On some inequalities for submanifolds of Bochner Kaehler manifolds

0.11 MB·

by  Mehraj Ahmad Lone

On some inequalities for submanifolds of Bochner Kaehler
manifolds
Mehraj AhmadLonea,∗,MohammedJamalib, MohammadHasan Shahidc
aDepartmentofMathematics,CentralUniversityofJammu,Jammu-180011,India.
bDepartmentofMathematics,Al-FalahUniversity,Haryana-121004,India.
cDepartmentofMathematics,JamiaMilliaIslamia,NewDelhi-110025,India.
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1 Abstract
0
2 B.Y.ChenestablishedsharpinequalitiesbetweencertainRiemannianinvariantsandthesquared

r
p mean curvature for submanifolds in real space form as well as in complex space form. In this
A
paperwe generalizeChen inequalitiesforsubmanifoldsof BochnerKaehler manifolds. More-

6 over,weconsiderCR-warpedproductsubmanifoldsofBochnerKaehlermanifoldandestablish
2
an inequalityforscalar curvature.

]
G Keywords: Bochner Kaehlermanifold,CR-warped product,slantsubmanifolds,Einstein
D
manifold,Chen inequality.
.
h
2010MathematicsSubject Classification: 53C15,53C25, 53C40.
t
a
m
[

