D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 5 2 IS S N 1 3 0 3 –5 9 9 1
ON CR SUBMANIFOLDS OF A S MANIFOLD ENDOWED
WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION
MEHMET AKIF AKYOL AND RAMAZAN SARI
Abstract. In this paper, we study CR submanifolds of an S manifold en-dowed with a semi-symmetric non-metric connection. We give an example, investigating integrabilities of horizontal and vertical distributions of CR sub manifolds endowed with a semi-symmetric non-metric connection. We also consider parallel horizontal distributions of CR submanifolds.
1. Introduction
In 1963, Yano [23] introduced the notion of f -structure on a C1 m-dimensional
manifold M , as a non-vanishing tensor …eld f of type (1; 1) on M which satis…es f3+ f = 0 and has constant rank r. It is known that r is even, say r = 2n.
Moreover, T M splits into two complementary subbundles Imf and ker f and the restriction of f to Imf determines a complex structure on such subbundle. It is also known that the existence of an f -structure on M is equivalent to a reduction of the structure group to U (n) O(s) (see [9]), where s = m 2n. In 1970, Goldberg and Yano [12] introduced globally frame f -manifolds (also called metric f - manifolds and f .pk-manifolds). A wide class of globally frame f -manifolds was introduced in [9] by Blair according to the following de…nition: a metric f -structure is said to be a K-structure if the fundamental 2-form , de…ned usually as (X; Y ) = g(X; f Y ), for any vector …elds X and Y on M , is closed and the normality condition holds, that is, [f; f ]+2Psi=1d i
i = 0, where [f; f ] denotes the Nijenhuis torsion of f . A
K-manifold is called an S-manifold if d k = , for all k = 1; : : : ; s. The S-manifolds
have been studied by several authors (see, for instance, [2],[3],[5],[10],[11]).
On the other hand, the notion of a CR submanifold of Kaehlerian manifolds was introduced by A. Bejancu in [6]. Later, the concept of CR submanifolds has been developed by [4], [8], [13], [14], [16], [18], [19], [20], [22] and others.
Received by the editors: Feb 04, 2016, Accepted: March 22, 2016. 2010 Mathematics Subject Classi…cation. 53C15, 53C40.
Key words and phrases. CR submanifold, S manifold, Semi-symmetric non-metric connection.
c 2 0 1 6 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis t ic s .
Let r be a linear connection in an n dimensional di¤erentiable manifold M : The torsion tensor T and the curvature tensor R of r are given respectively by [7]
T (X; Y ) = rXY rYX [X; Y ] ;
R (X; Y ) Z = rXrYZ rYrXZ r[X;Y ]Z:
The connection r is symmetric if the torsion tensor T vanishes, otherwise it is non-symmetric. The connection r is a metric connection if there is a Riemannian metric g in M such that rg = 0; otherwise it is non-metric. It is well known that a linear connection is symmetric and metric if it is the Levi-Civita connection. In [17], Friedmann and Schouten introduced the notion of semi-symmetric linear connections. More precisely, if r is a linear connection in a di¤erentiable manifold M , the torsion tensor T of r is given by T (X; Y ) = rXY rYX [X; Y ]; for
any vector …elds X and Y on M . The connection r is said to be symmetric if the torsion tensor T vanishes, otherwise it is said to be non-symmetric. In this case, r is said to be a semi-symmetric connection if its torsion tensor T is of the form T (X; Y ) = (Y )X (X)Y , for any X; Y , where is a 1-form on M . Moreover, if g is a (pseudo)-Riemannian metric on M , r is called a metric connection if rg = 0, otherwise it is called non-metric. It is well known that the Riemannian connection is the unique metric and symmetric linear connection on a Riemannian manifold. In 1932, Hayden [15] de…ned a metric connection with torsion on a Riemannian manifold. In [1] Agashe and Cha‡e de…ned a semi-symmetric non-metric connection on a Riemannian manifold and studied some of its properties. Later, the concept of semi-symmetric non-metric connection has been developed by (see, for instance, [3], [21]) and others. In this paper we study CR submanifolds of an S manifold endowed with a semi-symmetric non-metric connection. We consider integrabilities of horizontal and vertical distributions of CR submanifolds with a semi-symmetric non-metric connection. We also consider parallel horizontal distributions of CR submanifolds.
