RICCI CURVATURE OF SUBMANIFOLDS IN KENMOTSU SPACE FORMS

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RICCI CURVATURE OF SUBMANIFOLDS IN KENMOTSU SPACE FORMS

KADRI ARSLAN, RIDVAN EZENTAS, ION MIHAI, CENGIZHAN MURATHAN, and CIHAN ÖZGÜR Received 20 April 2001 and in revised form 21 August 2001

In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension.

Similar problems for submanifolds in complex space forms were studied by Matsumoto et al. In this paper, we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in Kenmotsu space forms.

2000 Mathematics Subject Classiﬁcation: 53C25, 53C40.

1. Preliminaries. Let(M, <, >)˜ be a Hermitian manifold and denote byJthe canon- ical almost complex structure on ˜M. According to the behavior of the tangent bundle T Mwith respect to the action ofJ, we may distinguish two special classes of subman- ifoldsMin ˜M:

(a) complex submanifolds, that is,J(TpM)=TpM, for allp∈M.

(b) totally real submanifolds, that is,J(TpM)⊂T_p^⊥M, for all p∈M, whereTpM (resp.,T_p^⊥M) is the tangent (resp., the normal) vector space ofMatp. Such submanifolds were deﬁned and studied by Chen and Ogiue [4].

On the other hand, Yano and Ishihara [8] considered a submanifoldMwhose tangent bundleT M splits into a complex subbundleᏰand a totally real subbundleᏰ^⊥. Later, such a submanifold was called a CR-submanifold [1, 2]. Blair and Chen [1]

proved that aCR-submanifold of a locally conformal Kaehler manifold is a Cauchy- Riemann manifold in the sense of Greenﬁeld.

The ﬁrst main result on CR-submanifolds was obtained by Chen [2]: any CR- submanifold of a Kaehler manifold is foliated by totally real submanifolds (i.e., the totally real subbundle is involutive).

As nontrivial examples ofCR-submanifolds, we can mention the (real) hypersur- faces of Hermitian manifolds.

2. Kenmotsu manifolds and their submanifolds. Tanno [7] has classiﬁed, into three classes, the connected almost contact Riemannian manifolds whose automor- phisms groups have the maximum dimensions:

(1) homogeneous normal contact Riemannian manifolds with constant φ- holomorphic sectional curvature;

(2) global Riemannian products of a line or circle and a Kaehlerian space form;

(3) warped product spacesL×fF, whereLis a line andF a Kaehlerian manifold.

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Kenmotsu [5] studied the third class and characterized it by tensor equations. Later, such a manifold was called a Kenmotsu manifold.

A(2m+1)-dimensional Riemannian manifold(M, g)˜ is said to be a Kenmotsu manifold if it admits an endomorphismφof its tangent bundleTM, a vector ﬁeld˜ ξ, and a 1-formη, which satisfy:

φ²= −Id+η⊗ξ, η(ξ)=1, φξ=0, η◦φ=0, g(φX, φY )=g(X, Y )−η(X)η(Y ), η(X)=g(X, ξ),

∇Xφ

Y= −g(X, φY )ξ−η(Y )φX,

∇Xξ=X−η(X),

(2.1)

for any vector ﬁeldsX, Y on ˜M, where ˜∇denotes the Riemannian connection with respect tog.

We denote byωthe fundamental 2-form of ˜M, that is, ω(X, Y )=g(φX, Y ), for allX, Y ∈Γ(TM). It was proved that the pairing˜ (ω, η)deﬁnes a locally conformal cosymplectic structure, that is,

dω=2ω∧η, dη=0. (2.2)

A Kenmotsu manifold with constantφ-holomorphic sectional curvaturecis called a Kenmotsu space form and it is denoted by ˜M(c). Then its curvature tensor ˜R is expressed by (cf. [5])

4 ˜R(X, Y )Z=(c−3)

g(Y , Z)X−g(X, Z)Y +(c+1)

η(X)Y−η(Y )X

η(Z)+

g(X, Z)η(Y )−g(Y , Z)η(X) ξ +ω(Y , Z)φX−ω(X, Z)φY−2ω(X, Y )φZ

.

(2.3)

Let ˜Mbe a Kenmotsu manifold andMann-dimensional submanifold tangent toξ.

