Certain inequalities for submanifolds in ( K,μ)-contact space forms
Transcript of Certain inequalities for submanifolds in ( K,μ)-contact space forms
CERTAIN INEQUALITIES FOR SUBMANIFOLDS IN(k; �)-CONTACT SPACE FORMS
KADRI ARSLAN, RIDVAN EZENTAS, ION MIHAI,CENGIZHANMURATHAN AND CIHAN ÖZGÜR
Abstract. In [Ch3], B.Y. Chen established sharp relationship be-tween the Ricci curvature and the squared mean curvature for asubmanifold in a Riemanian space form with arbitrary codimen-sion. In [MMO], one dealt with similar problems for submanifoldsin complex space forms.In this article, we obtain sharp relationships between the Ricci
curvature and the squared mean curvature for submanifolds in(k; �)-contact space forms.
1. (k; �)-Contact Space Forms
1A di¤erentiable manifold fM2n+1is said to be a contact manifold if itadmits a global di¤erential 1-form � such that �^(d�)n 6= 0, everywhereon fM2n+1.Given a contact form �, one has a unique vector �eld �, which is
called the characteristic vector �eld, satisfying
�(�) = 1; d�(�;X) = 0; (1.1)
for any vector �eld X:It is well-known that, there exists a Riemannian metric g and a
(1; 1)-tensor �eld ' such that
�(X) = g(X; �); d�(X;Y ) = g(X;'Y ), '2X = �X + �(X)�; (1.2)
where X and Y are vector �elds on fM:From (1:2) it follows that
'� = 0 ; � � ' = 0 ; g('X;'Y ) = g(X; Y )� �(X)�(Y ): (1.3)
1This paper is prepared during the third named author�s visit to the UludagUniversity, Bursa, Turkey in July 2000. The third author is supported by theScienti�c and Technical Research Council of Turkey (TÜBITAK) for NATO-PCAdvanced Fellowships Programme.MSC 2000: 53C25, 53C40Keywords: Ricci curvature, Chen invariants, mean curvature, (k; �)-contact
space forms, contact CR-submanifold1
2 K. ARSLAN ET AL.
A di¤erentiable manifold fM2n+1 equipped with structure tensors('; �; �; g) satisfying (1.2) is said to be a contact metric manifold andis denoted by fM = (fM2n+1; '; �; �; g):
On a contact metric manifold fM; we can de�ne a (1,1)-tensor �eld hby h=1
2L�', where L denotes Lie di¤erentiation . Then we may observe
that h is symmetric and satis�es
h� = 0 and h' = �'h; erX� = �'X � 'hX; (1.4)
where er is Levi-Civita connection [BKP].For a contact metric manifold fM one may de�ne naturally an almost
complex structure on fM � R . If this almost complex structure isintegrable, fM is said to be a Sasakian manifold . A Sasakian manifoldis characterized by the condition
(erX')Y = g(X; Y )� � �(Y )X; (1.5)
for all vector �elds X and Y on the manifold [Bl].A contact metric manifold fM is Sasakian if and only if
eR(X; Y )� = �(Y )X � �(X)Y; (1.6)
for all vector �elds X and Y [Bl].Let fM be a contact metric manifold. The ( k; �)-nullity distribution
of fM for the pair (k; �) is a distribution
N(k; �) : p! Np(k; �) = fZ 2 TpM j eR(X; Y )Z = k[g(Y; Z)X � g(X;Z)Y ]+�[g(Y; Z)hX � g(X;Z)hY ]g; (1.7)