The paper is organized as follows: In section 2, we give a brief introduction to S manifolds. In section 3, we study CR submanifolds of S-manifolds. We …nd necessary conditions for the induced connection on a CR submanifold of an S manifold with symmetric non-metric connection to be also a semi-symmetric non-metric connection. In section 4, We study integrabilities of hori-zontal and vertical distributions of CR submanifolds with a semi-symmetric non-metric connection.
2. S Manifolds
A (2n+s) dimensional di¤erentiable manifold fM is called a metric f -manifold if there exist an (1; 1) type tensor …eld f , s vector …elds 1; : : : ; s, s 1-forms 1; : : : ; s
and a Riemannian metric g on fM such that f2= I + s X i=1 i i; i( j) = ij; f i= 0; i f = 0; (2.1) g(f X; f Y ) = g(X; Y ) s X i=1 i(X) i(Y ); (2.2)
for any X; Y 2 (T fM ); i; j 2 f1; : : : ; sg. In addition we have:
i(X) = g(X;
i); g(X; f Y ) = g(f X; Y ): (2.3)
Moreover, a metric f -manifold is normal if [f; f ] + 2
s
X
=1
d = 0
where [f; f ] is Nijenhuis tensor of f .
Then a 2-form F is de…ned by F (X; Y ) = g(X; f Y ), for any X; Y 2 (T fM ); called the fundamental 2-form. Then fM is said to be an S-manifold if the f structure is normal and
1^ ::: ^ s^ (d )n6= 0; F = d
for any = 1; :::; s: In the case s = 1; an S manifold is a Sasakian manifold. Now, if er denotes the Riemannian connection associated with g, then [7]
e rXf Y = s X =1 g (f X; f Y ) + (Y ) f2X ; (2.4)
for all X; Y 2 (T fM ). From (2.4), it is deduced that e
rX = f X; (2.5)
for any X; Y 2 (T fM ); 2 f1; :::; sg:
3. CR Submanifold of S Manifolds
De…nition 3.1. An (2m+s) dimensional Riemannian submanifold M of S manifold f
M is called a CR submanifold if 1; 2; ::: ; sis tangent to M and there exists on
M two di¤ erentiable distributions D and D? on M satisfying:
(i) T M = D D? spf
1; :::; sg;
(ii) The distribution D is invariant under f; that is f Dx= Dx for any x 2 M;
(iii) The distribution D? is anti-invariant under f; that is, f D?
