C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 67, N umb er 2, Pages 116–125 (2018) D O I: 10.1501/C om mua1_ 0000000866 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
SEMI-SLANT SUBMANIFOLDS OF (k; )- CONTACT MANIFOLD
M.S.SIDDESHA AND C.S BAGEWADI
Abstract. In the present paper, we study semi-slant submanifolds of (k; )-contact manifold and give conditions for the integrability of invariant and slant distributions which are involved in the de…nition of semi-slant submanifold. Further, we show the totally geodesicity of such distributions.
1. Introduction
The geometry of slant submanifolds was initiated by Chen [6] as a natural gen-eralization of both holomorphic and totally real submanifolds. Since then many geometers have studied such slant immersions in almost Hermitian manifolds. The contact version of slant immersions was introduced by Lotta [11]. Latter, Cabrerizo et al., [3] studied and characterized slant submanifolds of K-contact and Sasakian manifolds and have given several examples of such immersions.
In 1994, Papaghiuc [12] has introduced the notion of semi-slant submanifolds of almost Hermitian manifolds. Cabrerizo et al., [4] extended the study of semi-slant submanifolds to the setting of almost contact metric manifolds. They worked out the integrability conditions of the distributions involved on these submanifolds and studied the geometrical signi…cance of these distributions. Motivated by these stud-ies of the above authors [4, 9, 12], in the present paper we extend the study of the semi-slant submanifolds of (k; )-contact manifold, which consist of both Sasakian as well as non-Sasakian cases and are introduced in 1995 by Blair, Koufogiorgos and Papantoniou [2]. Hence it is worth studying and is a generalization of [4].
The paper is organized as follows: In section-2, we recall the notion of (k; )-contact manifold and some basic results of submanifolds, which are used for further study. Section-3 is devoted to study semi-slant submanifolds of (k; )-contact man-ifold. Lastly, in section-4 we consider totally umbilical and totally contact umbilical semi-slant submanifolds of (k; )-contact manifold and …nd the necessary conditions to be totally geodesic.
Received by the editors: May 05, 2016; Accepted: May 16, 2017. 2010 Mathematics Subject Classi…cation. 53C42, 53C25, 53C40.
Key words and phrases. Semi-slant submanifold; (k; )-contact manifold; totally umbilical submanifold and totally geodesic submanifold.
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2. Preliminaries
A contact manifold is a C1 (2n + 1) manifold ~M2n+1 equipped with a global
1-form such that ^ (d )n 6= 0 everywhere on ~M2n+1. Given a contact form
it is well known that there exists a unique vector …eld , called the characteristic vector …eld of , such that ( ) = 1 and d (X; ) = 0 for every vector …eld X on
~
M2n+1. A Riemannian metric g is said to be associated metric if there exists a
tensor …eld of type (1,1) such that
2
= I + ; ( ) = 1; = 0; = 0; (2.1)
g( X; Y ) = g(X; Y ) (X) (Y ); g(X; ) = (X); (2.2)
g(X; Y ) = g( X; Y ); (2.3)
for all vector …elds X; Y 2 T ~M . Then the structure ( ; ; ; g) on ~M2n+1 is called a contact metric structure and the manifold ~M2n+1equipped with such a structure is called a contact metric manifold [1].
Now we de…ne a (1; 1) tensor …eld h by h = 1
2L , where L denotes the Lie
dif-ferentiation, then h is symmetric and satis…es h = h. Further, a q-dimensional distribution on a manifold M is de…ned as a mapping D on M which assigns to each point p 2 M, a q-dimensional subspace Dp of TpM .