1. Introduction
2
v
0 In [6], B. Y. Chen established sharp inequality for a submanifold in a real space form in-
3
1 volvingintrinsicinvariantsofthesubmanifoldsandsquaredmeancurvature,themainextrinsic
4
invariant and in [3], B. Y. Chen obtained the same inequality for complex space form. After
0
.
1 that many research articles [7, 8, 9] havebeen publishedby different authors for different sub-
0
manifoldsandambientspacesincomplexaswellasincontactversion. Inthisarticleweobtain
6
1
theseinequalitiesforsubmanifoldsinBochner Kaehlermanifold.
:
v
In [2] Bishop and O’Neil initiated the thoery of warped product submanifold as a gener-
i
X
alization of pseudo-Riemannian product manifold. In [5] Chen introduced the notion of CR-
r
a
warped products. In This paper we study the CR-warped product submanifolds of Bochner
Kaehlermanifolds.
2. Preliminaries
LetW bean-dimensionalsubmanifoldofaBochnerKaehlermanifoldW ofdimension2m.
Let (cid:209) and (cid:209) be the Levi-Civita connection on W and W respectively. Let J be the complex
∗Correspondingauthor
Emailaddress:[email protected](MehrajAhmadLone)
structureon W . Then theGaussand Weingartenformulasare givenrespectivelyby
(cid:209) Y =(cid:209) Y +w (X,Y), (1)
X X
(cid:209) V =−B X+(cid:209) ⊥Y, (2)
X V X
for all X,Y tangent to W and vector field V normal to W . Where w , (cid:209) ⊥, B denotes the
X V
second fundamental form, normal connection and theshape operator respectively. The second
fundamentalform andtheshapeoperatorarerelated by
g(w (X,Y),V)=g(B X,Y). (3)
V
Let RbethecurvaturetensorofW ,Then theGaussequationisgivenby[6]
R(X,Y,Z,W)=R(X,Y,Z,W)+g(w (X,W),w (Y,Z))−g(w (X,Z),w (Y,W))
foranyvectorfields X,Y,Z,W tangenttoW .
ThecurvaturetensorofaBochner KaehlermanifoldW isgivenby[10]
R(X,Y,Z,W) = L(Y,Z)g(X,W)−L(X,Z)g(Y,W)+L(X,W)g(Y,Z)
−L(Y,W)g(X,Z)+M(X,W)g(JX,W)−M(X,Z)g(JY,W)
+M(X,W)g(JY,Z)−M(Y,W)g(JX,Z)
−2M(X,Y)g(JZ,W)−2M(Z,W)g(JX,Y) (4)
where
1 r
L(Y,Z)= Ric(Y,Z)− g(Y,Z), (5)
2n+4 2(2n+2)(2n+4)
M(Y,Z)=−L(Y,JZ), (6)
L(Y,Z)=L(Z,Y), L(Y,Z)=L(JY,JZ), L(Y,JZ)=−L(JY,Z), (7)
Ricandr are theRicci tensorand scalarcurvatureofW .
Letx∈W and{e ,...,e }beanorthonormalbasisofthetangentspaceT W and{e ,...,e }
1 n x n+1 2m
betheorthonormalbasisofT⊥W . Wedenoteby H , themean curvaturevectorat x, that is
H (x)= 1 (cid:229) n w (e,e), (8)
i i
n
i=1
Also,weset
w r =g(w (e,e ),e ), i, j∈{1,...,n}, r∈{n+1,...,2m}
ij i j r
2
and
n
kw k2 = (cid:229) (w (e,e ),w (e,e )). (9)
i j i j
i,j=1
Foranyx∈W andX ∈T W ,weputJX =TX+FX,whereTX andFX arethetangentialand
x
normalcomponentsofJX,respectively.
We denoteby
n
kTk2 = (cid:229) g2(Te,e ).
i j
i,j=1
Let W bea Riemannianmanifold. Denoteby K (p )thesectionalcurvatureofW oftheplane
section p ⊂T W ,x∈W . The scalarcurvature r for an orthonormalbasis{e ,e ,...,e } of the
x 1 2 n
tangentspaceT W at x isdefined by
x
r (x)= (cid:229) K(e ∧e ).
i j
i<j
Lemma 2.1. [6] Let n≥2 andx ,x ,...,x , b bereal numberssuchthat
1 2 n
n n
((cid:229) x)2 =(n−1)((cid:229) x2+b)
i i
i=1 i=1
then2x x ≥b, with equalityholdsif andonlyif
1 2
x +x =x =...=x .
1 2 3 n
In [1] A. Bejancu introduced the notion of CR-submanifolds, which is the generalization
of invariant and anti-invariant submanifolds. In [4] B. Y. Chen introduced the notion of slant
submanifoldsas ageneralizationofCR-submanifolds.
Definition 2.1. A submanifold W of a Bochner Kaehler manifold W is said to be a slant
submanifold if for any x∈W and X ∈T W , the angle between JX and T W is constant, i.e.,
x x
theangledoesnotdepend onthechoiceofx∈W andX ∈T W . Theangleq ∈[0,p ]iscalled
x 2
theslantangleofW inW .
Invariantandanti-invariantsubmanifoldsaretheslantsubmanifoldswithslantangleq =0
and q = p respectively and when 0 < q < p , then slant submanifold is called proper slant
2 2
submanifold.
Definition 2.2. Let (N ,g ) and (N ,g ) be two Riemannian manifolds and f, a positive dif-
1 1 2 2
ferentiable function on N . The warped product of N and N is the Riemannian manifold
1 1 2
M =N ×N =(N ×N ,g),whereg=g + f2g
1 2 1 2 1 2
Definition 2.3. A Riemannian manifold W is said to be Einstein manifold if the Ricci tensor
isproportionaltothemetrictensor, thatis,Ric(X,Y)=l g(X,Y) forsomeconstantl .
3
3. B. Y.Chen inequalities
In this section, we obtain B. Y. Chen inequalities for submanifolds of a Bochner Kaehler
manifolds.
First wehave,
Theorem 3.1. Let W be a submanifold of a Bochner Kaehler manifold W . Then, for each
pointx∈W and eachplanesectionp ⊂T W , we have
x
5n2+31n+26+3kTk2 n2(n−2) 6
K (p )≥ r − kH k2− Ric(e,Je )g(e,Je ).
i j i j
2(2n+2)(2n+4) 2(n−1) 2(2n+4)
(cid:18) (cid:19)
(10)
Equality holds if and only if there exists an orthonormal basis {e ,e ,...,e } of T W and or-
1 2 n x
thonormalbasis{e ,e ,...,e }ofT⊥W suchthattheshapeoperatorstakesthefollowing
n+1 n+2 2m
forms
a 0 0 ··· 0
0 b 0 ··· 0
 
B = 0 0 x ··· 0 ,a +b =x (11)
n+1
 
... ... ... ... ...
 