x Tx?M for
any x 2 M; where TxM and TxM? are the tangent space of M at x:
We denote by 2p and q the real dimensions of Dx and Dx? respectively, for any
x 2 M: Then if p = 0 we have an anti-invariant submanifold tangent to 1; 2; :::; s
Example 3.1. In what follows, (R2n+s; f; ; ; g) will denote the manifold R2n+s
with its usual S-structure given by =1 2(dz n X i=1 yidxi); = 2 @ @z f ( n X i=1 (Xi @ @xi + Yi @ @yi ) + s X =1 Z @ @z ) = n X i=1 (Yi @ @xi Xi @ @yi ) + s X =1 n X i=1 Yiyi @ @z g = s X =1 +1 4( n X i=1 dxi dxi+ dyi dyi);
(x1; :::; xn; y1; :::; yn; z1; :::; zs) denoting the Cartesian coordinates on R2n+s. The
consider a submanifold of R8 de…ned by
M = X(u; v; k; l; t1; t2) = 2(u; 0; k; v; l; 0; t1; t2):
Then local frame of T M e1 = 2 @ @x1 ; e2= 2 @ @y1 ; e3 = 2 @ @x3 ; e4= 2 @ @y2 ; e5 = 2 @ @z1 = 1; e6= 2 @ @z2 = 2 and e1 = 2 @ @x2 ; e2= 2 @ @y3
from a basis of T?M . We determine D
1 = spfe1; e2g and D2 = spfe3; e4g, then
D1, D2 are invariant and anti-invariant distribution. Thus T M = D1 D2
spf 1; 2g is a CR submanifold of R8:
Let er be the Levi-Civita connection of fM with respect to the induced metric g. Then Gauss and Weingarten formulas are given by
e
rXY = rXY + h(X; Y ) (3.1)
and
e
rXN = ANX + rX?N (3.2)
for any X; Y 2 (T M) and N 2 (T?M ). r ? is the connection in the normal
bundle, h is the second fundamental from of fM and AN is the Weingarten
endomor-phism associated with N . The second fundamental form h and the shape operator A related by
g(h(X; Y ); N ) = g(ANX; Y ) (3.3)
A CR submanifold is said to be D totally geodesic if h(X; Y ) = 0 for any X; Y 2 (D) and it is said to be D? totall geodesic if h(Z; W ) = 0 for any
Z; W 2 (D?):
The projection morphisms of T M to D and D? are denoted by P and Q
respec-tively. For any X; Y 2 (T M) and N 2 (T?M ) we have
X = P X + QX + s X =1 (X) ; 1 s (3.4) f N = BN + CN (3.5)
where BN (resp. CN ) denotes the tangential (resp. normal) component of 'N: Now, we de…ne a connectionr as
rXY = erXY + s
X
=1
(Y ) X:
Theorem 3.1. Let er be the Riemannian connection on a S manifold fM . Then the linear connection which is de…ned as
rXY = erXY + s
X
=1
(Y )X; 8X; Y 2 (T M) (3.6)
is a semi-symmetric non metric connection on fM :
Proof. Using new connection and the fact that the Riemannian connection is torsion free, the torsion tensor T of the connection r is given by
T (X; Y ) =
s
X
=1
f (Y )X (X)Y g (3.7)
for all X; Y 2 (T M). Moreover, by using (3.6) again, for all X; Y; Z 2 (T M) and since er is a metric connection, we have
(rXg)(Y; Z) = s
X
=1
fg(X; Y ) (Z) g(X; Z) (Y )g: (3.8)
From (3.7) and (3.8) we conclude that the linear connection r is a semi-symmetric non-metric connection on fM .
Theorem 3.2. Let M be a CR submanifold of S-manifold fM . Then (rXf )Y = s X =1 fg(X; Y ) (Y )(X + f X)g (3.9) for all X; Y 2 (T M).
Proof. From (3.6), we get (rXf )Y = s X =1 fg(X; Y ) (Y ) X (Y ) f Xg
for all X; Y 2 (T M). Therefore we obtain the result from (2.4).
Corollary 3.1. Let M be a CR submanifold of S-manifold fM with semi-symmetric non-metric connectionr: Then
rX = f X + X (3.10)
for all X 2 (T M):
We denote by same symbol g both metrics on fM and M . Let r be the semi-symmetric non-metric connection on fM and r be the induced connection on M. Then,
rXY = rXY + m(X; Y ) (3.11)
where m is a (T?M ) valued symmetric tensor …eld on CR- submanifold M . If
r denotes the induced connection from the Riemannian connection r; then
rXY = rXY + h(X; Y ); (3.12)
where h is the second fundamental form. Using (3.1) and (3.4), we have rXY + m(X; Y ) = rXY + h(X; Y ) +
s
X
=1
(Y )X: Equating tangential and normal components from both the sides, we get
m(X; Y ) = h(X; Y ) and rXY = rXY + s X i=1 (Y )X: (3.13)
Thus r is also a semi-symmetric non-metric connection. From (3.2) and (3.13), we have rXN = rXN + s X =1 (N )X = ANX + r?XN + s X =1 (N )X; where X 2 (T M) and N 2 (T?M ).