where k; � 2 R and k � 1 (see [Kou]).If k = 1, then h = 0 and fM is a Sasakian manifold [BKP]. Also one
has trh = 0, trh' = 0 and h2 = (k � 1)'2: So if the characteristicvector �eld � belongs to the (k; �)-nullity distribution then we have
eR(X; Y )� = k [�(Y )X � �(X)Y ] + � [�(Y )hX � �(X)hY ] : (1.8)
Moreover, if M has constant '-sectional curvature c then it is calleda (k; �)-contact space form and is denoted by fM(c):The curvature tensor of fM(c) is given by [Kou]:
CERTAIN INEQUALITIES FOR SUBMANIFOLDS 3
4 eR(X; Y )Z = (c+ 3)fg(Y; Z)X � g(X;Z)Y g (1.9)
+(c+ 3� 4k)f�(X)�(Z)Y � �(Y )�(Z)X+g(X;Z)�(Y )� � g(Y; Z)�(X)�g+(c� 1)f2g(X;'Y )'Z + g(X;'Z)'Y � g(Y; 'Z)'Xg�2fg(hX;Z)hY � g(hY; Z)hX + g(X;Z)hY�2g(Y; Z)hX � 2�(X)�(Z)hY + 2�(Y )�(Z)hX+2g(hX;Z)Y � 2g(hY; Z)X + 2g(hY; Z)�(X)��2g(hX;Z)�(Y )� � g('hX;Z)'hY + g('hY; Z)'hXg+4�f�(Y )�(Z)hX � �(X)�(Z)hY+g(hY; Z)�(X)� � g(hX;Z)�(Y )�g:
If k 6= 1; then � = k + 1 and c = �2k � 1:
2. Riemannian invariants
The Riemannian invariants of a Riemannian manifold are the intrin-sic characteristics of the Riemannian manifold. In this section we recalla string of Riemannian invariants on a Riemannian manifold [Ch2].Let M be a Riemannian manifold. Denote by K(�) the sectional
curvature of M associated with a plane section � � TpM; p 2 M andby r the Riemannian connection on M:For any orthonormal basis fe1; e2;:::; eng of the tangent space TpM ,
the scalar curvature � at p is de�ned by
�(p) =Xi<j
K(ei ^ ej):
We denote by
(infK)(p) = inffK(�); � � TpM; dim � = 2g;and we introduce the �rst Chen invariant
�M(p) = �(p)� (infK)(p):Let L be a subspace of TpM of dimension r � 2 and fe1; e2;:::; erg
an orthonormal basis of L. We de�ne the scalar curvature �(L) of ther-plane section L by
�(L) =X�<�
K(e� ^ e�); �; � = 1; :::; r:
Given an orthonormal basis fe1; e2;:::; eng of the tangent space TpM ,we simply denote by � 1:::r the scalar curvature of r -plane section
4 K. ARSLAN ET AL.
spanned by e1; :::; er: The scalar curvature �(p) of M at p is noth-ing but the scalar curvature of the tangent space of M at p. And if Lis a 2-plane section, �(L) is nothing but the sectional curvature K(L)of L.For an integer l � 0, we denote by S(n; l) the �nite set which consists
of k-tuples (n1; n2; :::; nl) of integers � 2 satisfying n1 < n and n1 +:::+nl � n. Denote by S(n) the set of l-tuples with l � 0 for a �xed n.For each l-tuples (n1; :::; nl) 2 S(n), we introduce a Riemannian
invariant de�ned by
�(n1; :::; nl) = �(p)� S(n1; :::; nl)(p);
whereS(n1; :::nl) = inff�(L1) + :::+ �(Ll)g:
L1; :::; Ll run over all l mutually orthogonal subspaces of TpM such thatdimLj = nj; j = 1; :::; l:We de�ne:
d(n1; :::nl) =
n2(n+ l � 1�lP
j=1
nj)
2(n+ l �lP
j=1
nj)
;
b(n1; :::nl) =1
2[n(n� 1)�
lXj=1
nj(nj � 1)]:
We recall the following,
Lemma 2.1 (Ch1). Let a1; :::; an; b 2 R such that
(nXi=1
ai)2 = (n� 1)(
nXi=1
a2i + b):
Then we have 2a1a2 � b. Moreover, 2a1a2 = b , a1 + a2 = a3 = ::: =an.