For any vector ﬁeldXtangent toM, we put

φX=P X+F X, (2.4)

whereP X(resp.,F X) denotes the tangential (resp., normal) component ofφX. ThenP is an endomorphism of tangent bundleT MandF is a normal bundle valued 1-form onT M.

The equation of Gauss is given by R(X, Y , Z, W )˜ =R(X, Y , Z, W )+g

h(X, W ), h(Y , Z) −g

h(X, Z), h(Y , W ) , (2.5) for any vectorsX,Y,Z,W tangent toM.

We denote byHthe mean curvature vector, that is, H(p)= 1

n n i=1

h

ei, ei , (2.6)

where{e1, . . . , en}is an orthonormal basis of the tangent spaceTpM,p∈M.

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Also, we set

h^r_ij=g h

ei, ej , er ,

h²= n i,j=1

g h

ei, ej , h ei, ej

. (2.7)

Let{e1, . . . , en}be an orthonormal basis ofTpM. We put P²=

n i,j=1

g²

P ei, ej . (2.8)

By analogy with submanifolds in a Kaehler manifold, diﬀerent classes of submanifolds in a Kenmotsu manifold were considered (cf. [6]).

A submanifoldMtangent toξis said to be invariant (resp., anti-invariant) ifφ(TpM)

⊂TpM, for allp∈M(resp.,φ(TpM)⊂T_p^⊥M, for allp∈M).

A submanifoldMtangent toξis called a contactCR-submanifold [9] if there exists a pair of orthogonal diﬀerentiable distributionsᏰandᏰ^⊥onM, such that,

(i) T M=Ᏸ⊕Ᏸ^⊥⊕{ξ}, where{ξ}is the 1-dimensional distribution spanned byξ;

(ii) Ᏸis invariant byφ, that is,φ(Ᏸp)⊂Ᏸp, for allp∈M;

(iii) Ᏸ^⊥is anti-invariant byφ, that is,φ(Ᏸ^⊥p)⊂T_p^⊥M, for allp∈M.

In particular, ifᏰ^⊥= {0}(resp., Ᏸ= {0}),M is an invariant (resp., anti-invariant) submanifold.

Next, recall some notions introduced by Chen (see [3]).

LetLbe ak-plane section ofTpMandXa unit vector inL. We choose an orthonormal basis{e1, . . . , ek}ofLsuch thate1=X.

Deﬁne the Ricci curvature RicLofLatXby

RicL(X)=K12+K13+···+K1k, (2.9) whereKijdenotes the sectional curvature of the 2-plane section spanned byei,ej. We simply called such a curvature ak-Ricci curvature.

The scalar curvatureτ of thek-plane sectionLis given by τ(L)=

1≤i<j≤k

Kij. (2.10)

For each integerk, 2≤k≤n, the Riemannian invariantΘk on ann-dimensional Riemannian manifoldMis deﬁned by

Θk(p)= 1 k−1inf

L,XRicL(X), p∈M, (2.11)

whereLruns over allk-plane sections inTpMandXruns over all unit vectors inL.

Recall that for a submanifoldMin a Riemannian manifold, the relative null space ofMat a pointp∈Mis deﬁned by

ᏺp=

X∈TpM|h(X, Y )=0, Y∈TpM

. (2.12)

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3. Ricci curvature and squared mean curvature. Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for submanifolds in real space forms (see [3]).

We prove similar inequalities for certain submanifolds of a Kenmotsu space form M(c). We will consider submanifolds˜ Mtangent to the Reeb vector ﬁeldξ.

Theorem3.1. LetM(c)˜ be a(2m+1)-dimensional Kenmotsu space form andMan n-dimensional submanifold tangent toξ. Then

(i) for each unit vectorX∈TpMorthogonal toξ, Ric(X)≤1

4

(n−1)(c−3)+1 2

3P X²−2 (c+1)+n²H²

; (3.1)

(ii) ifH(p)=0, then a unit tangent vectorX∈TpM orthogonal toξsatisfies the equality case of (3.1) if and only ifX∈ᏺp;

(iii) the equality case of (3.1) holds identically for all unit tangent vectors orthogonal toξatpif and only ifpis a totally geodesic point.

