J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
194Ae202.
CR-WARPED PRODUCT SUBMANIFOLDS OF
SASAKIAN MANIFOLDS ADMITTING CERTAIN
CONNECTIONS
Shyamal Kumar Hui1,a .
Joydeb Roy1,b Department of Mathematics.
The University of Burdwan.
Burdwan, 713104.
West Bengal.
India, a skhui@math.
roy8@gmail.
Abstract.
The existence or non-existence of a warped product CR-submanifolds of Sasakian manifolds admitting certain connections have been studied.
Among others.
Ricci solitons of such notions have been investigated.
Key words and Phrases: Sasakian manifold.
CR-warped product, semisymmetric metric connection, semisymmetric non-metric connection.
Abstract.
Dalam paper ini, dikaji eksistensi atau non-eksistensi dari suatu hasil kali warped CR-submanifold dari manifold Sasaki dengan koneksi tertentu.
Lebih jauh, dikaji juga Ricci soliton dari hasil kali tersebut.
Kata Kunci: Manifold Sasaki, hasil kali CR-warped, koneksi metrik semisimetrik, koneksi non- metrik semisimetrik.
INTRODUCTION
The semisymmetric connection was introduced in .
If such connection is metric compatible then it is called semisymmetric metric connection otherwise semisymmetric non-metric connection.
For details we may refer .
, .
, .
, .
, .
, .
The concept of CR-submanifold was introduced in .
It is the generalization of invariant and anti-invariant submanifold.
For detail study, see .
, .
, .
, .
Warped product is a generalization of Riemannian product introduced in .
2010 Mathematics Subject Classification: 53C25, 53C40.
Received: 20-06-2018, accepted: 17-04-2019.
Warped product CR-submanifolds Its existence and non-existence are very much significant.
Here, such existence and non-existence are studied in case of CR-submanifolds of Sasakian manifolds.
The Ricci soliton on a Riemannian manifold M with Riemannian metric g is a triplet .
V, ) such that AV g S g = 0.
Here S is the Ricci tensor of M .
AV denotes the Lie derivative operator along V OO N(M ) and OO R .
Nature of such soliton depends on .
Such soliton have been studied in section 4 in case of CR-warped product submanifold of Sasakian We summarize all the obtained results of this paper in section 5 in tabular form.
Throughout the paper we denote:
Ricci Tensor by AuRTAy, .
Warped product by AuWPAy, .
Semisymmetric metric connection by AuSMAy, .
Semisymmetric non-metric connection by AuSNMAy, .
Levi-Civita connection by AuLCAy, .
Almost contact manifold AuACAy.
BACKGROUND
An odd dimensional C O manifold ME 2n 1 .
> .
is called an AC .
if for all X OO N(ME ) I2 = OeI O , () = 1, .
where I is a tensor field of type .
, .
and is a vector field .
nown as characteristic vector fiel.
and is an 1-form such that (X) = g(X, ).
From .
we get
I = 0
If M (IX) = 0.
if it admits a g such that g(X.
Y ) = g(IX.
IY ) (X)(Y ) then it is called almost contact metric manifold .
, denoted by ME From .
we get g(X.
IY ) = Oeg(IX.
Y ).
Now ME (I, , , .
is called Sasakian .
if A X I)(Y ) g(X.
Y ) = (Y )X, (ON .
(I, , , .
A Y = IY
A is the LC.
Throughout the paper we denote Sasakian manifold OAX.
Y OO N(ME ) and ON O eE and SNM ON A with of dimension .
by ME .
The correspondence between SM ON A are ON eE Y = ON A X Y (Y )X Oe g(X.
Y ).
ON .
Hui and J.
Roy
A XY = ON
A X Y (Y )X
e and ON are respectively the induced Let M be a submanifold of ME and ON.
ON A semisymmetric connection ON connection of M from the Riemannian connection ON.
O A Then Gauss formula with respect to above three connections are given by and ON.
A X Y Oe ONX Y = h(X.
Y ).
ON .
eE Y Oe ON e XY = e h(X.
Y ), .
A X Y Oe ONX Y = h(X.
Y ).
e ON).
Again Here h.
h, .
is the second fundamental form admitting ON.
ON,
M tangent to is called .
invariant if I(Tp M ) OC Tp M , .
anti-invariant if I(Tp M ) OC TpOu M , .
CR if OE two orthogonal distribution D and DOu so that D is invariant and DOu is anti-invariant and T M = D Oi DOu Oi < > OA p OO M .
The WP .
of two RM (N1 , g1 ) and (N2 , g2 ) is the RM N1 yf N2 = (N1 y N2 , .