Let M be an n-dimensional submanifold of fM(c): We denote by ehthe second fundamental form and by R the Riemann curvature tensorof M . Then the equation of Gauss is given by
~R(X; Y; Z;W ) = R(X; Y; Z;W )+ (2.1)
+g(eh(X;W );eh(Y; Z))� g(eh(X;Z);eh(Y;W ));for any vectors X; Y; Z;W tangent to M .
CERTAIN INEQUALITIES FOR SUBMANIFOLDS 5
We denote by H the mean curvature vector, i.e.
H(p) =1
n
nXi=1
eh(ei; ei)where fe1; :::; eng is an orthonormal basis of the tangent space TpM; p 2M:Also, we set ehrij = g(eh(ei; ej); er)
and
kehk2 = nXi;j=1
g(eh(ei; ej);eh(ei; ej))For any tangent vector �eld X to M , we put 'X = PX + FX,
where PX and FX are the tangential and normal components of 'X,respectively.Let fe1; :::; eng be an orthonormal basis of TpM . We denote by
kPk2 =nX
i;j=1
g2(Pei; ej)
Suppose L is a k-plane section of TpM and X a unit vector in L. Wechoose an orthonormal basis fe1; :::; ekg of L such that e1 = X:De�ne the Ricci curvature RicL of L at X by
RicL(X) = K12 +K13 + :::+K1k;
whereKij denotes the sectional curvature of the 2-plane section spannedby ei; ej. We simply called such a curvature a k-Ricci curvature.Recall that for a submanifold M in a Riemannian manifold, the
relative null space of M at a point p 2M is de�ned by
Np = fX 2 TpM jeh(X;Y ) = 0; Y 2 TpMg:3. Ricci curvature and squared mean curvature
B.Y. Chen established a sharp relationship between the Ricci cur-vature and the squared mean curvature for submanifolds in real spaceforms (see [Ch3]).We prove similar inequalities for certain submanifolds of a (k; �)-
contact space form fM(c).We consider the submanifold M is normal to � in a (k; �)-contact
space form fM(c) for the case k < 1. The case k = 1 is the Sasakiancase which has been considered by the third author in [Mi].
6 K. ARSLAN ET AL.
Theorem 3.1. Let M be an n-dimensional submanifold normal to �of a (2m+ 1)-dimensional (k; �)-contact space form ~M(c). Then:
i) For each unit vector X 2 TpM , we have
Ric(X) � 1
4fn2kHk2�khXk2�3(k+1)kPXk2+2(1�k)(n�1)g: (3.1)
ii) IfH(p) = 0, then a unit tangent vectorX at p satis�es the equalitycase of (3.1) if and only if X 2 Np.iii) The equality case of (3.1) holds identically for all unit tangent
vectors at p if and only if either p is a totally geodesic point or n = 2and p is a totally umbilical point.