The (k; )-nullity distribution of a contact metric manifold ~M ( ; ; ; g) is a distri-bution
N (k; ) : p ! Np(k; ) = fZ 2 TpM : ~R(X; Y )Z
= k[g(Y; Z)X g(X; Z)Y ] + [g(Y; Z)hX g(X; Z)hY ]g; for all X; Y 2 T ~M . Hence if the characteristic vector …eld belongs to the (k; ) nullity distribution, then we have
~
R(X; Y ) = k[ (Y )X (X)Y ] + [ (Y )hX (X)hY ]: (2.4) The contact metric manifold satisfying the relation (2.4) is called (k; ) contact metric manifold [2]. It consists of both k-nullity distribution for = 0 and Sasakian for k = 1. In (k; )-contact manifold the following relation holds:
( ~rX )(Y ) = g(X + hX; Y ) (Y )(X + hX); (2.5)
for all X; Y 2 T ~M , where ~r denotes the Levi-Civita connection on ~M . We also have on (k; )-contact manifold ~M
~
rX = X hX: (2.6)
Let M be a submanifold of a (k; )-contact manifold ~M ; we denote by the same symbol g the induced metric on M . Let T M be the set of all vector …elds tangent to M and T?M is the set of all vector …elds normal to M . Then, the Gauss and
Weingarten formulae are given by ~
for any X; Y 2 T M, V 2 T?M , where r (resp. r?) is the induced connection on
the tangent bundle T M (resp. normal bundle T?M ) [7]. The shape operator A is
related to the second fundamental form of M by
g(AVX; Y ) = g( (X; Y ); V ): (2.8)
Now, for any x 2 M; X 2 TxM and V 2 Tx?M , we put
X = T X + F X; V = tV + f V; (2.9)
where T X (resp. F X) is the tangential (resp. normal) component of X, and tV (resp. f V ) is the tangential (resp. normal) component of V . The relation (2.9) gives rise to an endomorphism T : TxM ! TxM whose square (T2) will be denoted
by Q. The tensor …elds on M of type (1; 1) determined by these endomorphisms will be denoted by the same letters T and Q respectively. From (2.3) and (2.9)
g(T X; Y ) + g(X; T Y ) = 0; (2.10)
for each X; Y 2 T M. The covariant derivatives of the tensor …elds T; Q and F are de…ned as
(rXT )Y = rXT Y T (rXY ); (2.11)
(rXQ)Y = rXQY Q(rXY ); (2.12)
(rXF )Y = rXF Y F (rXY ): (2.13)
Using (2.5), (2.6), (2.7), (2.9), (2.11), and (2.12), we obtain
(rXT )Y = AF YX + t (X; Y ) + g(X + hX; Y ) (Y )(X + hX); (2.14)
(rXF )Y = (X; T Y ) + f (X; Y ): (2.15)
3. Semi-slant submanifolds of a (k; )-contact manifold
As a generalization of slant and CR-submanifolds, Papaghiuc [12] introduced the notion of semi-slant submanifolds of an almost Hermitian manifolds. Cabrerizo et al., [4] gave the contact version of semi-slant submanifold and they obtained several interesting results. The purpose of the present section is to study semi-slant submanifolds of a (k; )-contact manifold.
A submanifold M of an almost contact metric manifold ~M is said to be a slant submanifold if for any x 2 M and any X 2 TxM , the Wirtinger’s angle, the angle
between X and TxM , is constant 2 [0; 2 ]. Here the constant angle is called
the slant angle of M in ~M . The invariant submanifolds are slant submanifolds with slant angle 0 and anti-invariant submanifolds are slant submanifolds with slant angle 2. A slant submanifold is called proper, if it is neither invariant nor anti-invariant. Recently, we have de…ned and studied slant submanifolds of a (k; )-contact manifold in [13].
A submanifold M of an almost contact metric manifold ~M is said to be a semi-slant submanifold of ~M [4] if there exist two orthogonal distributions D1 and D2
(i) T M admits the orthogonal direct decomposition T M = D1 D2 < >.
(ii) The distribution D1 is an invariant distribution, i.e., (D1) = D1.
(iii) The distribution D2is slant with slant angle 6= 0.
