C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 67, N umb er 2, Pages 264–281 (2018) D O I: 10.1501/C om mua1_ 0000000880 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
ON THE GEOMETRY OF PSEUDO-SLANT SUBMANIFOLDS OF A NEARLY SASAKIAN MANIFOLD
SÜLEYMAN DIRIK, MEHMET ATÇEKEN, AND ÜM ·IT YILDIRIM
Abstract. In this paper, we study the pseudo-slant submanifolds of a nearly Sasakian manifold. We characteterize a totally umbilical properpseudo-slant submanifolds and …nd that a necessary and su¢ cient condition for such sub-manifolds totally geodesic. Also the integrability conditions of distributions of pseudo-slant submanifolds of a nearly Sasakian manifold are investigated.
1. Introduction
The di¤erential geometry of slant submanifolds has shown an increasing develop-ment since B.Y. Chen de…ned slant submanifolds in complex manifolds as a natural generalization of both the invariant and anti-invariant submanifolds [3], [4]. Many research articles have been appeared on the existence of these submanifolds in dif-ferent knows spaces. The slant submanifolds of an almost contact metric manifolds were de…ned and studied by A. Lotta [2]. After, such submanifolds were studied by J.L Cabrerizo et. al[6], in Sasakian manifolds . Recently, in [9],[10],[11],[13] M. Atçeken studied slant and pseudo-slant submanifold in (LCS)n manifold and
other manifolds. The notion of semi-slant submanifolds of an almost Hermitian manifold was introduced by N. Papagiuc [12]. Recently, A. Carrizo [5],[6] de…ned and studied bi-slant immersions in almost Hermitian manifolds and simultaneously gave the notion of pseudo-slant submanifolds in almost Hermitian manifolds. The contact version of pseudo-slant submanifolds has been de…ned and studied by V. A. Khan and M. A Khan [16].
The present paper is organized as follows.
In section 1, the notions and de…nitions of submanifolds of a Riemannian mani-fold were given for later use. In this paper, we study pseudo-slant submanimani-folds of
Received by the editors: February 13, 2017; Accepted: August 02, 2017. 2010 Mathematics Subject Classi…cation. 53C15, 53C25, 53C17, 53D15, 53D10.
Key words and phrases. Nearly Sasakian manifold, Slant submanifold, Proper-slant submani-fold, Pseudo-slant submanifold.
c 2 0 1 8 A n ka ra U n ive rsity. C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a tic s a n d S t a tis tic s . 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 t h e m a tic s a n d S t a tis t ic s .
a nearly Sasakian manifold. In section 2, we review basic formulas and de…nitions for a nearly Sasakian manifold and their submanifolds. In section 3, we recall the de…nition and some basic results of a pseudo-slant submanifold of almost contact metric manifold. We study characterization of totally umbilical proper-slant sub-manifolds and …nd that a necessary and su¢ cient condition for such subsub-manifolds is to be totally geodesic. In section 4, the integrability conditions of distributions of pseudo-slant submanifolds of a nearly Sasakian manifold are investigated.
2. Preliminaries
In this section, we give some notations used throughout this paper. We recall some necessary fact and formulas from the theory of nearly Sasakian manifolds and their submanifolds.
Let fM be an (2m + 1) dimensional almost contact metric manifold with an almost contact metric structure ('; ; ; g); that is, ' is a (1; 1) tensor …eld , is a vector …eld; is 1-form and g is a compatible Riemanian metric such that
'2X = X + (X) ; (2.1)
' = 0; ( ) = 1; ' = 0; (X) = g(X; ) (2.2)
and
g('X; 'Y ) = g(X; Y ) (X) (Y ); g('X; Y ) = g(X; 'Y ) (2.3) for any vector …elds X; Y 2 (T fM ), where ( fM ) denotes the set of all vector …elds on fM . If in addition to above relations,
( erX')Y = g(X; Y ) (Y )X (2.4)
then fM is called a Sasakian manifold, where er is the Levi-Civita connections of g: The almost contact metric manifold fM is called a nearly Sasakian manifold if it satisfy the following condition
( erX')Y + ( erY')X = 2g(X; Y ) (Y )X (X)Y (2.5)
for any X; Y 2 (T fM ).