Proof. Let X 2 TpM be a unit tangent vector at p. We choose an or-thonormal basis fe1; :::; en; 'e1; :::; 'en; e2n+1; :::; e2m+1 = �g in TpfM(c),such that e1; :::; en are tangent to M at p, with e1 = X.Then, from the equation of Gauss, we have
2� = n2 kHk2� eh 2+1� k
2n(n�1)�3
2(k+1)kPk2�1
2khk2�
nXi;j=1
g2(hei; ej)
(3.2)From (3.2), we get
n2kHk2 = 2� +2m+1Xr=n+1
[(hr11)2 + (hr22 + :::+ h
rnn)
2 + 2Xi<j
(hrij)2]� (3.3)
�22m+1Xr=n+1
X2�i<j�n
hriihrjj�
1� k2n(n�1)+3
2(k+1)kPk2+1
2khk2+
nXi;j=1
g2(hei; ej)
= 2�+1
2
2m+1Xr=n+1
[(hr11+:::+hrnn)
2+(hr11�hr22�:::�hrnn)2]+22m+1Xr=n+1
Xi<j
(hrij)2
�22m+1Xr=n+1
X2�i<j�n
hriihrjj�
1� k2n(n�1)+3
2(k+1)kPk2+1
2khk2+
nXi;j=1
g2(hei; ej):
It follows that1
2n2kHk2 � 2� � 1
2(1� k)n(n� 1) + 3
2(k + 1)kPk2 +
Xi<j
g2(hei; ej)(3.4)
+1
2khk2 � 2
2m+1Xr=n+1
X2�i<j�n
hehriiehrjj � (ehrij)2i
CERTAIN INEQUALITIES FOR SUBMANIFOLDS 7
From the equation of Gauss, we �nd
2X
2�i<j�nKij =
2m+1Xr=n+1
X2�i<j�n
hehriiehrjj � (ehrij)2i+ 12(1� k)(n� 1)(n� 2)(3.5)
�32(k + 1)
nXi=2
kPeik2 �1
2
nXi=2
kheik2 �nX
i;j=2
g2(hei; ej):
Substituting (3.5) in (3.4), one gets
1
2n2kHk2 � 2Ric(X) + 1
2khXk2 + 3
2(k + 1)kPXk2 � (1� k)(n� 1);
or equivalently (3.1).ii) Assume H(p) = 0. Equality holds in (3.1) if and only if
hr12 = ::: = hr1n = 0; h
r11 = h
r22 + :::+ h
rnn; r 2 fn+ 1; :::; 2mg:
Then hr1j = 0;8j 2 f1; :::; ng; r 2 fn+ 1; :::; 2mg, i.e. X 2 Np.iii) The equality case of (3.1) holds for all unit tangent vectors at p
if and only if
hrij = 0; i 6= j; r 2 fn+ 1; :::; 2mg; hr11 + :::+ hrnn � 2hrii = 0; i 2 f1; :::; ng; r 2 fn+ 1; :::; 2mg:We distinguish two cases:a) n 6= 2, then p is a totally geodesic point;b) n = 2, it follows that p is a totally umbilical point.The converse is trivial.
A submanifold M normal to � is said to be invariant (resp. anti-invariant) if '(TpM)� TpM , 8p 2 M (resp '(TpM)� T?p M , 8p 2 M). It is known that an invariant submanifold of a (k; �)-contact spaceform is minimal.
Corollary 3.2. Let M be an n-dimensional invariant submanifold ofa (k; �)-contact space form fM(c), (k < 1). Then:i) For each unit vector X 2 TpM orthogonal to �, we have
Ric(X) � �14(5k + 2) + 2n(1� k)g: (3.6)
ii) A unit tangent vector X 2 TpM orthogonal to � satis�es theequality case of (3.6) if and only if X 2 Np.iii) The equality case of (3.6) holds identically for all unit tangent
vectors orthogonal to � at p if and only if either p is a totally geodesicpoint or n = 2 and p is a totally umbilical point.
8 K. ARSLAN ET AL.
Corollary 3.3. LetM be an n-dimensional anti-invariant submanifoldof a (k; �)-contact space form fM(c); (k < 1). Then:i) For each unit vector X 2 TpM orthogonal to �, we have
Ric(X) � 1
4fn2kHk2 � 2(n� 1)(k + 1) + (k � 1)g: (3.7)
ii) If H(p) = 0, then a unit tangent vector X 2 TpM orthogonal to� satis�es the equality case of (3.7) if and only if X 2 Np.iii) The equality case of (3.7) holds identically for all unit tangent
vectors orthogonal to � at p if and only if either p is a totally geodesicpoint or n = 2 and p is a totally umbilical point.
4. B.Y. Chen inequalities.
B.Y. Chen proved a sharp inequality for submanifolds M in realspace forms fM(c) involving the scalar curvature and sectional curvatureof M (intrinsic invariants) and the squared mean curvature (extrinsicinvariant).