In particular, if = 2, then a semi-slant submanifold reduces to a semi-invariant submanifold. On a semi-slant submanifold M , for any X 2 T M, we write
X = P1X + P2X + (X) ; (3.1)
where P1X 2 D1and P2X 2 D2. Now by equations (2.9) and (3.1)
X = P1X + T P2X + F P2X: (3.2)
Then, it is easy to see that
P1X = T P1X; F P1X = 0; T P2X 2 D2: (3.3)
Thus
T X = P1X + T P2X and F X = F P2X: (3.4)
Let denote the orthogonal complement of D2 in T?M i.e., T?M = D2 .
Then it is easy to observe that is an invariant subbundle of T?M .
Now, we are in a position to workout the integrability conditions of the distrib-utions D1 and D2 on a semi-slant submanifold of a (k; )-contact manifold.
Lemma 3.1. Let M be a semi-slant submanifold of a (k; )-contact manifold ~M , then
g([X; Y ]; ) = 2g( X; Y ) + g(Y; hX) g(X; hY ); (3.5) for any X; Y 2 D1 D2.
The assertion can be proved by using the fact that rX = X hX for
X 2 D1 and (2.3). Since for any X 2 D1
g([X; X]; ) 6= 0, we have
Corollary 3.1. Let M be a semi-slant submanifold of a (k; )-contact manifold ~M such that dim(D1) 6= 0. Then, the invariant distribution D1 is not integrable.
Now for the slant distribution, we have
Theorem 3.1. Let M be a semi-slant submanifold of a (k; )-contact manifold ~M . Then the slant distribution D2 is integrable if and only if slant angle of D2 is 2
i.e., M is semi-invariant submanifold. Proof. For any Z; W 2 D2, by (3.5) we have
g([Z; W ]; ) = 2g(T Z; W ) + g(W; T hZ) g(Z; T hW ):
If D2 is integrable, then T j D2 0 and so = 2. Hence M is a semi-invariant
submanifold.
Conversely, if sla(D2) = 2, then Z = F Z for each Z 2 D2and by equations (2.5)
and (2.7)
for each Z; W 2 D2. Interchanging Z and W in the above equation and subtracting
the obtained relation from the same, we obtain
[Z; W ] = AF ZW AF WZ + r?ZF W r?WF Z g(hZ; W ) + g(hW; Z) : (3.6)
Further, by using equations (2.3), (2.7) and (2.8) in (2.5), it is easy to obtain that
AF ZW = AF WZ; (3.7)
for each Z; W 2 D2. In view of (3.5), (2.1) and (3.7), equation (3.6) yields
[Z; W ] = (r?ZF W r?WF Z): (3.8)
The right hand side of the above lies in D2 because on using equations (2.5), (2.7)
and (2.10), we observe that
g(V; r?WF Z) = g(A VW; Z)
for all V 2 and Z; W 2 D2. This shows that
g(r?ZF W r?WF Z; V ) = 0:
i.e., r?ZF W r?WF Z lies in F D2 for each Z; W 2 D2, and thus from equation
(3.8), [Z; W ] 2 D2.
Now, for Y 2 D1 D2, by equation (2.5), we have
~
r Y = ~r Y: In particular, for Y 2 D1, the above equation yields
r Y = r Y: This implies r Y 2 D1 for any Y 2 D1.
The above observation together with the fact that (X; ) = 0 for X 2 D1yields
Lemma 3.2. On a semi-slant submanifold M of a (k; )-contact manifold ~M , [X; ] 2 D1 and [Z; ] 2 D2
for any X 2 D1 and Z 2 D2.
Lemma 3.3. Let M be a semi-slant submanifold of a (k; )-contact manifold ~M . Then, for any X; Y 2 T M, we have
P1(rX P1Y ) + P1(rXT P2Y ) = P1(rXY ) + P1AF P2YX (Y )P1X (3.9)
Proof. By using equations (2.1), (2.7), (3.1), (3.2) and (3.3) we obtain
rX P1Y + ( P1Y; X) + rXT P2Y + (T P2Y; X) AF P2YX + r?XF P2Y
= P1rXY + T P2rXY + F P2rXY + t (X; Y ) + f (X; Y )
+g(X + hX; Y ) (Y )P1(X + hX) (Y )P2(X + hX) (Y ) (X) :
Proposition 3.2. Let M be a semi-slant submanifold of (k; )-contact manifold ~
M . Then
(i) D1 < > is integrable if and only if
(X; Y ) = (Y; X); (3.10)
(ii) D2 < > is integrable if and only if
P1(rZT W AN WZ rWT Z + AN ZW ) = 0; (3.11)
for any X; Y 2 D1 and Z; W 2 D2.