Now, let M be a submanifold of an almost contact metric manifold fM , we denote the induced connections on M and the normal bundle T?M by r and r?, respectively, then the Gauss and Weingarten formulas are given by
e
rXY = rXY + h(X; Y ) (2.6)
and
e
for any X; Y 2 (T M), V 2 (T?M ), where h is the second fundamental form
and AV is the Weingarten map associated with V as
g(AVX; Y ) = g(h(X; Y ); V ): (2.8)
for all X; Y 2 (T M) and V 2 (T?M ):
The mean curvature vector H of M is given by H = 1 m m X i=1 h(ei; ei); (2.9)
where m is the dimension of M and fe1; e2; :::; emg is a local orthonormal frame of
M . A submanifold M of a Riemannian manifold fM is said to be totally umbilical if
h(X; Y ) = g(X; Y )H; (2.10)
for any X; Y 2 (T M). A submanifold M is said to be totally geodesic if h = 0 and M is said to be minimal if H = 0:
Let M be a submanifold of an almost contact metric manifold fM : Then for any X 2 (T M); we can write
'X = T X + N X; (2.11)
where T X is the tangential component and N X is the normal component of 'X: Similarly, for V 2 (T?M ); we can write
'V = tV + nV; (2.12)
where tV is the tangential component and nV is the normal component of 'V: Thus by using (2.1), (2.2), (2.11) and (2.12 ), we obtain
T2+ tN = I + ; N T + nN = 0; (2.13)
T t + tn = 0; n2+ N t = I (2.14)
and
T = 0 = N ; T = 0 = N: (2.15)
Furthermore, for any X; Y 2 (T M); we have g(T X; Y ) = g(X; T Y ) and V; U 2 (T?M ); we get g(U; nV ) = g(nU; V ). These show that T and n are also skew-symmetric tensor …elds. Moreover, for any X 2 (T M) and V 2 (T?M ); we have
g(N X; V ) = g(X; tV ) (2.16)
The covariant derivatives of the tensor …eld T; N; t and n are, respectively, de…ned by (rXT )Y = rXT Y T rXY; (2.17) (rXN )Y = r?XN Y N rXY; (2.18) (rXt)V = rXtV tr?XV (2.19) and (rXn)V = r?XnV nr?XV (2.20)
for any X; Y 2 (T M) and V 2 (T?M ).
Now, for any X; Y 2 (T M); let us denote the tangential and normal parts of ( erX')Y by PXY and FXY , respectively. Then we decompose
( erX')Y = PXY + FXY (2.21)
thus, by an easy computation, we obtain the formulae
PXY = (rXT )Y AN YX th(X; Y ) (2.22)
and
FXY = (rXN )Y + h(X; T Y ) nh(X; Y ): (2.23)
Similarly, for any V 2 (T?M ), denoting tangential and normal parts of ( erX')V
by PXV and FXV; respectively, we obtain
PXV = (rXt)V AnVX + T AVX (2.24)
and
FXV = (rXn)V + h(tV; X) + N AVX: (2.25)
Now, for any X; Y 2 (T M), from (2.5), we have
( erX')Y + ( erY')X = 2g(X; Y ) (Y )X (X)Y (2.26)
that is,
( erX')Y + ( erY')X = erX'Y ' erXY + erY'X ' erYX:
By using (2.6), (2.7), (2.11) and (2.12), we get
( erX')Y + ( erY')X = reXT Y + erXN Y '(rXY + h(X; Y )) + erYT X + erYN X '(rYX + h(X; Y )): = rXT Y + h(X; T Y ) AN YX + r?XN Y T rXY N rXY th(X; Y ) nh(X; Y ) +rYT X + h(Y; T X) AN XY + r?YN X T rYX N rYX th(X; Y ) nh(X; Y ): (2.27)
Making use of (2.27) and (2.26), we obtain
rXT Y + h(X; T Y ) AN YX + r?XN Y T rXY N rXY
th(X; Y ) nh(X; Y ) + rYT X + h(Y; T X) AN XY
+r?YN X T rYX N rYX th(X; Y ) nh(X; Y )
2g(X; Y ) + (Y )X + (X)Y = 0:
By taking tangential and normal parts of the above equation, respectively, we have equation
(rXT )Y + (rYT )X = AN XY + AN YX + 2th(X; Y )
+2g(X; Y ) (Y )X (X)Y (2.28)
and
(rXN )Y + (rYN )X = h(X; T Y ) h(Y; T X) + 2nh(X; Y ) : (2.29)
On the other hand, for Y = in (2.5) and by using (2.2), (2.6) and (2.7), we see
T [X; ] = (r T )X T r X 2th(X; ) AN X + X (X) (2.30)
and
N [X; ] = (r N)X N r X 2nh(X; ) + h(T X; ): (2.31) In contact geometry, A. Lotta introduced slant submanifolds as follows [2]: De…nition 2.1. Let M be a submanifold of an almost contact metric manifold (M ; '; ; ; g). Then M is said to be a slant submanifold if the angle (X) between 'X and TM(p) is constant at any point p 2 M for any X linearly independent
of . Thus the invariant and anti-invariant submanifolds are special class of slant submanifolds with slant angles = 0 and = 2, respectively. If the slant angle is neither zero nor 2, then slant submanifold is said to be proper slant submanifold. If M is a slant submanifold of an almost contact metric manifold, then the tangent bundle T M of M can be decomposed as
T M = D ;
where denotes the distribution spanned by the structure vector …eld and D is the complementary distribution of in T M; known as the slant distribution on M: For a slant submanifold M of an almost contact metric manifold fM ; the normal bundle T?M of M is decomposed as
T?M = N (T M )
In an almost contact metric manifold. In fact, J. L Cabrerizo obtained the fol-lowing theorem[6].
