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.