Theorem 4.1 (Ch1). Given an m-dimensional real space form ~M(c)and an n-dimensional submanifold M , n � 3; we have:
infK � � � n� 22f n2
n� 1 kHk2 + (n+ 1)cg: (4.1)
The equality case of inequality (4.1) holds at a point p 2 M if andonly if there exists an orthonormal basis fe1; e2; :::; eng of TpM and anorthonormal basis fen+1; :::; emg of T?p M such that the shape operatorsof M in ~M(c) at p have the following forms:
An+1 =
0BBBBB@a 0 0 � � � 00 b 0 � � � 00 0 � � � � 0� � � � � � �� � � � � � �0 0 0 � � � �
1CCCCCA , a+ b = � (4.2)
Ar =
0BBBBBB@
ehr11 ehr12 0 � � � 0ehr12 �ehr11 0 � � � 00 0 0 � � � 0� � � � � � �� � � � � � �0 0 0 � � � 0
1CCCCCCA (4.3)
CERTAIN INEQUALITIES FOR SUBMANIFOLDS 9
where we denote by
Ar = Aer ; r = n+ 1; :::;m;ehrij = g(eh(ei; ej); er); r = n+ 1; :::;m:By an analogous way we prove an inequality for submanifolds M
normal to � in (k; �)-contact space forms ~M(c). The Sasakian case wasstudied in [DMV].
Theorem 4.2. Given a (2m+1)-dimensional (k; �)-contact space form~M(c) and a submanifold M normal to �,dimM = n; n � 3, we have:
i) For any invariant plane section � � TpM
K(�) � ��(n� 2)n2
2(n� 1) kHk2�14
�n(1� k)(n� 1) + 3(k + 1)kPk2
�(1+2k):
(4.4)ii) For any anti-invariant plane section � � TpM
K(�) � ��(n� 2)n2
2(n� 1) kHk2�14
�n(1� k)(n� 1) + 3(k + 1)kPk2
+1
2(1�k):
(4.5)The equality case of inequalities (4.4) and (4.5) holds at a point
p 2 M if and only if there exists an orthonormal basis fe1; e2; :::; engof TpM and an orthonormal basis fen+1; :::; e2m+1 = �g of T?p M suchthat the shape operators of M in ~M(c) at p have the forms (4.2) and(4.3).
Proof. From the equation (2.1) we have
2� = n2 kHk2� eh 2+1� k
2n(n�1)�3
2(k+1)kPk2�1
2khk2�
nXi;j=1
g2(hei; ej):
(4.6)Putting
" = 2��n2(n� 2)n� 1 kHk2�1� k
2n(n�1)+3
2(k+1)kPk2+
nXi;j=1
g2(hei; ej);
(4.7)it follows that
n2 kHk2 = ("+ eh2 )(n� 1) (4.8)
Let � � TpM , � = spfe1; e2g, fe1; :::; eng � TpM , en+1 = 1kHkH;
fen+1; :::; e2m; e2m+1 = �g � T?p M: The equation (4.8) becomes
10 K. ARSLAN ET AL.
(nXi=1
ehn+1ii )2 = (n� 1)(Xi;j
X(ehn+1ii )2+
Xi6=j
(ehn+1ij )2+Xr�n+2
Xi;j
(ehrij)2+ ")By Chen Lemma, one obtains
2ehn+111ehn+122 �
Xi6=j
(ehn+1ij )2 +Xr�n+2
Xi;j
(ehrij)2 + "):Gauss equation gives
K(�) = eR(e1; e2; e1; e2) + 2m+1X(
r�n+2
ehr11ehr22 � (ehr12)):By using (1.9) we get
K(�) =1
2(1� k)� 3
2(1 + k)g2(e1; P e2)�
1
2fg2(he1; e2)� g(he2; e2)g(he1; e1)
�2g(he1; e1)� 2g(he2; e2)� g2('he1; e2) + g('he2; e2)g('he1; e1)
=1
2(1� k)� 3
2(1 + k)g2(e1; P e2) + ehn+111
ehn+122 � (ehn+112 )2
+2mX(
r=n+1
ehr11ehr22 � (ehr12)2) + 12[eh2m+111eh2m+122 � (eh2m+112 )2]
� 1
2(1� k)� 3
2(1 + k)g2(e1; P e2) +
"
2:
The equality case follows from the above equations and Chen Lemma.