Theorem 2.2. [6]. Let M be a slant submanifold of an almost contact metric manifold fM such that 2 (T M). Then M is slant submanifold if and only if there exists a constant 2 [0; 1] such that
T2= (I ) (2.32)
Furthermore, the slant angle of M satis…es = cos2 .
Hence, for a slant submanifold M of an almost contact metric manifold fM ; the following relations are consequences of the above theorem.
g(T X; T Y ) = cos2 fg(X; Y ) (X) (Y )g (2.33) and
g(N X; N Y ) = sin2 fg(X; Y ) (X) (Y )g : (2.34) for any X; Y 2 (T M).
Lemma 2.3. [6]. Let D be a distribution on M , orthogonal to . Then, D is a slant if and only if there is a constant 2 [0; 1] such that
(T P2)2X = X (2.35)
for all X 2 (D ), where P2denotes the orthogonal projection on D . Furthermore,
the slant angle of M satis…es = cos2 .
3. Pseudo-slant submanifolds of a nearly Sasakian manifold In this section, we will obtain the integrability condition of the distributions of pseudo-slant submanifold of a nearly Sasakian manifold.
De…nition 3.1. We say that M is a pseudo-slant submanifold of an almost contact metric manifold (M ; '; ; ; g) if there exists two orthogonal distributions D and D? on M such that
i. T M admits the orthogonal direct decomposition T M = D? D ; 2 (D )
ii. The distribution D? is anti-invariant(totally-real) i.e., 'D? (T?M ),
iii. The distribution D is a slant with slant angle 6= 0;2; that is, the angle between D and '(D ) is a constant [16].
From the de…nition, it is clear that if = 0; then the pseudo-slant submanifold is a semi-invariant submanifold. On the other hand, if = 2, submanifold becomes an anti- invariant.
We suppose that M is a pseudo-slant submanifold of an almost contact metric manifold fM and we denote the dimensions of distributions D?and D by d1 and
d2; respectively, then we have the following cases:
i. If d2= 0 then M is an anti-invariant submanifold,
ii. If d1= 0 and = 0; then M is an invariant submanifold,
iii. If d1= 0 and 2 (0;2)then M is a proper slant submanifold
with slant angle ;
iv. If = 2; then M is an anti-invariant submanifold,
v. If d1:d26= 0 and = 0; then fM is a semi-invariant submanifold,
vi. If d1:d26= 0 and 2 (0;2), then M is a proper pseudo-slant submanifold.
If we denote the projections on D?and D by P1and P2; respectively, then for
any vector …eld X 2 (T M), we can write
X = P1X + P2X + (X) : (3.1)
On the other hand, applying ' on both sides of equation (3.1), we have 'X = 'P1X + 'P2X and T X + N X = N P1X + T P2X + N P2X; T P1X = 0; (3.2) from which T X = T P2X; N X = N P1X + N P2X and 'P1X = N P1X, T P1X = 0; 'P2X = T P2X + N P2X (3.3) T P2X 2 (D ):
For a pseudo-slant submanifold M of a nearly Sasakian manifold fM , the normal bundle T?M of a pseudo-slant submanifold M is decomposable as
T?M = '(D?) N (D ) '(D?) ? N(D ) (3.4)
where is an invariant subbundle of T?M .
Now, we construct an example of a pseudo-slant submanifold in an almost contact metric manifold.