, g = g1 f 2 g2 .
f is the positive definite smooth function on N1 .
If M = N1 yf N2 is a WP submanifold of ME then we have .
ONU X = ONX U = X.
nf )U .
OAX OO e(T N1 ) and U OO e(T N2 ).
e Here N1 and N2 are orthogonal.
Let M be of the form N1 yf N2 admitting ON.
If OO e(T N1 ) then e X U = X.
nf )U ON e U X = X.
nf )U (X)IU ON .
OAX OO e(T N1 ) and U OO e(T N2 ).
If OO e(T N2 ) then we have .
e X U = X.
nf )U (U )X ON e U X = X.
nf )U.
Plugging .
yields e X Y = ONX Y (Y )X Oe g(X.
Y ) ON .
h(X.
Y ) = h(X.
Y ) .
WP admitting SNM have been studied in .
Let M = N1 yf N2 is a WP submanifold of ME admitting SNM, where N1 and N2 are orthogonal.
If OO e(T N1 ) then we have .
O ONX U = X.
nf )U O ONU X = X.
nf )U (X)IU .
OAX OO e(T N1 ) and U OO e(T N2 ).
If is tangent to N2 then we have .
O ONX U = X.
nf )U (U )X O ONU X = X.
nf )U.
Warped product CR-submanifolds Then by virtue of .
we have from .
that O ONX Y Oe ONX Y = (Y )X .
O h = h.
CR WARPED PRODUCT SUBMANIFOLD
eE It may be Let M = N1 yf N2 is a WP CR-submanifold of ME admitting ON.
noted that such submanifold are also tangent to .
So two cases may arise:
case I: OO e(T N1 ) case II: OO e(T N2 ).
Again, case I has two subcases:
N1 invariant and N2 anti-invariant submanifold of ME .
N1 anti-invariant and N2 invariant submanifold of ME .
Now for the subcase .
, we state the following:
Theorem 3.
There exist WP submanifold M = N1 yf N2 of ME with respect to eE so that N is invariant and N is anti-invariant and OO e(T N ).
e such Proof.
Let M = N1 yf N2 is the given submanifold of ME with respect to ON that N1 is invariant and N2 is anti-invariant and OO e(T N1 ).
Then from .
e U = .
nf )U IU.
Out of .
one can get IU U Oe (U ) IU U.
From .
we get .
nf )U = U i.
, .
nf ) = 1.
Thus such warped product exists and hence the proof.
Now for subcase .
, we state the following:
Theorem 3.
There does not exist any such submanifold M = N1 yf N2 of ME with eE such that N is anti-invariant and N is invariant and OO e(T N ).
respect to ON Proof.
By virtue of .
we have eE I)Y g(X.
Y ) g(X.
IY ) = (Y )X Oe (Y )I(X) (ON .
for all X.
Y OO N(M ).
Now if OO e(T N1 ) and U OO e(T N2 ) then by virtue of .
we get
e U I) = U Oe IU
(ON
Hui and J.
Roy e I)U = 0.
(ON
e U I) (ON e I)U = U Oe IU.
(ON
e X I)Y = ON e X IY Oe I(ON e X Y ).
(ON
Hence Again we know On account of .
it is found, e U I) (ON e I)U = .
nf )IU.
(ON
Hence from .
we get the result.
We now consider case II.
In that case two sub cases also arises in similar to case I.
For the sub case .
we have:
Theorem 3.
There does not exist any such submanifold M = N1 yf N2 of ME with eE such that N is invariant and N is anti-invariant and OO e(T N ).
respect to ON Proof.
Let us assume that M = N1 yf N2 is a warped product submanifold of eE such that OO e(T N ).
Then from .
we get ME with respect to ON e X = X.
nf ) X.
e X = IX X.
From .
we obtain From .
it ensures that X.
nf ) = IX.
Now taking inner product with respect to one can get:
nf )g(, ) = (IX) = 0 Ne X.
nf ) = 0.
So f is constant and the case is trivial.
Hence the warped product does not exist.
Now for subcase .
, we state the following:
Theorem 3.
There does not exist any such submanifold M = N1 yf N2 of ME with eE such that N is anti-invariant and N is invariant and OO e(T N ).
respect to ON Proof.
If OO e(T N2 ).
X OO e(T N1 ), then by virtue of .
we obtain e X I) = X Oe (X) Oe IX
(ON
e I)X = 0.
(ON
e X I) (ON e I)X = X Oe IX Oe (X).
(ON
Hence from .
we get Warped product CR-submanifolds Again, we know that e X I)Y = ON e X IY Oe I(ON e X Y ).
