J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
75Ae83.
HEMI-SLANT SUBMANIFOLD OF (LCS)n -MANIFOLD Payel Karmakar1 and Arindam Bhattacharyya2 Department of Mathematics.
Jadavpur University.
Kolkata-700032.
India payelkarmakar632@gmail.
Department of Mathematics.
Jadavpur University.
Kolkata-700032.
India bhattachar1968@yahoo.
Abstract.
In this paper, we analyse briefly some properties of hemi-slant submanifold of (LCS)n -manifold.
Here we discuss about some necessary and sufficient conditions for distributions to be integrable and obtain some results in this direction.
We also study the geometry of leaves of hemi-slant submanifold of (LCS)n -manifold.
At last, we give an example of a hemi-slant submanifold of an (LCS)n -manifold.
Key words and Phrases: (LCS)n -manifold, hemi-slant submanifold, integrablity, leaves of distribution.
INTRODUCTION
An n-dimensional Lorentzian manifold ME is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric gE, that is ME admits a smooth symmetric tensor field gE of type .
such that for each point.
the tensor gEp :
Tp ME y Tp ME Ie R is a non-degenerate inner-product of signature (-, ,.
, ).
Tp ME denotes the tangent vector space of ME at p and R is the real no.
A non-zero vector Xp OO Tp ME is known to be spacelike, null or lightlike, or timelike according as gEp (Xp .
Xp ) > 0, = 0 or < 0 respectively.
If ME is a differentiable manifold of dimension n, and there exists a (I, , ) structure satisfying I2 = I O , () = Oe1.
I() = 0.
I = 0, then M is called an almost paracontact manifold.
In an almost paracontact structure (I, , , gE), gE(X.
IY ) = gE(IX.
Y ), 2020 Mathematics Subject Classification: 53C05, 53C15, 53C40, 53C50.
Received: 12-03-2021, accepted: 17-01-2022.
Karmakar and A.
Bhattacharyya
A X )Y (ON
A Y )X,
2g(IX.
Y ) = (ON I2 X = X (X).
I = 0.
I() = 0, () = Oe1, .
where I is a tensor of type .
, is a vector field, is a 1-form and gE is Lorentzian metric satisfying gE(IX.
IY ) = gE(X.
Y ) (X)(Y ), gE(X, ) = (X) .
for all vector fields X.
Y on ME .
In a Lorentzian manifold (ME , gE), a vector field P defined by gE(X.
P ) = A(X) for any X OO e(T ME ), is called con-circular if A X A)(Y ) = .
E(X.
Y ) O(X)A(Y )}, (ON A denotes the operator where is a non-zero scalar and O is a closed 1-form and ON of covariant differentiation of ME with respect to gE.
Let ME admits a unit timelike concircular vector field , called the structure vector field of the manifold, then gE(, ) = Oe1, since is a unit concircular vector field, it follows that OE a non-zero 1-form such that gE(X, ) = (X).
The following equations holdOe A X )Y = .
E(X.
Y ) (X)(Y )], 6= 0, (ON A X = X = d(X) = A(X).
ON for all vector fields X.
Y on ME and is a non-zero scalar function related to A, by A = Oe().
A X , from which it follows that I is a symmetric .
tensor Let IX = 1 ON and call it the structure tensor on the manifold.
Thus the Lorentzian manifold ME together with unit timelike concircular vector field , its associated 1-form and a .
tensor field I is called a Lorentzian Concircular Structure manifold i.
(LCS)n -manifold.
Specially, if = 1, then we obtain LP-Sasakian structure of Matsumoto .
In an (LCS)n -manifold .
> .
, the following relations holdOe I2 = I O , () = Oe1, where I denotes the identity transformation of the tangent space T ME .
I = 0.
I = 0, gE(X.
IY ) = gE(IX.
Y ), rankI = 2n, .
gE(IX.
IY ) = gE(X.
Y ) (X)(Y ), gE(X, ) = (X), .
RE(X.
Y ) = .
Oe A)[(Y )X Oe (X)Y ] .
OA X.
Y OO T ME .
Also (LCS)n -manifold satisfiesOe A X I)Y = .
E(X.
Y ) 2(X)(Y ) (Y )X], (ON .
A X = IX.
Hemi-Slant Submanifold of (LCS)n -Manifold Let M be a submanifold of ME with (LCS)n -structure (I, , , gE) with induced metric g and let ON is the induced connection on the tangent bundle T M and ONOu is the induced connection on the normal bundle T Ou M of M .
The Gauss and Weingarten formulae are characterized byOe u X Y = ONX Y h(X.
Y ).
ON .
Ou ONX N = OeAN X ONX N, .
OA X.
Y OO T M.