Example 3.2. Let M be a submanifold of R7 de…ned by
(u; v; s; z; w) = (p3u; v; v sin ; v cos ; s cos z; s cos z; w):
We can easily to see that the tangent bundle of M is spanned by the tangent vectors e1= p 3 @ @x1 ; e2= @ @y1 + sin @ @x2 + cos @ @y2 ; e5= = @ @w; e3= cos z @ @x3 cos z @ @y3 ; e4= s sin z @ @x3 + s sin z @ @y3 :
For the almost contact structure of ' of R7; choosing '( @ @xi ) = @ @yi ; '( @ @yj ) = @ @xj ; '( @ @w) = 0; 1 i; j 3 and = @
@w; = dw. For any vector …eld W = i@x@i + j
@ @yj + @ @w 2 T (R 7); then we have 'Z = i @ @yj j @ @xi ; g('Z; 'Z) = 2i + 2j; g(Z; Z) = 2i + 2j+ 2; (Z) = g(Z; ) = and '2Z = i @ @xi j @ @yj @ @w + @ @w = Z + (Z)
for any i; j = 1; 2; 3. It follows that g('Z; 'Z) = g(Z; Z) 2(Z): Thus ('; ; ; g)
is an almost contact metric structure on R7. Thus we have
'e1= p 3 @ @y1 ; 'e2= @ @x1 + sin @ @y2 cos @ @x2 'e3= cos z @ @y3 + cos z @ @x3 ; 'e4= s sin z @ @y3 s sin z @ @x3 :
By direct calculations, we can infer D = spanfe1; e2g is a slant distribution with
slant angle cos = g(e2;'e1)
ke2kk'e1k =
p 2
2 , = 45 . Since
g('e3; e1) = g('e3; e2) = g('e3; e4) = g('e3; e5) = 0;
g('e4; e1) = g('e4; e2) = g('e4; e3) = g('e4; e5) = 0:
Thus e3 and e4 are orthogonal to M , D? = spanfe3; e4g is an anti-invariant
dis-tribution. Thus M is a 5-dimensional proper pseudo-slant submanifold of R7 with
its usual almost contact metric structure.
Theorem 3.3. Let M be a pseudo-slant of a nearly Sasakian manifold fM : Then the anti-invariant distribution D? is integrable if and only if
AN XY = AN YX (3.5)
for any X; Y 2 (D?):
Proof. By using (2.3), (2.6) and (2.8), we can write
2g(A'YX; Z) = g(h(Z; X); 'Y ) + g(h(Z; X); 'Y )
= g( erXZ; 'Y ) + g( erZX; 'Y )
= g(' erXZ; Y ) g(' erZX; Y )
= g( erX'Z; Y ) g( erZ'X; Y )
for any X; Y 2 (D?) and Z 2 (T M): By using (2.5), (2.7) and (2.16) we have 2g(AN YX; Z) = g( erX'Z; Y ) g( erZ'X; Y ) +g(2g(Z; X) (Z)X (X)Z; Y ) = g( erXY; 'Z) + g(AN XZ; Y ) (Z)g(X; Y ) = g( erXY; 'Z) + g(AN XY; Z) g(X; Y )g( ; Z) = g(rXY; T Z) + g(h(X; Y ); N Z) +g(AN XY; Z) g(X; Y )g( ; Z) = g(T rXY; Z) + g( th(X; Y ); Z) +g(AN XY; Z) g(X; Y )g( ; Z):
This implies that
2AN YX = AN XY T rXY th(X; Y ) g(X; Y ) (3.6)
interchanging X by Y in (3.6) , we have
2AN XY = AN YX T rYX th(X; Y ) g(X; Y ) (3.7)
then from (3.6) and (3.7), we can derive.
2AN YX 2AN XY = AN XY AN YX + T rYX T rXY
= AN XY AN YX + T (rYX rXY )
here
3(AN YX AN XY ) = T [X; Y ] :
For [X; Y ] 2 (D?), ' [X; Y ] = N [X; Y ]. Since the tangent component of
' [X; Y ] is the zero, the anti- invariant distribution D? is integrability if and only if (3.5) is satis…ed.
Theorem 3.4. Let M be a pseudo-slant of a nearly Sasakian manifold fM : Then the anti-invariant distribution D? is integrable if and only if
AN XY + T rXY + th(Y; X) + g(X; Y ) = 0 (3.8)
for any X; Y 2 (D?):
Proof. For any X; Y 2 (D?), from (2.5), we have ( erX')Y + ( erY')X = 2g(X; Y )
which is equivalent to e
rX'Y ' erXY + erY'X ' erYX 2g(X; Y ) = 0:
By using (2.6), (2.7), (2.11) and (2.12), we can write
AN YX + r?XN Y T rXY N rXY 2th(Y; X) AN XY
Then from the tangent components of the last equation and (3.5), we conclude that 2AN XY + T rXY + T rYX + 2th(Y; X) + 2g(X; Y ) = 0;
which implies
T [X; Y ] = 2AN XY + 2T rXY + 2th(Y; X) + 2g(X; Y ) :
This proves our assertion.