Now, Gauss and Weingarten formulas for a CR-submanifolds of a S-manifold with a semi-symmetric non-metric connection is
rXN = ANX + r?XN + s
X
=1
(N )X (3.15)
for any X; Y 2 (T M) and N 2 (T?M ); h second fundamental form of M and
AN is the Weingarten endomorphism associated with N:
Theorem 3.3. The connection induced on CR submanifold of a S-manifold with semi-symmetric non-metric connection is also a semi-symmetric non-metric con-nection.
Proof. From (3.7) and (3.8), we get
T (X; Y ) = T (X; Y ) and (rXg)(Y; Z) = (rXg)(Y; Z)
for any X; Y 2 (T M); where T is the torsion tensor of r:
4. Integrability and Parallel of Distributions
Lemma 4.1. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. Then,
P rXf P Y P Af QYX f P rXY = s X =1 (Y ) (P X + f P X) ; (4.1) QrXf P Y QAf QYX th (X; Y ) = s X =1 (Y ) QX; (4.2) h (X; f P Y ) f QrXY + r?Xf QY = nh (X; Y ) s X =1 (Y ) f QX; (4.3) for all X; Y 2 (T M):
Proof. By direct covariant di¤erentiation, we have rXf Y = rXf Y + f rXY :
for any X; Y 2 (T M): By virtue of (3.4),(3.9),(3.14) and (3.15) we get
rXf P Y + h (X; f P Y ) + Af QYX + r?Xf QY = s P =1fg (X; Y ) (Y ) (f X + X)g + fr XY + f h (X; Y ) :
Using (3.4) again, we have P rXf P Y + QrXf P Y + h (X; f P Y ) P Af QYX QAf QYX + r?Xf QY = s P =1fg (X; Y ) P + g (X; Y ) Q (Y ) P X (Y ) QX (Y ) f P X (Y ) f QXg +f P rXY + f QrXY + th (X; Y ) + nh (X; Y ) :
Equations (4.1)-(4.3) follow by comparing the horizontal, vertical and normal com-ponents.
Lemma 4.2. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. Then,
Af WZ f P rZW th (Z; W ) = s X =1 g (Z; W ) ; (4.4) r?Zf W = f QrZW + nh (Z; W ) (4.5) for any Z; W 2 (D?):
Proof. From (3.9), we have rZf W =
s
X
=1
g (f Z; f W ) + (W ) f2Z f Z
for any Z; W 2 (D?): Since (W ) = 0 for W 2 (D), using (2.2) we get
rZf W = s X =1 g (f Z; f W ) = s X =1 g (Z; W ) : Therefore rZf W f rZW = s X =1 g (Z; W ) : In above equation, using (3.14) and (3.15), we have
Af WZ + r?Zf W f rZW f h (Z; W ) = s X =1 g (Z; W ) Af WZ + r?Zf W f P rZW f QrZW th (Z; W ) nh (Z; W ) = s P =1 g (Z; W ) : Now comparing tangent and normal parts in above equation, we obtain (4.4) and (4.5).
Lemma 4.3. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. Then,
rXf Y f P rXY = s X =1 g (X; Y ) + th (X; Y ) ; (4.6) h (X; f Y ) = f QrXY + nh (X; Y ) (4.7) for any X; Y 2 (D): Proof. From (3.9), we have
rXf Y = s
X
=1
g (f X; f Y ) + (Y ) f2X f X
for any X; Y 2 (D). Using (Y ) = 0 for each Y 2 (D) and (2.2) we obtain rXf Y = s X =1 g (f X; f Y ) = s X =1 g (X; Y ) : Moreover, we have rXf Y f rXY = s X =1 g (X; Y ) : Now using (3.14), we have
rXf Y + h (X; f Y ) f rXY f h (X; Y ) = s X =1 g (X; Y ) rXf Y + h (X; f Y ) f P rXY f QrXY th (X; Y ) nh (X; Y ) = s X =1 g (X; Y ) :
Now comparing tangent and normal parts, we obtain (4.6) and (4.7).