Proof. Now, for any X; Y 2 D1 < > and V 2 T?M
g( ~rX Y r~Y X; V ) = g( (X; Y ) ( X; Y ); V );
after simpli…cation, we get
g(( ~rX )Y ( ~rY) X + [X; Y ]; V ) = g( (X; Y ) ( X; Y ); V ):
Now using (2.5) and (3.2), we obtain
g(F P2[X; Y ]; V ) = g( (X; Y ) ( X; Y ); V ):
Removing inner product, we get
F P2[X; Y ] = (X; Y ) ( X; Y ): (3.12)
Hence, if D1 < > is integrable then (3.10) holds directly from (3.12).
Conversely, by using (3.10), it is easy to prove that
(X; Y ) (Y; X) = (P1X; P1Y ) (P1Y; P1X) = 0;
for any X; Y 2 D1 < >. Thus, by applying (3.12) it follows that F P2[X; Y ] = 0.
So, we can easily deduce that P2[X; Y ] must vanish. Since D2is a slant distribution
with nonzero slant angle. Hence [X; Y ] 2 D1 < > and statement (i) holds.
With regards to statement (ii), by virtue of (3.9) we have
P1[Z; W ] = P1(rZT W rWT Z AF WZ + AF ZW ):
for any Z; W 2 D2 < >. Hence (3.11) holds if and only if
P1[Z; W ] = 0; (3.13)
for any Z; W 2 D2 < >. But it can be showed that (3.13) is equivalent to
D2 < > being an integrable distribution.
The Nijenhuis tensor …eld S of the tensor T is given by
S(X; Y ) = [T X; T Y ] + T2[X; Y ] T [T X; Y ] T [X; T Y ];
for X; Y 2 T M. In particular, for X 2 D1 and Z 2 D2, the above equation on
simpli…cation takes the form
Using (2.14) the above equation becomes
S(X; Z) = AF ZT X + t (T X; Z) t (T Z; X) T (AF ZX): (3.14)
Theorem 3.3. If the invariant distribution D1 on a semi-slant submanifold M of
a (k; )-contact manifold ~M is integrable and its leaves are totally geodesic in M, then
(i) (D1; D1) 2 ;
(ii) S(D1; D2) 2 D2.
Proof. By hypothesis, for any X; Y in D1 and Z in D2
g(rXY; Z) = 0;
and therefore by Gauss formula, we have
g( ~rXY; Z) = 0:
The above equation on making use of equations (2.5), (2.7) and (2.9) yields g( (X; Y ); F Z) = 0:
This proves statement (i). To prove statement (ii), use (3.14) to get g(S(X; Z); Y ) = g(AF ZT X + t (T X; Z) t (T Z; X) T AF ZX; Y ):
The right hand side of the above equation is zero in view of statement (i) and thus (ii) is established.
Next for the slant distribution, we have:
Theorem 3.4. If the slant distribution D2 on a semi-slant submanifold M of a
(k; )-contact manifold ~M is integrable and its leaves are totally geodesic in M, then (i) (D1; D2) 2 ;
(ii) S(D1; D2) 2 D1.