Theorem 3.5. Let M be a pseudo-slant submanifold of a nearly Sasakian manifold f
M . Then the slant distribution D is integrable if and only if
2(rYN )X + r?YN X r?XN Y + h(Y; T X) h(X; T Y ) + 2nh(X; T Y ) 2 N (D )
for any Y; X 2 (D ):
Proof. For any Y; X 2 (D ) and Z 2 (D?); we have
g([Y; X] ; Z) = g( erYX; Z) g( erXY; Z)
= g(' erYX; 'Z) g(' erXY; 'Z):
Since of (Z) = 0 and 'Z = N Z for any Z 2 (D?); we obtain
g([Y; X] ; Z) = g( erY'X; N Z) g( erX'Y; N Z)
+g(( erX')Y ( erY')X; N Z)
= g( erY'X; N Z) g( erX'Y; N Z)
+g(( erX')Y + ( erY')X; N Z) 2g(( erY')X; N Z):
By using (2.5) in this equation, we have
g([Y; X] ; Z) = g( erY'X; N Z) g( erX'Y; N Z) 2g(( erY')X; N Z)
+g(2g(Y; X) (Y )X (X)Y; N Z): Also using (2.11) in this equation, we have
g([Y; X] ; Z) = g( erYT X; N Z) + g( erYN X; N Z) g( erXT Y; N Z)
g( erXN Y; N Z) 2g(( erY')X; N Z):
From the Gauss and Weingarten formulas, the above equation takes the form g([Y; X] ; Z) = 2g(( erY')X; N Z) + g(r?YN X; N Z) g(r?XN Y; N Z)
+g(h(Y; T X); N Z) g(h(X; T Y ); N Z): (3.9) Substituting 2g(( erY )X; N Z) into the (3.9), we get
2g(( erY')X; N Z) = 2g((rYT )X AN XY th(Y; X); N Z)
+2g((rYN )X + h(Y; T X) nh(X; T Y ); N Z)
Substituting (3.10) in the equation (3.9), we have
g([Y; X] ; Z) = g( 2(rYN )X 2h(Y; T X) + 2nh(X; T Y ); N Z)
+g(r?YN X r?XN Y + h(Y; T X) h(X; T Y ); N Z);
= g( 2(rYN )X + r?YN X r?WN Y
+h(Y; T X) h(X; T Y ) + 2nh(X; T Y ); N Z): Thus we conclude [Y; X] 2 (D ) if and only if
2(rYN )X +r?YN X r?XN Y +h(Y; T X) h(X; T Y )+2nh(X; T Y ) 2 N (D ):
Theorem 3.6. Let M be a pseudo-slant submanifold of a nearly Sasakian manifold f
M : Then the slant distribution D is integrable if and only if P1f(rYT )X + rXT Y T rYX AN XY AN YX
2th(X; Y ) + (Y )X + (X)Y g = 0 (3.11)
for any X; Y 2 (D ):
Proof. For any X; Y 2 (D ) and we denote the projections on D? and D by P1
and P2; respectively, then for any vector …elds X; Y 2 (D ); by using equation
(2.5), we obtain
( erX')Y + ( erY )X = 2g(X; Y ) (Y )X (X)Y;
that is, e
rX'Y ' erXY + erY'X ' erYX = 2g(X; Y ) (Y )X (X)Y:
By using equations (2.6), (2.7), (2.11) and (2.12), we can write
rXT Y + h(X; T Y ) AN YX + r?XN Y T rXY N rXY
th(X; Y ) nh(X; Y ) + rYT X + h(Y; T X) AN XY
+r?YN X T rYX N rYX th(X; Y ) nh(X; Y )
2g(X; Y ) + (Y )X + (X)Y = 0
From the tangential components of the last equation, we conclude that rXT Y T rXY + (rYT )X AN XY AN YX
2th(X; Y ) 2g(X; Y ) + (Y )X + (X)Y = 0 which implies that
T [X; Y ] = rXT Y T rYX + (rYT )X AN XY AN YX
2th(X; Y ) 2g(X; Y ) + (Y )X + (X)Y: (3.12) Applying P1 to (3.12), we get (3.11)
Theorem 3.7. Let M be a proper pseudo-slant submanifold of a nearly Sasakian manifold fM . Then D is integrable if and only if
2g(rXY; Z) =
n
g(AN ZX; T Y ) + g(AN ZY; T X) + g(r?XN Y; N Z) + g(r?YN X; N Z)
o for any X; Y 2 (D ) and Z 2 (D?).