 
0 0 0 ··· x 
 
 
and
w r w r 0 ··· 0
11 12
w r −w r 0 ··· 0
 12 11 
B = 0 0 0 ··· 0 ,r=n+2,...,2m. (12)
r
 ... ... ... ... ...
 
 
 0 0 0 ··· 0
 
 
Proof. UsingGaussequation,theRiemanniancurvaturetensorofW isgivenby
R(X,Y,Z,W) = L(Y,Z)g(X,W)−L(X,Z)g(Y,W)+L(X,W)g(Y,Z)
−L(Y,W)g(X,Z)+M(Y,Z)g(JX,W)−M(X,Z)g(JY,W)
−M(X,W)g(JY,Z)−M(Y,W)g(JX,Z)−2M(X,Y)(JZ,W)
−2M(Z,W)g(JX,Y)+g(w (X,W),w (Y,Z))−g(w (X,Z),w (Y,W))
foranyX, Y, Z, W∈TW .
(cid:229)
R(e,e ,e ,e) = L(e ,e )g(e,e)−L(e ,e )g(e ,e)+L(e,e)g(e ,e )
i j j i j j i i i j j i i i j j
i,j
−L(e ,e)g(e,e )+M(e ,e )g(Je,e)−M(e,e )g(Je ,e)
j i i j j j i i i j j i
−M(e,e)g(Je ,e )−M(e ,e)g(Je,e )−2M(e,e )(Je ,e)
i i j j j i i j i j j i
−2M(e ,e)g(Je,e )+g(w (e,e),w (e ,e ))−g(w (e,e ),w (e ,e))
j i i j i i j j i j j i
4
= L(e ,e )g(e,e)−L(e,e )g(e ,e)+L(e ,e)g(e ,e )
j j i i i j j i i i j j
−L(e ,e)g(e,e )−L(e ,Je )g(Je,e)+L(e,Je )g(Je ,e)
j i i j j j i i i j j i
+L(e ,Je)g(Je ,e )+L(e ,Je)g(Je,e )+2L(e,Je )(Je ,e)
i i j j j i i j i j j i
+2L(e ,Je)g(Je,e )+g(w (e,e),w (e ,e ))−g(w (e,e ),w (e ,e)).
j i i j i i j j i j j i
(13)
Using(7), (8)and (9)in(13), wehave
(cid:229)
R(e,e ,e ,e) = 2nL(e,e)−2L(e,e )g(e,e )+6L(e,Je )g(e,Je )
i j j i i i i j i j i j i j
i,j
+n2kH k2−kw k2.
Which simplifiesto,
2r = 2(n−1)L(e,e)+6L(e,Je )g(e,Je )+n2kH k2−kw k2. (14)
i i i j i j
Combining(5)and (14), wehave
2(n−1) 2(n−1)r
2r = Ric(e,e)− g(e,e)
i i i i
2n+4 2(2n+2)(2n+4)
6 6r
+ Ric(e,Je )g(e,Je )− g(e,Je )g(e,Je )
i j i j i j i j
2n+4 2(2n+2)(2n+4)
+n2kH k2−kw k2.
or
6n2+2n−8−6kTk2 6
2r = r + Ric(e,Je )g(e,Je )
i j i j
2(2n+2)(2n+4) 2n+4
+n2kH k2−kw k2.
or
6n2+2n−8−6kTk2 6
(2− )r = Ric(e,Je )g(e,Je )+n2kH k2−kw k2.
i j i j
2(2n+2)(2n+4) 2n+4
Denotingby
6n2+2n−8−6kTk2 n2(n−2) 6
e =(2− )r − kH k2− Ric(e,Je )g(e,Je ),
i j i j
2(2n+2)(2n+4) n−1 2n+4
weobtain
n2(n−2)
e =n2kH k2−kw k2− kH k2.
n−1
or
n2kH k2 =(n−1)(e +kw k2). (15)
5
Forchosen orthonormalbasis,theaboveequationtakestheform
n n 2m n
((cid:229) w n+1)2 =(n−1) (cid:229) (w n+1)2+(cid:229) (w n+1)2+ (cid:229) (cid:229) (w )2+e . (16)
ii ii ij ij
i=1 "i=1 i6=j r=n+1i,j=1 #
Usinglemma1 in(16), wehave
2m n
2w n+1w n+1 ≥ (cid:229) (w n+1)2+ (cid:229) (cid:229) (w r)2+e . (17)
11 22 ij ij
i6=j r=n+1i,j=1
On theotherhand, fromGauss equationweobtain
K (p )=L(e ,e )+L(e ,e )+g(w (e ,e ),w (e ,e )−g(w (e ,e ),w (e ,e )). (18)
2 2 1 1 1 1 2 2 1 2 2 1
Combing(5)and (18), we derive
K (p )= 4n+3 r +w n+1w n+1+ (cid:229)2m w r w r − (cid:229)2m (w r )2. (19)
(2n+2)(2n+4) 11 22 11 22 12
r=n+2 r=n+1
Incorporating(17)in (19), wearriveat theinequality
K (p ) ≥ 1 (cid:229) (w n+1)2+1 (cid:229)2m (cid:229) n (w r)2+1e
2 ij 2 ij 2
i6=j r=n+1i,j=1
+ 4n+3 r + (cid:229)2m w r w r − (cid:229)2m (w r )2.
(2n+2)(2n+4) 11 22 12
r=n+2 r=n+1
Which impliesthat
4n+3 1
K (p )≥ r + e .
(2n+2)(2n+4) 2
or
5n2+31n+26+3kTk2 n2(n−2) 6
K (p )≥( )r − kH k2− Ric(e,Je )g(e,Je ).
i j i j
2(2n+2)(2n+4) 2(n−1) 2(2n+4)
(20)
Iftheequalityin (10)at a point p holds,then theinequality(20)becomeequality. In thiscase,
wehave
w n+1 =w n+1 =w n+1 =0, i6= j>2,
1j 2j ij
w r =0,∀i6= j, i, j=3,...,2m, r=n+1,...,2m,
ij