(ON
So, in view of .
from the above expression we get e I)X (ON e X I) = (IXlnf ) Oe IX.
(ON
Now, from .
we have (IXlnf ) = X Oe (X), .
which implies:
(IXlnf )g(, ) Ne IXlnf = g(X, ) Oe (X)g(, ) So f is constant and the warped product does not exist.
Which completes the Remark 1: For the warped product CR-submanifold M of ME with respect to Levi-Civita connection .
, semisymmetric non-metric connection, all the results remain the same.
RICCI SOLITON
Theorem 4.
If .
, , ) is a RS on a warped product submanifold M = N1 y N2 eE such that N invariant.
N anti-invariant and OO e(T N ) of ME with respect to ON then N2 is nearly quasi Einstein.
Proof.
Let .
, , ) be a Ricci soliton on a warped product CR-submanifold M of ME .
Then we have:
(X.
Y ) 2S(X, (A Y ) 2g(X.
Y ) = 0
e and A is the Lie for all X.
Y OO M , where Se is the Ricci tensor of M admitting ON derivative along on M .
From .
we get:
e U = U IU where is tangent to N1 and U OO N2 .
Therefore for U.
V OO N2 we have e .
(U.
V ) = g(U IU.
V ) g(U.
V IV ) = 2g(U.
V ).
Hence from .
we get:
e V ) ( .
g(U.
V ) = 0.
S(U,
Again, using .
we can compute that e Z) = S(Y.
Z) .
m Oe .
[(Y )(Z) Oe g(Y.
Z) Oe g(IY.
Z)] S(Y, .
Hui and J.
Roy for all Y.
Z OO e(T M ).
Hence by virtue of .
we have from .
that S(U.
V ) = .
m Oe 2 Oe )g(U.
V ) .
m Oe .
g(IU.
V ) .
for all U.
V OO e(T N2 ), so N2 is nearly quasi Einstein.
Theorem 4.
If .
V, ) is a RS on a WP submanifold M of ME with respect to O ON such that OO e(T N1 ) then N2 is Einstein.
Proof.
For the connection ON we have from .
we get ONU = U IU.
Therefore by virtue of .
we have O (A .
(U.
V ) g(U IU.
V ) g(U.
V IV ) 2g(U.
V ) .
for all U.
V OO e(T N2 ).
Since .
, , ) is a Ricci soliton, we get (A .
(U.
V ) 2S(U.
V ) 2g(U.
V ) = 0.
Again using .
we can obtain that O S(U.
V ) = S(U.
V ) 2m.
(U.
V ) (U )(V )].
So, by virtue of .
we get S(U.
V ) = Oe.
g(U.
V ) .
for all U.
V OO e(T N2 ), so N2 is Einstein.
CONCLUSION
A comparative study on the existence and non-existence of WP CR-submanifold of Sasakian manifold with respect to SM and SNM is considered.
The results with respect to LC in this context was studied in .
Here we summarize the facts in next two theorems.
From Theorem 3.
1- 3.
4 and the Remark 1, we can state that:
Theorem 5.
Let M = N1 yf N2 be a WP CR-subamnifold of ME .
Then we have:
Warped product CR-submanifolds Levi-Civita nature of tangent to N1 tangent to N1 tangent to N2 tangent to N2 semisymmetric tangent to N1 tangent to N1 tangent to N2 tangent to N2 semisymmetric tangent to N1 non-metric tangent to N1 tangent to N2 tangent to N2 nature of N1 and N2 N1 invariant.
N2 anti-invariant N1 anti-invariant.
N2 invariant N1 invariant.
N2 anti-invariant N1 anti-invariant.
N2 invariant N1 invariant.
N2 anti-invariant N1 anti-invariant.
N2 invariant N1 invariant.
N2 anti-invariant N1 anti-invariant.
N2 invariant N1 invariant.
N2 anti-invariant N1 anti-invariant.
N2 invariant N1 invariant.
N2 anti-invariant N1 anti-invariant.
N2 invariant existance/non-existance warped product exist warped product does not exist warped product does not exist warped product does not exist warped product exist warped product does not exist warped product does not exist warped product does not exist warped product exist warped product does not exist warped product does not exist warped product does not exist From Theorem 4.
1 and 4.
2 following theorem can be stated:
Theorem 5.
If .
, , ) is a RS of WP submanifold M = N1 yf N2 of ME such that N1 invariant.
N2 anti-invariant submanifold of ME and OO e(T N1 ).
Then we have:
connection on ME Einstein nearly quasi Einstein Einstein Acknowledgement.
This work acknowledges to the SERB (File No: EMR/2015/002.
Government of India for providing financial support.
REFERENCES