Proof. For any X; Y 2 (D ) and Z 2 (D?), by using (2.3), we have g(rXY; Z) = g( erXY; Z) = g(' erXY; 'Z) + ( erXY ) (Z):
Since (Z) = 0; we get
g(rXY; Z) = g(' erXY; N Z);
from which
g(rXY; Z) = g( erX'Y; N Z) g(( erX')Y; N Z):
From the Gauss and Weingarten formulas and structure equation (2.5), we get g(rXY; Z) = g( erXT Y; N Z) + g( erXN Y; N Z) g(( erX')Y; N Z) = g(h(X; T Y ); N Z) + g(r?XN Y; N Z) + g(( erY')X; N Z) 2g(X; Y )g( ; N Z) + (X)g(Y; N Z) + (Y )g(X; N Z) = g(h(X; T Y ); N Z) + g(r?XN Y; N Z) + g(( erY')X; N Z)(3.13) Interchanging X by Y in (3.13), we have g(rYX; Z) = g(h(Y; T X); N Z) + g(r?YN X; N Z) + g(( erX')Y; N Z): (3.14)
From (3.13) and (3.14), we can derive
g(rXY; Z) + g(rYX; Z) = g(h(X; T Y ); N Z) + g(r?XN Y; N Z) + g(( erY')X; N Z) +g(h(Y; T X); N Z) + g(r?YN X; N Z) + g(( erX')Y; N Z) = g(h(X; T Y ); N Z) + g(h(Y; T X); N Z) +g(r?YN X; N Z) + g(r?XN Y; N Z) +g(( erY')X + ( erX')Y; N Z) By using (2.5), we obtain g(rXY; Z) + g(rYX; Z) = g(h(X; T Y ); N Z) + +g(h(Y; T X); N Z) +g(r?YN X; N Z) + g(r?XN Y; N Z)
Using the property of Lie bracket, we derive
2g(rXY; Z) + g([Y; X] ; Z) = g(h(X; T Y ); N Z) + g(h(Y; T X); N Z)
which implies that 2g(rXY; Z) = n g(AN ZX; T Y ) + g(AN ZY; T X) + g(r?XN Y; N Z) + g(r?YN X; N Z) o : This proves our assertion.
Theorem 3.8. Let M be a totally umbilical proper pseudo-slant submanifold of a nearly Sasakian manifold fM . Then the endomorphism T is parallel on M if and only if M is anti-invariant submanifold of fM .
Proof. If T is parallel, then from (2.10) and (2.28), we have
AN XX + th(X; X) + g(X; X) (X)X = 0 (3.15)
Interchanging X by T X in (3.15), we drive
AN T XT X + th(T X; T X) + g(T X; T X) = 0: (3.16)
Taking the inner product of (3.16) by , we get
0 = g(AN T XT X + th(T X; T X) + g(T X; T X) ; )
= g(h(T X; ); N T X) + g(T X; T X) = g(g(T X; )H; N T X) + g(T X; T X)
= g(T X; T X) = cos2 fg(X; X) (X) (X)g
for any vector …eld X on M . This implies that M is an anti-invariant submanifold. If M is an anti-invariant submanifold, then it is obvious that rT = 0.
Theorem 3.9. Let M be totally umbilical proper pseudo-slant submanifold of a nearly Sasakian manifold fM : Then M is a totally geodesic submanifold if H, r?XH 2 ( ).
Proof. For any X; Y 2 (T M), we have e
rX'Y = ( erX')Y + ' erXY: (3.17)
Making use of (2.6), (2.7), (2.10), (2.11) and (3.17) equation takes the form rXT Y + g(X; T Y )H AN YX + r?XN Y = ( erX')Y + T rXY
+N rXY + g(X; Y )'H:
Taking the inner product with 'H the last equation, we obtain
g(r?XN Y; 'H) = g(( erX')Y; 'H) + g(N rXY; 'H) + g(X; Y ) kHk2
by using (2.7), we get
g( erXN Y; 'H) = g(( erX')Y; 'H) + g(X; Y ) kHk2: (3.18)
In same way, we have
Then from (3.18) and (3.19), we can derive g( erXN Y + erYN X; 'H) = g(( erX')Y + ( erY')X; 'H) + 2g(X; Y ) kHk2: From (2.5), we obtain g( erXN Y + erYN X; 'H) = g(2g(X; Y ) (Y )X (X)Y; 'H) + 2g(X; Y ) kHk2: Hence g( erXN Y + erYN X; 'H) = 2g(X; Y ) kHk2: (3.20)
Now, for any X 2 (T M); we have e
rX'H = ( erX')H + ' erXH
by means of (2.6), (2.7), (2.11), (2.12), ( 2.24) and (2.25), we obtain
A'HX + r?X'H = PXH + FXH T AHX N AHX + nr?XH: (3.21)
Taking the inner product with N Y and taking into account that nr?XH 2 ( ),
we see that
g(r?X'H; N Y ) = g(FXH; N Y ) g(N AHX; N Y ): (3.22)
From (2.8), (2.10), (2.34) and (3.22), we obtain g(r?X'H; N Y ) = sin2
n
g(X; Y ) kHk2 (AHX) (Y )
o +g(FXH; N Y ):
Since r is metric connection, NY and 'H are mutually orthogonal, by using (2.2), (2.7), (2.8) and (2.10), we get g( erXN Y; 'H) = sin2 fg(X; Y ) (X) (Y )g kHk2 g(FXH; N Y ): (3.23) Similarly, we have g( erYN X; 'H) = sin2 fg(X; Y ) (X) (Y )g kHk2 g(FYH; N X): (3.24)
From (3.23) and (3.24), we obtain
g( erXN Y + erYN X; 'H) = 2 sin2 fg(X; Y ) (X) (Y )g kHk2
g(FXH; N Y ) g(FYH; N X): (3.25)
Thus (3.20) and (3.25) imply