N OO T Ou M, h is the 2nd fundamental form and AN is the Weingarten mapping associated with N via g(AN X.
Y ) = g.
(X.
Y ).
N ).
The mean curvature H is given by h.
i , ei ), k i=1 .
where k is the dimension of M and .
i }ki=1 is the local orthonormal frame on M .
For any X OO e(T M ).
IX = T X F X, .
where T X is the tangential component and F X is the normal component of IX.
Similarly, for any V OO e(T Ou M ).
IV = tV f V, .
where tV, f V are the tangential component and the normal component of IV respectively.
The covariant derivatives of the tensor fields T.
F, t, f are defined asOe (ONX T )Y = ONX T Y Oe T ONX Y, .
(ONX F )Y = ONOu X F Y Oe F ONX Y, (ONX .
V = ONX tV Oe tONOu
X V,
(ONX f )V = ONOu OA X.
Y OO T M.
V OO T Ou M.
A submanifold is calledOe .
invariant if OA X OO e(T M ).
IX OO e(T M ), i.
anti-invariant if OA X OO e(T M ).
IX OO e(T Ou M ), .
totally umbilical if h(X.
Y ) = g(X.
Y )H OA X.
Y OO e(T M ).
H is the mean curvature, i.
totaly geodesic if h(X.
Y ) = 0 OA X.
Y OO e(T M ), .
minimal if H = 0 on M .
Karmakar and A.
Bhattacharyya Let M be a Riemannian manifold isometrically immersed in an almost contact metric manifold (ME .
I, , , .
and be tangent to M .
Then the tangent bundle T M decomposes as T M = DOi < >, where D is the orthogonal distribution to .
Now for each non-zero vector X tangent to M at x, such that X is not proportional to x , we denote the angle between IX and Dx by (X).
M is called slant submanifold if the angle (X) is constant, which is independent of the choice of x OO M and X OO Tx M Oe < x >.
The constant angle OO .
A2 ] is then called the slant angle of M in ME .
If = 0, then the submanifold is invariant, if = A2 , then the submanifold is anti-invariant and if 6= 0.
A2 , then the submanifold is proper According to A.
Lotta .
, when M is a proper slant submanifold of ME with slant angle , then OA X OO e(T M ).
T 2 (X) = Oecos2 (X Oe (X)).
Carriazo .
introduced hemi-slant submanifolds as a special case of bislant submanifolds and he called them pseudo-slant submanifolds.
A submanifold M of an (LCS)n -manifold is called hemi-slant if there exist two orthogonal distributions D and DOu satisfying .
Oe .
T M = D Oi DOu Oi < >, i.
D is a slant distribution with slant angle 6= A2 , .
DOu is totally real i.
IDOu OI T Ou M.
A hemi-slant submanifold is called proper if 6= 0.
A2 .
CR-submanifolds and slant submanifolds are hemi-slant submanifolds with slant angle = A2 and D = 0 respectively.
In the rest of this paper, we use M as a hemi-slant submanifold of an (LCS)n manifold ME .
If we denote the dimensions of the distributions DOu and D by m1 , m2 respectively, then we haveOe .
if m2 = 0, then M is anti-invariant, i.
if m1 = 0, = 0, then M is invariant, .
if m1 = 0, 6= 0, then M is proper-slant with slant angle , i.
if m1 m2 6= 0.
OO .
A2 ), then M is proper hemi-slant.
Let M be hemi-slant submanifold of an (LCS)n -manifold ME , then for any X OO TM.
X = P1 X P2 X (X), .
where P1 .
P2 are projection maps on the distributions DOu .
D respectively.
Now operating I on .
, we get IX = IP1 X IP2 X (X)I.
Hemi-Slant Submanifold of (LCS)n -Manifold Using .
, we obtain T X F X = F P1 X T P2 X F P2 X.
On comparing, we get
T X = T P2 X,
F X = F P1 X F P2 X.
If we denote the orthogonal complement of I(T M ) in T Ou M by AA, then the normal bundle T Ou M can be decomposed as T Ou M = F (DOu ) Oi F (D )Oi < AA > .
Since F (DOu ) and F (D ) are orthogonal distributions, g(X.
Y ) = 0 for each X OO DOu and Y OO D .
Hence by .
, we have OA Z OO DOu .
W OO D , g(F Z.
F W ) = g(IZ.
IW ) = g(Z.
W ) = 0, which shows that F (DOu ).
F (D ) are mutually perpendicular.
So, .
is an orthogonal direct decomposition.
There are various types of works done on hemi-slant submanifolds.
Abutuqayqah worked on geometry of hemi-slant submanifolds of almost contact manifolds .
Khan et al.
discussed about totally umbilical hemi-slant submanifolds of Kahler manifolds .
and of cosymplectic manifolds .