Lemma 4.4. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. Then,
rX = f P X + X; 8X 2 (T M) (4.8)
h (X; ) = f QX; 8X 2 (T M) (4.9)
Proof. Using (3.14) in (3.10), we easily obtain
rX = f X + X ) rX + h (X; ) = f X + X
which gives
rX + h (X; ) = f P X f QX + X:
Now comparing tangent and normal parts, we get
rX = f P X + X and h (X; ) = f QX:
On the other hand, using (3.3) we have
g (AV ; X) = g (h (X; ) ; V ) = g (0; V ) = 0
for X 2 (D) and V 2 (T?M ). Using (4.9) in the above equation, we get g (AV ; X) = 0; 8X 2 (D) which leads to AV 2 (D?)
also
g (AV ; X) = 0; 8X 2 (D) ) g (AV ; X) = (AVX) = 0
which gives (4.10).
Theorem 4.1. Let M be a CR submanifold of a S-manifold fM with semi-symmetric non-metric connection. Then the distribution D is not integrable.
Proof. For any X; Y 2 (D), we have
g([X; Y ]; i) = g(Y; erX i) + g(X; erY i):
Using (3.10) and (3.14), we have
g([X; Y ]; i) = g(Y;rX i X) + g(X; rY i Y )
= g(Y; f X) + g(X; f Y ):
Thus D is integrable if and only if g(X; f Y ) = g(Y; f X). From (2.3), the proof is complete.
Theorem 4.2. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. The distribution D Spf 1; :::; sg is integrable
if and only if
h (X; f Y ) = h (Y; f X) for any X; Y 2 (D Spf 1; :::; sg):
Proof. From (4.7), we have
h (X; f P Y ) = f QrXY + nh (X; Y ) ; 8X; Y 2 (D spf 1; :::; sg): (4.11)
Interchanging X and Y; we have
h (Y; f P X) = f QrYX + nh (Y; X) ; 8X; Y 2 (D spf 1; :::; sg): (4.12)
Adding (4.11) and (4.12), we obtain
Then we have [X; Y ] 2 (D spf 1; :::; sg) if and only if h (X; fY ) = h (Y; fX) :
Corollary 4.1. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. The distribution D Spf 1; :::; sg is integrable if and only if
ANf X = f ANX
for any X 2 (D spf 1; :::; sg):
De…nition 4.1. A CR submanifold is said to be mixed totally geodesic if h(X; Z) = 0, for any X 2 (D) and Z 2 (D?).
Lemma 4.5. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. Then M is mixed totaly geodesic if and only if one of the following satis…ed;
AVX 2 D 8X 2 (D); V 2 (T?M ) ; (4.13)
AVX 2 D? 8X 2 (D?); V 2 (T?M ) : (4.14)
Proof. For X 2 (D), V 2 (T?M ) and Y 2 (D?); consider AVX; then from
(3.3) we get g (AVX; Y ) = g (h (X; Y ) ; V ) = 0 , AVX 2 (D): Hence, we have g (h (X; Y ) ; V ) = 0 , h (X; Y ) = 0 , AVX 2 (D) 8X 2 (D); V 2 (T?M );
which gives (4.13). In a similar way is deduced relation (4.14).