Theorem 4.3. Given a (2m+1)-dimensional (k; �)-contact space form~M(c) and a submanifold M normal to �; dimM = n; n � 3, we have:
�(n1; :::; nr) � d(n1; :::; nr) kHk2 +1
2
�(c+ 3)
4n(n� 1) + 1
2(k � 1)n
�+
+(1� k)4
rXj=1
nj(nj � 1)(4.9)
Proof. Let n1; :::; nl � 2, n1 < n, n1 + :::+ nl � n. The equation (4.6)may be written as
2� = n2 kHk2 � eh 2 + 1
2(1� k)n(n� 1)� 3
2(k + 1) kPk2
�12khk2 �
nXi;j=1
g2(hei; ej); (4.10)
CERTAIN INEQUALITIES FOR SUBMANIFOLDS 11
and denote
" = 2� � 2d(n1; :::; nl) kHk2 �1
2(1� k)n(n� 1)� 3
2(k + 1) kPk2
+1
2khk2 �
nXi;j=1
g2(hei; ej);
and = n+ l �lP
j=1
nj. We have
n2 kHk2 = ("+ eh 2) :
Let p 2M , fe1; :::; eng � TpM , L1; :::; Lk � TpM such thatL1=Spfe1; :::; en1g, L2=Spfen1+1; ::::; en1+n2g,..., Ll=Spfen1+:::+nl�1+1 ; :::; en1+:::+nlg:
Then�(Lj) =
X�;�2�j�<�
K(e� ^ e�)
where �j = (n1 + ::: + nj�1 + 1; :::; n1 + ::: + nj): By Gauss equationone has
K(e� ^ e�) = R(e�; e�; e�; e�)= eR(e�; e�; e�; e�) + g(eh(e�; e�);eh(e�; e�))� g(eh(e�; e�);eh(e�; e�)):
By (1.9), we get
eR(e�; e�; e�; e�) = 1�12(k+1)�1
2[g(eh(e�; e�); �)g(eh(e�; e�); �)�g2(eh(e�; e�); �)];
which implies
�(Lj) =12(1� k)� 3
2(k + 1)
P�;�2�j�<�
g2(Pe�; e�)]
+P�;�
2m+1Pr=n+1
(ehr��ehr�� � (ehr��)2):So
lXj=1
�(Lj) � 1
4(1� k)
lXj=1
nj(nj � 1)�3
2(k + 1) kPk2 + "
2
� 1
4(1� k)
lXj=1
nj(nj � 1)�3
4(k + 1) kPk2
+� � d(n1; :::; nl) kHk2 �(1� k)2
n(n� 1) + 14khk2 :
12 K. ARSLAN ET AL.
Therefore
�(n1; :::; nl) � d(n1; :::; nl) kHk2 +1
4(1� k)
(n(n� 1)�
lXj=1
nj(nj � 1))
+3
4(k + 1) kPk2 � 1
4khk2 :
References
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[YK] K. Yano and M. Kon, Structures on Manifolds, World Scienti�c, Singapore,1984.
Kadri Arslan, Ridvan Ezentas, CengizhanMurathan and Cihan Özgür,Uludag University,Faculty of Art and Sciences,Department of Mathematics,Görükle 16059, Bursa, Turkey.
Ion Mihai,Faculty of Mathematics,Str. Academiei 14,70109 Bucharest, Romania.