w r +w r =0,∀r=n+2,...,2m,
 11 22
w n+2+w n+1 =...=w m +w m =0.
 11 22 11 22

Now, if we choose e ,e such that w n+1= 0 and we denote by a = w r ,b = w r , x =
 1 2 12 11 22
w n+1 = ... = w r . Therefore by choosing the suitable orthonormal basis the shape operators
33 33
takethedesiredforms.
6
We concludethefollowingcorollaryfrom thistheorem.
Corollary 3.2. Let W be a submanifold of a Bochner Kaehler manifold W which is Einstein.
Then, foreach pointx∈W andeach planesectionp ⊂T W , wehave
x
5n2+31n+26+3kTk2 n2(n−2) 6l
K (p )≥( )r − kH k2− kTk2.
2(2n+2)(2n+4) 2(n−1) 2(2n+4)
Theequalityatapointx∈W holdsiffthereexistsanorthonormalbasis{e ,e ,...,e }ofT W
1 2 n x
and orthonormal basis {e ,e ,...,e } of T⊥W such that shape operators of W in W at
n+1 n+2 2m
xhavetheforms(11)and(12).
Similarly, in case if W is a slant submanifoldof a Bochner Kaehler manifoldW . We have
thefollowingtheorem
Theorem 3.3. Let W be a slant submanifold of a Bochner Kaehler manifold W . Then, for
each pointx∈W andeach planesectionp ⊂T W , wehave
x
5n2+31n+26+3cos2q n2(n−2) 6
K (p )≥( )r − kH k2− Ric(e,Je )cosq .
i j
2(2n+2)(2n+4) 2(n−1) 2(2n+4)
Equality holds if and only if there exists an orthonormal basis {e ,e ,...,e } of T W and or-
1 2 n x
thonormalbasis{e ,e ,...,e }ofT⊥W suchthattheshapeoperatortakesthefollowing
n+1 n+2 2m
forms
a 0 0 ··· 0
0 b 0 ··· 0
 
B = 0 0 x ··· 0 ,a +b =x (21)
n+1
 
... ... ... ... ...
 