2g(X; Y ) kHk2 = 2 sin2 fg(X; Y ) (X) (Y )g kHk2 g(FXH; N Y ) g(FYH; N X):
Thus we have
cos2 g(X; Y ) kHk2+ sin2 (X) (Y ) kHk2= 1
In view of (2.20) and (2.25) the fact that H 2 ( ); then the above equation takes the form
cos2 g(X; Y ) kHk2+ sin2 (X) (Y ) kHk2 = sin2 g(X; Y ) kHk2
+ sin2 (X) (Y ) kHk2: (3.26) From (3.26), we conclude that g(X; Y ) kHk2 = 0, 8X; Y 2 (T M): Since M is a proper-slant, we obtain H = 0: This tells us that M is totally geodesic in fM :
Theorem 3.10. Let M be a totally umbilical pseudo-slant submanifold of a nearly Sasakian manifold fM . Then at least one of the following statements is true; i. dim(D?) = 1;
ii. H 2 ( );
iii. M is a proper pseudo-slant submanifold. Proof. For any X 2 (D?) from (2.5), we have
( erX')X = g(X; X) ;
or,
e
rXN X '(rXX + h(X; X)) kXk2 = 0:
From the last equation, we have
AN XX + r?XN X N rXX th(X; X) nh(X; X) kXk2 = 0:
The tangential components of (3.27), we obtain AN XX + th(X; X) + kXk2 = 0
Taking the inner product by Y 2 (D?), we have g(AN XX + th(X; X); Y ) = 0:
This implies that
g(h(X; Y ); N X) + g(th(X; X); Y ) = 0: Since M is totally umbilical submanifold, we obtain
g(g(X; Y )H; N X) + g(g(X; X)tH; Y ) = 0 or,
g(X; Y )g(H; N X) + g(X; X)g(tH; Y ) = 0; which implies that
g(tH; Y )X g(tH; X)Y = 0:
Here, tH is either zero or X and Y are linearly dependent. If tH 6= 0; then the vectors X and Y are linearly dependent and dim(D?) = 1.
On the other hand, tH = 0, i.e., H 2 ( ): Since dim (D ) 6= 0, M is a pseudo-slant submanifold. Since 6= 0 and d1:d2 6= 0; M is a proper pseudo-slant
submanifold.
Theorem 3.11. Let M be totally umbilical proper pseudo-slant submanifold of a nearly Sasakian manifold fM : Then following conditions are equivalent
i. H 2 ( ); ii. '2X = r
T X ;
iii. M is an anti-invariant submanifold, for any X 2 (T M):
Proof. For any X 2 (T M); from (2.5), we have
( erX')X = g(X; X) (X)X:
By means of (2.6), (2.7), (2.11) and (2.12), we obtain
0 = reXT X + erXN X '(rXX + h(X; X)) g(X; X) + (X)X
= rXT X + h(X; T X) AN XX + r?XN X T rXX N rXX
th(X; X) nh(X; X) g(X; X) + (X)X: (3.27)
The tangential components of (3.27), we obtain
rXT X T rXX th(X; X) AN XX g(X; X) + (X)X = 0:
Since M is a totally umbilical submanifold, we can derive AN XX = g(H; N X)X;
then we have
rXT X T rXX g(X; X)tH g(H; N X)X
g(X; X) + (X)X = 0: (3.28)
If H 2 ( ); then from (3.28), we conclude that
rXT X T rXX g(X; X) + (X)X = 0: (3.29)
Taking the inner product of (3.29) by , we get
g(rXT X; ) = g(X; X) 2(X): (3.30)
Interchanging X by T X in (3.30) and making use of (2.33), we derive g(rT XT2X; ) = g(T X; T X); or, g(rT X ; T2X) = cos2 g( X; 'X); g(rT X ; cos2 (X (X) )) = cos2 g('X; 'X); that is, cos2 fg('X; 'X) g(rT X ; (X (X) )g = 0:
Since M is a proper pseudo-slant submanifold, we have g('X; 'X) g(rT X ; (X (X) )) = 0;
that is,
g('X; 'X) g(rT X ; X) + (X)g(rT X ; ) = 0: (3.31)
Taking the covariant derivative of above equation with respect to T X for any X 2 (T M ); we obtain g(rT X ; ) + g( ; rT X ) = 0; which implies g(rT X ; ) = 0
and then (3.31) becomes
g(X; X) 2(X) g(rT X ; X) = 0: (3.32)
This proves ii: of the Theorem. So if (3.32) is satis…ed, then (3.28) implies H 2 ( ): Now, interchanging X by T X in (3.32), we derive
g(T X; T X) g(rT2X ; T X) = 0;
that is,
cos2 g('X; 'X) + cos2 g(r(X (X) ) ; T X) = 0;
from which
cos2 g('X; 'X) + cos2 g(rX ; T X) cos2 (X)g(r ; T X) = 0:
Since r = 0; we obtain
cos2 fg('X; 'X) + g(rX ; T X)g = 0: (3.33)
We note
g('X; 'X) + g(rX ; T X) 6= 0
from (3.33, we can derive if cos = 0, then M is an anti-invariant submanifold. Acknowledgment
The authors sincerely thank the referee(s) for the corrections and comments in the revision of this paper. This work is supported by Scienti…c Research Project in Gaziosmanpasa University (BAP 2015-43).