De…nition 4.2. The horizontal (resp.vertical) distribution on D (resp. D?) is
said to be parallel with respect to the connection r on M if
rXY 2 (D) resp: rZW 2 (D?) for any X; Y 2 (D) (resp: Z; W 2 (D?)):
Theorem 4.3. Let M be a horizontal CR submanifold of an S manifold fM with semi-symmetric non-metric connection. Then, the horizontal distribution D is parallel if and only if
h (X; f Y ) = h (Y; f X) = f h (X; Y ) (4.15) for all X; Y 2 (D):
Proof. Since every parallel is involutive then the …rst equality follows immediately. Now since D is parallel, we have
Then from (4.2), we have
th (X; Y ) = 0 8X; Y 2 (D): (4.16)
From (4.3), D is parallel if and only if
h (X; f Y ) = nh (X; Y ) : But we have
f h (X; Y ) = th (X; Y ) + nh (X; Y ) ;
and from (4.9), f h (X; Y ) = nh (X; Y ) ; which completes the proof.
Theorem 4.4. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. The distribution D? Spf
1; :::; sg is integrable if and only if Af XY Af YX = s X =1 f (X) Y (Y ) Xg (4.17) for all X; Y 2 (D? spf 1; :::; sg):
Proof. If X; Y 2 (D? spf 1; :::; sg); then from (4.1) and (4.2) we have
P Af QYX f P rXY = 0; (4.18) QAf QYX th (X; Y ) = s X =1 (Y )X: (4.19)
Adding (4.18) and (4.19), we have
Af YX f P rXY th (X; Y ) = s
X
=1
(Y )X: (4.20)
Now interchanging X and Y , we have
Af XY f P rYX th (X; Y ) = s
X
=1
(X)Y: (4.21)
Subtracting (4.20) and (4.21), we obtain Af YX + Af XY f P [X; Y ] = s X =1 f (Y ) X + (X) Y g : Hence P [X; Y ] = 0; we obtain , Af XY Af YX = s X =1 f (X) Y (Y ) Xg :
Corollary 4.2. Let M be CR submanifold of an S manifold fM with semi-symmetric non metric connection. Then, the distribution D? is integrable if and only if
Af YX = Af XY (4.22)
for all X; Y 2 (D?):
Proof. The proof can be obtained directly from (4.17).
Lemma 4.6. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. Then, the distribution D? is parallel if and only if
Af WZ = s X =1 g (Z; W ) + th (Z; W ) (4.23) for all Z; W 2 (D?): Proof. From (4.4), we have
Af WZ f P rZW = s
X
=1
g (X; Y ) + th (Z; W ) 8Z; W 2 (D?): If D? is parallel then we get
rZW 2 (D?) , P rZW = 0;
which gives (4.23).
Lemma 4.7. Let M be a CR submanifold of an S manifold fM with semi-symmetric non-metric connection. Then the distribution D? is parallel if and only if
Af WZ 2 (D?) (4.24)
for any Z; W 2 (D?):
Proof. For any Z; W 2 (D?); from (3.9) we have rZf W =
s
X
=1
g (f Z; f W ) + (W ) f2Z f Z : Using (3.14) and (3.15) we obtain
rZf W f rZW = s X =1 g (f Z; f W ) + (W ) f2Z f Z Af WZ + r?Zf W f rZW f h (Z; W ) = s X =1 g (f Z; f W ) + (W ) f2Z f Z :
Taking inner product with Y 2 (D) in the above equation, we have g ( Af WZ; Y ) + g r?Zf W; Y g (f rZW; Y ) g (f h (Z; W ) ; Y ) = s X =1 g (f Z; f W ) g ( ; Y ) + (W ) g f2Z; Y (W ) g (f Z; Y ) : Then we have g(Af WZ; Y ) = g(f rZW; Y ) = g(rZW; f Y ):
This imply that
g (Af WZ; Y ) = 0 , Af WZ 2 (D?):
Therefore we obtain
rZW 2 D?, Af WZ 2 (D?); 8Z; W 2 (D?):
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Current address : Mehmet Akif AKYOL. Bingöl University, Faculty of Arts and Sciences, Deparment of Mathematics, 12000, Bingöl, Turkey
E-mail address : makyol@bingol.edu.tr
Current address : Ramazan SARI. Merzifon Vocational Schools, Amasya University, Amasya, Turkey