 
0 0 0 ··· x 
 
 
and
w r w r 0 ··· 0
11 12
w r −w r 0 ··· 0
 12 11 
B = 0 0 0 ··· 0 ,r=n+2,...,2m. (22)
r
 
 ... ... ... ... ...
 
 
 0 0 0 ··· 0
 
 
From thistheorem,followingcorollaries can beeasilydeduced.
Corollary 3.4. Let W be a slant submanifold of a Bochner Kaehler manifold W , which is
Einstein. Then, for eachpointx∈W and eachplanesectionp ⊂T W , we have
x
5n2+31n+26+3cos2q n2(n−2) 6l
K (p )≥( )r − kH k2− cos2q .
2(2n+2)(2n+4) 2(n−1) 2(2n+4)
7
Theequalityholdsatapointx∈W ifandonlyifthereexistsanorthonormalbasis{e ,e ,...,e }
1 2 n
of T W and orthonormal basis {e ,e ,...,e } of T⊥W such that shape operators of W
x n+1 n+2 2m
inW atx havetheforms(21)and (22).
Corollary 3.5. Let W be a invariant submanifold of a Bochner Kaehler manifold W . Then,
foreach pointx∈W andeach planesectionp ⊂T W , wehave
x
5n2+31n+26+3 n2(n−2) 6
K (p )≥( )r − kH k2− Ric(e,Je ).
i j
2(2n+2)(2n+4) 2(n−1) 2(2n+4)
Theequalityatapointx∈W holdsiffthereexistsanorthonormalbasis{e ,e ,...,e }ofT W
1 2 n x
and orthonormal basis {e ,e ,...,e } of T⊥W such that shape operators of W in W at
n+1 n+2 2m
xhavetheforms(21)and(22).
Corollary 3.6. Let W be a anti-invariant submanifold of a Bochner Kaehler manifold W .
Then, foreach pointx∈W andeach planesectionp ⊂T W , wehave
x
5n2+31n+26 n2(n−2)
K (p )≥( )r − kH k2.
2(2n+2)(2n+4) 2(n−1)
Theequalityatapointx∈W holdsiffthereexistsanorthonormalbasis{e ,e ,...,e }ofT W
1 2 n x
and orthonormal basis {e ,e ,...,e } of T⊥W such that shape operators of W in W at
n+1 n+2 2m
xhavetheforms(21)and(22).
4. Warpedproduct ofCR-submanifolds ofBochner Kaehlermanifolds
LetW =W × W bethewarpedproductCR-submanifoldsofBochnerKaehlermanifold
T f ⊥
W suchthattheinvariantdistributionisD=TW andanti-invariantdistributionisD⊥=TW ,
T ⊥
where f :W −→R. Then themetricg onW isgivenby [5]
T
g(X,Y)=hp X,p Yi+(f ◦p )2hs X,s Yi
∗ ∗ ∗ ∗
wherep and s are theprojectionmapsfromW ontoW and W respectively.
T ⊥
It iseasy to seethat
TW =D⊕D⊥ and T⊥W =JD⊥⊕n , (23)
wheren is theorthogonaldistributionto JD⊥ inthenormalbundleT⊥W .
From (23), wecan write
w (X,Y)=w JD⊥(X,Y)+w n (X,Y)
Alsoforwarped product submanifoldW ofW ,wehave[5]
X(f)
(cid:209) Z =X(logf)Z= Z (24)
X
f
8
foranyvectorfields X ∈D and Z ∈D⊥.