, and they also discussed about a classification on totally umbilical proper slant and hemislant submanifolds of a nearly trans-Sasakian manifold .
Laha et al.
totally umbilical hemi-slant submanifolds of LP-Sasakian manifold .
and hemislant submanifold of Kenmotsu manifold .
Tastan et al.
discussed about hemi-slant submanifolds of a locally product Riemannian manifold .
and of a locally conformal Kahler manifold .
Another important works on hemi-slant submanifolds were done by A.
Lotta in 1996 .
, by M.
Lone et al.
in 2016 .
and by M.
Siddesha et al.
in 2018 .
Motivated from these works, in this paper, we analyse some properties regarding distributions and leaves of hemi-slant submanifold of (LCS)n -manifold.
MAIN RESULTS
In this section, we discuss about some necessary and sufficient conditions for distributions to be integrable and obtain some results in this direction.
We also study the geometry of leaves of hemi-slant submanifold of (LCS)n -manifold.
Theorem 2.
Let M be a hemi-slant submanifold of an (LCS)n -manifold ME , then OA Z.
W OO DOu .
AIW Z = AIZ W Oe (W )Z Oe (Z)W Oe 2(Z)(W ).
Proof.
On using .
, we have u X Z.
W ) Oe g(IONX Z.
W ) g(AIW Z.
X) = g.
(Z.
X).
IW ) = g(Ih(Z.
X).
W ) = g(ION Karmakar and A.
Bhattacharyya u X Z.
W ) = g(ON u X IZ.
W ) Oe g((ON u X I)Z.
W ).
= g(ION Again using .
, we get g(AIW Z.
X) = g(AIZ X ONOu X IZ.
W ) Oe g.
(X.
Z) 2(X)(Z) (Z)X.
W ) = g(AIZ X.
W ) Oe g(X.
Z)(W ) Oe 2(X)(Z)(W ) Oe (Z)g(X.
W ) = g.
(W.
X).
IZ) Oe g(X.
Z)(W ) Oe (Z)g(X.
W ) Oe 2(X)(Z)(W ) = g(AIZ W Oe (W )Z Oe (Z)W Oe 2(Z)(W ).
X) Ne AIW Z = AIZ W Oe (W )Z Oe (Z)W Oe 2(Z)(W ).
Theorem 2.
Let M be a hemi-slant submanifold of an (LCS)n -manifold ME .
Then the distribution D Oi DOu is integrable if and only if g([X.
Y ], ) = 0 OA X.
Y OO D Oi DOu .
Proof.
For X.
Y OO D Oi DOu , u X Y, ) Oe g(ON u Y X, ) g([X.
Y ], ) = g(ON = Oeg(ONX .
Y ) g(ONY .
X) = Oeg(IX.
Y ) g(IY.
X) = 0.
Since T M = D Oi DOu Oi < >, therefore [X.
Y ] OO D Oi DOu .
So.
D Oi DOu is Conversely, let D Oi DOu is integrable.
Then OA X.
Y OO D Oi DOu , [X.
Y ] OO D Oi DOu .
As T M = D Oi DOu Oi < >, therefore g([X.
Y ], ) = 0.
Theorem 2.
Let M be a hemi-slant submanifold of an (LCS)n -manifold ME .
Then the anti-invariant distribution DOu is integrable if and only if OA W OO DOu .
W is a scalar multiple of .
Proof.
For Z.
W OO DOu , from .
, we have u Z I)W = .
(Z.
W ) 2(Z)(W ) (W )Z].
(ON
After some calculations and using .
, .
, we get OeAF W Z ONOu Z F W Oe T ONZ W Oe F ONZ W Oe th(Z.
W ) Oe f h(Z.
W ) = .
(Z.
W ) 2(Z)(W ) (W )Z].
Comparing tangential components, we have OeAF W Z OeT ONZ W Oeth(Z.
W ) = .
(Z.
W ) 2(Z)(W ) (W )Z].
Interchanging Z.
W , we obtain OeAF Z W OeT ONW Z Oeth(W.
Z) = .
(W.
Z) 2(W )(Z) (W )Z].
Subtracting .
and using the fact that h is symmetric, we have AF W ZOeAF Z W T (ONZ W OeONW Z) = [(Z)W Oe(W )Z].
Hemi-Slant Submanifold of (LCS)n -Manifold From .
, we have AF W Z Oe AF Z W T ([Z.
W ]) = [(Z)W Oe (W )Z].
Now DOu is integrable if and only if [Z.
W ] OO DOu and as DOu is anti-invariant.
ID OI T Ou M and so.
T [Z.
W ] = 0.
Ou Hence from .