Proof. By hypothesis,
g(rZW; X) = 0;
for any Z; W 2 D2 and X 2 D1. By applying (2.5), (2.7) and (2.9)
g( (X; Z); F W ) = 0: That proves (i). Now by using equation (3.14)
g(S(X; Z); W ) = g(AF ZT X + t (T X; Z) t (T Z; X) T AF ZX; W );
for X 2 D1 and Z; W 2 D2. The right hand side of the above equation is zero by
part (i). This proves (ii) and the theorem. Example: For any 2 [0;2]
de…nes a …ve dimensional semi-slant submanifold M , with slant angle , in R9with
its usual (k; )-contact structure ( 0; ; ; g) [13]. Further, e1 = 2( @ @x1 + x5 @ @t); e2= 2 @ @x5 ; e3= 2( @ @x3 + x7 @ @t); e4 = cos (2 @ @x7 + sin (2 @ @x8 ); e5= @ @t = ; (3.15)
form a local orthonormal frame of T M . If we de…ne the distribution D1=< e1; e2>
and D2=< e3; e4>, then it is easy to check that the distribution D1 is invariant
under and D2is slant with slant angle . That is M is semi-slant submanifold.
4. Totally umbilical submanifolds of (k; )-contact manifold De…nition 1. A submanifold M is said to be totally umbilical submanifold if its second fundamental form satis…es
(X; Y ) = g(X; Y )H; for all X; Y 2 T M, where H is the mean curvature vector.
To investigate totally umbilical submanifolds of a (k; )-contact manifold, we …rst establish the following preliminary result.
Proposition 4.5. Let M be a semi-slant submanifold of a (k; )-contact manifold ~
M with (X; T X) = 0 for each X 2 D1 < >. If D1 < > is integrable then
each of its leaves are totally geodesic in M as well as in ~M . Proof. For X 2 D1 < >, by equation (2.15)
(rXF )X = (X; T X) + f (X; X);
by using (2.13) and the fact that F X = 0 for each X 2 D1, we get
F rXX = f (X; X): (4.1)
Now, making use of Proposition 3.2 and the assumption that (X; T X) = 0, we obtain (X; T Y ) = 0 i.e., (X; Y ) = 0 for each X; Y 2 D1 < >. This proves
that the leaves of D1 < > are totally geodesic in ~M . Thus by (4.1), we obtain
that rXY 2 D1 < > i.e., the leaves of D1 < > are totally geodesic in
M .
As an immediate consequence of the above, we have
Corollary 4.2. Let M be a totally umbilical semi-slant submanifold of a (k; )-contact manifold ~M . If D1 < > is integrable, then each of its leaves are totally
geodesic in M as well as in ~M .
De…nition 2. [10] A submanifold M of an almost contact metric manifold is said to be totally contact umbilical submanifold if
for all X; Y 2 T M, where K is a normal vector …eld on M. If K = 0 then M is said to be a totally contact geodesic submanifold. For a submanifold of a (k; )-contact manifold, the condition for totally contact umbilicalness reduces to
(X; Y ) = g( X; Y )K:
Theorem 4.6. Let M be a totally contact umbilical semi-slant submanifold of a (k; )-contact manifold ~M , with dim(D1) 6= 0. Then the mean curvature vector is
a global section of F D2.
Proof. Let X 2 D1be a unit vector …eld and V 2 , then
g(H; V ) = g( (X; X); V ) = g( ~rX X; V ) = g( (X; X; V )) = 0
=) H 2 F D2.
In view of Theorem 4.6, we have the following:
Theorem 4.7. A totally contact umbilical semi-slant submanifold of a (k; )-contact manifold is totally )-contact geodesic if the invariant distribution D1 is
inte-grable.
Acknowledgement: The authors are grateful to the referee for his valuable suggestions towards improvement of this paper.
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Current address : M.S.Siddesha: Department of Mathematics, Kuvempu University, Shanka-raghatta-577 451, Shimoga, Karnataka, INDIA.
E-mail address : mssiddesha@gmail.com
ORCID Address: http://orcid.org/0000-0003-2367-0544
Current address : C.S Bagewadi (Corresponding author): Department of Mathematics, Ku-vempu University, Shankaraghatta - 577 451, Shimoga, Karnataka, INDIA.
E-mail address : prof_bagewadi@yahoo.co.in