Further, we can decompose((cid:209) J)Y intothetangentialand normalcomponentsasunder
X
((cid:209) J)Y =P Y +Q Y (25)
X X X
whereP Y and Q Y denotes thetangentialand normalcomponentsof((cid:209) J)Y
X X X
First weprovethefollowinglemma
Lemma 4.1. Let W =W × W beaCR-warped productsubmanifoldofa BochnerKaehler
T f ⊥
manifoldW . Then wehave
w (JX,Z)=JP JX+X(logf)JZ
JD⊥ Z
g(P JX,W)=g(Q X,JW)
Z Z
and
g(w (JX,Z),Jw (X,Z))−kw n (X,Z)k2=g(QZX,Jw n (X,Z))
forX ∈D andZ ∈D⊥.
Proof. From Gaussequation,wehave
(cid:209) JX+w (JX,Z)−J((cid:209) X)−Jw (X,Z)=P X+Q X
Z Z Z Z
Using(24), weinfer
w (JX,Z)=P X+Q X+J[X(logf)Z]+Jw (X,Z)−JX(logf)Z.
Z Z
Replace X byJX,we get
−w (X,Z)=P JX+Q JX+JX(logf)JZ+Jw (JX,Z)+X(logf)Z.
Z Z
Wecan writetheaboveequationas
−w (X,Z)=PZJX+QZJX+JX(logf)JZ+Jw JD⊥(JX,Z)+Jw n (JX,Z)+X(logf)Z. (26)
On comparingthetangentialcomponents,we obtain
P JX+Jw (JX,Z)+X(logf)Z=0,
Z JD⊥
or
Jw (JX,Z)=−P JX−X(logf)Z
JD⊥ Z
whichshowsthat
w (JX,Z)=JP JX+X(logf)JZ, (27)
JD⊥ Z
forX ∈D and Z ∈D⊥.
9
Againon comparingthenormal componentsin(26), wehave
−w (X,Z)=QZJX+JX(logf)JZ+Jw n (JX,Z)
from whichweconcludethat
w (JX,Z)=QZX+X(logf)JZ+Jw n (X,Z)
or
w (JX,Z)−Jw n (X,Z)=QZX+X(logf)JZ (28)
By takingtheinnerproduct (28) withJW, weget
g(w (JX,Z),JW)=g(Q X,JW)+X(logf)g(JZ,JW) (29)
JD⊥ Z
Furtherusing(27)in(29), wehave
g(P JX,W)+X(logf)g(Z,W)=g(Q X,JW)+X(logf)g(Z,W)
Z Z
from whichweconcludethat
g(P JX,W)=g(Q X,JW) (30)
Z Z
Also,by takingtheinnerproductof(28)withJw (X,Z),wefind
g(w (JX,Z),Jw (X,Z))−kw n (X,Z)k2=g(QZX,Jw n (X,Z)).
Theorem 4.2. LetW =W × W beawarpedproductCR-submanifoldsofBochnerKaehler
T f ⊥
manifold W with P D∈D, then the squared norm of second fundamental form of W in W
D⊥
satisfiesthefollowinginequality
kw k2 ≥kP Dk2+qkgrad (logf)k2.
D⊥ D
Proof. Let{X ,...,X ,X =JX ,...,X =JX }bealocalorthonormalframeofvectorfields
1 p p+1 1 2p p
on N and {Z ,...,Z } be a local orthonormalframe of vectorfields on N , where 2p+q=n.
T 1 q ⊥
Then wehave
2p q 2p
kw k2 = (cid:229) g(w (Xi,Xj),w (Xi,Xj))+ (cid:229) (cid:229) g(w (Xi,Za ),w (Xi,Za ))
i,j=1 a =1j=1
q
+ (cid:229) g(w (Za ,Zb ),w (Za ,Zb ))
a ,b =1
from aboveequationwecan saythat
2p q
kw k2 ≥ (cid:229) (cid:229) g(w (Xi,Za ),w (Xi,Za ))
j=1a =1
10