DOu is integrable if and only if AF W ZOeAF Z W = [(Z)W Oe (W )Z].
From Theorem 2.
1, we have as T W = 0 = T Z.
AIW Z Oe AIZ W = Oe(W )Z Oe (Z)W Oe 2(Z)(W ) Ne [(Z)W Oe (W )Z] = Oe(W )Z Oe (Z)W Oe 2(Z)(W ) Ne 2(Z)W 2(Z)(W ) = 0 Ne (Z)W (Z)(W ) = 0 Ne W (W ) = 0.
Hence the result is proved.
Theorem 2.
Let M be a hemi-slant submanifold of an (LCS)n -manifold ME .
Then the slant distribution D is integrable if and only if OA X.
Y OO D .
P1 (ONX T Y Oe ONY T X) = [(Y )P1 X Oe (X)P1 Y ].
Proof.
We denote by P1 .
P2 the projections on DOu .
D respectively.
OA X.
Y OO D ,
we have from .
u X I)Y = .
E(X.
Y ) 2(X)(Y ) (Y )X].
(ON
On applying .
, .
, .
, .
, we have u X I)Y = ONX T Y h(X.
T Y ) Oe AF Y X ONX F Y Oe (T ONX Y F ONX Y ) Oe .
h(X.
Y ) (ON f h(X.
Y )) = .
(X.
Y ) 2(X)(Y ) (Y )X].
Comparing tangential components, we get ONX T Y Oe AF Y X Oe T ONX Y Oe th(X.
Y ) = .
(X.
Y ) 2(X)(Y ) (Y )X].
Interchanging X.
Y in .
and subtracting the resultant from .
, we ONX T Y OeONY T X OeAF Y X AF X Y OeT ONX Y T ONY X = [(Y )X Oe(X)Y ].
Since X.
Y OO D .
F X = 0 = F Y , applying P1 to both sides of .
, we P1 (ONX T Y Oe ONY T X) = [(Y )P1 X Oe (X)P1 Y ].
Theorem 2.
Let M be a hemi-slant submanifold of an (LCS)n -manifold ME .
the leaves of DOu are totally geodesic in M , then OA X OO D and Z.
W OO DOu , g.
(Z.
X).
F W ) g.
h(Z.
W ).
X) = 0.
Karmakar and A.
Bhattacharyya Proof.
From .
, .
, .
, we have ONZ IW h(Z.
IW ) Oe AF W Z ONOu Z F W Oe IONZ W Oe Ih(Z.
W ) = .
(Z.
W ) 2(W )(Z) (W )Z].
Comparing tangential components and on taking inner product with X OO D , we obtain Oeg(AF W Z.
X) Oe g.
h(Z.
W ).
X) Oe g(T ONZ W.
X) = 0.
The leaves of DOu are totally geodesic in M if for Z.
W OO DOu .
ONZ W OO DOu .
So.
T ONZ W = 0.
Thus g(AF W Z.
X) g.
h(Z.
W ).
X) = 0.
Example.
Now we give an example of a hemi-slant submanifold of an (LCS)n manifold.
Let ME (R9 .
I, , , .
denote the manifold R9 with the (LCS)-structure given byOe = 3 OCz , = 13 (Oedz i=1 bi dai ), g = 19 i=1 .
ai O dai Oi dbi O dbi ) Oe O .
I( OCz ) = 0.
I( OCa i ) = OCbi , i = 1, 2, 3, 4, and OC I( OCbi ) = OCai for i = 1, 2 and I( OCbOC i ) = Oe OCa i for i = 3, 4, where .
1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 , .
OO R9 .
Let us consider a 5-dimensional submanifold M of ME defined by .
1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 , .
7Ie .
osa1 sina2 , cosb1 sinb2 , a oeb , a oeb , 3.
Then it can be easily proved that M is a hemi-slant submanifold of ME by choosing the slant distribution D =< e1 , e2 > with slant angle | Oe | and the totally real distribution DOu =< e3 , e4 >, where e1 = sin OCaOC 1 Oe cos OCaOC 2 , e2 = sin OCbOC1 Oe cos OCbOC2 , e3 = OCaOC 3 OCbOC3 , e4 = OCaOC 4 OCbOC4 such that .
1 , e2 , e3 , e4 , } forms an orthogonal frame on T M so that T M = D Oi DOu Oi < >.
Acknowledgement.
The first author is the corresponding author and has been sponsored by University Grants Commission (UGC) Junior Research Fellowship.
India.
UGC-Ref.
No.
: 1139/(CSIR-UGC NET JUNE 2.
The authors would like to thank the referee for the valuable suggestions to improve the paper.
Hemi-Slant Submanifold of (LCS)n -Manifold
REFERENCES