfying VP 0 and F O.
2. It is to be noted that many examples of generic surfaces in C which are not CR-submanifolds can be found in Chen's book [3].

[
field X tangent to M and N normal to M.Comparing tangential components, we get (xt)N AfNX-PANX+ (JN)X-(N)PX-g(X,tN)B M be any complex submanifold in complex number space C