References
[1] Ali, A., Othman W. A. M. and Ozel, C., Some inequalities for warped product pseudo-slant submanifolds of nearly Kenmotsu manifolds, Journal of Inequalities and Its Applications 1(2015), 1-7, DOI-10.1186/s13660-015-0802-5.
[2] Lotta, A., Slant submanifolds in contact geometry, Bull. Math. Soc. Sci. Math. Roum., Nouv. Ser. 39(1996), 183-198.
[3] Chen, B.Y., Geometry of Slant Submanifold, Kath. Üniv. Leuven, Dept. of Mathematics, Leuven, 1990.
[4] Chen, B. Y., Slant immersions, Bull. Austral. Math. Soc. 41(1990), 135-147.
[5] Cabrerizo, J. L., Carriazo, A., Fernandez L.M., and Fernandez, M., Structure on a slant submanifold of a contact manifold, Indian J. Pure Appl. Math. 31(2000), 857-864.
[6] Cabrerizo, J. L., Carriazo, A., Fernandez L. M, and Fernandez, M., Slant Submanifolds in Sasakian manifolds, Glasgow Math. J. 42(2000), 125-138.
[7] Khan, M. A., Uddin S. and Singh, K., A classi…cation on totally umbilical proper slant and hemi-slant submanifolds of a nearly trans-Sasakian manifold, Di¤ erential Geometry-Dynamical Systems, Vol.13(2011), 117-127.
[8] Atçeken, M., On Geometry of Submanifolds of (LCS)n-Manifolds. International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 304647, 11 pages doi:10.1155/2012/304647.
[9] Atçeken M. and Hui, S. K., Slant and Pseudo-Slant Submanifolds in (LCS)n-manifolds, Czechoslovak M. J, 63(138)(2013), 177-190.
[10] Atçeken, M. and Dirik, S., On the geometry of pseudo-slant submanifolds of a Kenmotsu manifold, Gulf Joural of Mathematics 2(2)(2014), 51-66.
[11] Atçeken, M. and Dirik, S., Pseudo-slant submanifolds of nearly Kenmotsu manifold. Serdica Mathematical Journal 41, 243-262,(2015).
[12] Papaghuic, N., Semi-slant submanifolds of a Kaehlarian manifold, An. St. Univ. Al. I. Cuza. Univ. Iasi. 40(2009), 55-61.rr
[13] Dirik S. and Atçeken, M., Pseudo-slant submanifolds of nearly Cosymplectic manifold, Turk-ish Journal of Mathematics and Computer Science, vol. 2014. Article ID 20140035, 14 pages, (2013).
[14] Uddin, S., Ozel, C., Khan M. A. and Singh, K., Some classi…cation result on totally umbilical proper slant and hemi slant submanifolds of a nearly Kenmotsu manifold, International Journal of Physical Scienses, Vol. 7(40)(2012), 5538-5544.
[15] Uddin, S., Bernardine, R., Wong and Mustafa, A. A., Warped Product Pseudo-Slant Sub-manifolds of a Nearly Cosymplectic Manifold, Hindawi Publishing Corporation Abstract and Applied Analysis, Volume, Article ID 420890, 13 pp, doi:10.1155/2012/420890 (2012). [16] Khan V. A. and Khan, M. A., Pseudo-slant submanifolds of a Sasakian manifold, Indian J.
Pure Appl. Math. 38(1)(2007), 31-42.
Current address : Süleyman Dirik: Amasya University Faculty of Arts and Sciences, Depart-ment of Mathematics, 05200 Amasya /TURKEY.
E-mail address : suleyman.dirik@amasya.edu.tr
ORCID Address: http://orcid.org/0000-0001-9093-1607
Current address : Mehmet Atçeken: Gaziosmanpasa University, Faculty of Arts and Sciences, Department of Mathematics, 60100 Tokat /TURKEY.
E-mail address : mehmet.atceken382@gmail.com
ORCID Address: http://orcid.org/0000-0001-8665-5945
Current address : Ümit Y¬ld¬r¬m: Gaziosmanpasa University Faculty of Arts and Sciences, Department of Mathematics, 60100 Tokat /TURKEY.
E-mail address : umit.yildirim@gop.edu.tr
