Certain inequalities for submanifolds in ( K,μ)-contact space forms

12
CERTAIN INEQUALITIES FOR SUBMANIFOLDS IN (k;)-CONTACT SPACE FORMS KADRI ARSLAN, RIDVAN EZENTAS, ION MIHAI,CENGIZHAN MURATHAN AND CIHAN ZGR Abstract. In [Ch3], B.Y. Chen established sharp relationship be- tween the Ricci curvature and the squared mean curvature for a submanifold in a Riemanian space form with arbitrary codimen- sion. In [MMO], one dealt with similar problems for submanifolds in 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 1 A di/erentiable manifold f M 2n+1 is said to be a contact manifold if it admits a global di/erential 1-form such that ^(d) n 6=0, everywhere on f M 2n+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 ), 2 X = X + (X ); (1.2) where X and Y are vector elds on f M: From (1:2) it follows that =0 ; =0 ;g(’X; ’Y )= g(X; Y ) (X )(Y ): (1.3) 1 This paper is prepared during the third named authors visit to the Uludag University, Bursa, Turkey in July 2000. The third author is supported by the Scientic and Technical Research Council of Turkey (TBITAK) for NATO-PC Advanced Fellowships Programme. MSC 2000: 53C25, 53C40 Keywords: Ricci curvature, Chen invariants, mean curvature, (k;)-contact space forms, contact CR-submanifold 1

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

[Bl] D.E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes inMath. 509, Springer, Berlin, 1976.

[BKP] D.Blair, T.Koufogiorgos and B.J.Papantoniou, Contact metric manifoldssatisfying a nullity condition, Israel J.Math. 91(1995),189-214.

[Ch1] B.Y. Chen, Some pinching and classi�cation theorems for minimal subman-ifolds, Archiv Math. 60 (1993), 568-578.

[Ch2] B. Y. Chen, Some new obstructions to minimal and Lagrangian isometricimmersions, Japan. J. Math., (to appear).

[Ch3] B.Y. Chen, Relations between Ricci curvature and shape operator for sub-manifolds with arbitrary codimensions, Glasgow Math. J. 41 (1999), 33-41.

[DMV] F. Defever, I. Mihai and L. Verstraelen, B.Y.Chen�s inequality for C-totallyreal submanifolds in Sasakian space forms, Boll.Un. Mat. Ital. 11 (1997),365-374.

[Kou] T. Koufogiorgos, Contact Riemannian manifolds with constant '� sec-tional curvature, Geometry and Topology of Submanifolds, 8 (1995) 195-197, World Sci., Singapore.

[MMO] K. Matsumoto, I. Mihai and A. Oiag¼a, Ricci curvature of submanifolds incomplex space forms, Rev. Roum. Math. Pures Appl., (to appear).

[Mi] I. Mihai, Ricci curvature of submanifolds in Sasakian space forms, submit-ted.

[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.