Proof. LetX∈TpMbe a unit tangent vectorXatp. We choose an orthonormal basise1, . . . , en=ξ,e_n+1, . . . , e_2m+1inTpM(c)˜ such thate1, . . . , enare tangent toMatp, withe1=X.

Then, from the equation of Gauss, we have n²H²=2τ+h²−n(n−1)c−3

4 −

3P²−2n+2 c+1

4 . (3.2)

From (3.2), we get n²H²=2τ+

2m+1 r=n+1

h^r₁₁ ²+

h^r₂₂+···+h^r_nn ²+2

i<j

h^r_ij2

−2

2m+1

r=n+1

2≤i<j≤n

h^r_iih^r_jj−n(n−1)c−3 4 −

3P²−2n+2 c+1 4

=2τ+1 2

2m+1 r=n+1

h^r₁₁+···+h^r_nn ²+

h^r₁₁−h^r₂₂−···−h^r_nn ² +2

2m+1 r=n+1

i<j

h^r_ij2

−2

2m+1 r=n+1

2≤i<j≤n

h^r_iih^r_jj

−n(n−1)c−3 4 −

3P²−2n+2 c+1 4 .

(3.3)

From the equation of Gauss, we ﬁnd

2≤i<j≤n

Kij=

2m+1

r=n+1

2≤i<j≤n

h^r_iih^r_jj− h^r_ij2 +(n−1)(n−2)

2

c−3 4 +

3P²−3P e1²−2n+4 c+1 8 .

(3.4)

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Substituting (3.4) in (3.3), we get 1

2n²H²=2 Ric(X)+1 2

2m+1

r=n+1

h^r₁₁−h^r₂₂−···−h^r_nn ²

+

2m+1 r=n+1

n j=1

h^r_1j2

−2(n−1)c−3 4 −

3P X²−2 c+1 4

≥2 Ric(X)−2(n−1)c−3 4 −

3P X²−2 c+1 4 ,

(3.5)

which is equivalent to (3.1).

For (ii) assume thatH(p)=0. Equality holds in (3.1) if and only if

h^r₁₂= ··· =h^r_1n=0, h^r₁₁=h^r₂₂+···+h^r_nn=0, r∈ {n+1, . . . ,2m}. (3.6) Thenh^r_1j=0, for allj∈ {1, . . . , n},r∈ {n+1, . . . ,2m}, that is,X∈ᏺp.

For (iii) the equality case of (3.1) holds for all unit tangent vectors atpif and only if

h^r_ij=0, i≠j, r∈ {n+1, . . . ,2m},

h^r₁₁+h^r₂₂+···+h^r_nn−2h^r_ii=0, i∈ {1, . . . , n}, r∈ {n+1, . . . ,2m}. (3.7) It follows thatpis a totally geodesic point.

The converse is trivial.

Corollary3.2. LetMbe ann-dimensional invariant submanifold tangent toξin a Kenmotsu space formM(c). Then,˜

(i) for each unit vectorX∈TpMorthogonal toξ, we have Ric(X)≤1

4

(n−1)(c−3)+1 2(c+1)

; (3.8)

(ii) a unit tangent vectorX∈TpMorthogonal toξsatisfies the equality case of (3.8) if and only ifX∈ᏺp;

Proof. It is known that every invariant submanifold of a Kenmotsu space form is minimal (cf. [6]).

On the other hand, for any unit tangent vectorX∈TpM orthogonal toξ, we have P X = φX = X =1.

Then, the inequality (3.1) implies (3.8).

Similarly, we can prove the following results.

Corollary3.3. LetMbe ann-dimensional anti-invariant submanifold tangent toξ in a Kenmotsu space formM(c). Then,˜

(i) for each unit vectorX∈TpMorthogonal toξ, we have Ric(X)≤1

4

(n−1)(c−3)−(c+1)+n²H²

; (3.9)

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(ii) ifH(p)=0, then a unit tangent vectorX∈TpM orthogonal toξsatisfies the equality case of (3.9) if and only ifX∈ᏺp;

Corollary3.4. LetMbe ann-dimensional contactCR-submanifold of a Kenmotsu space formM(c). Then˜

(i) for each unit vectorX∈Ᏸp, Ric(X)≤1

4

(n−1)(c−3)+1

2(c+1)+n²H²

; (3.10)

(ii) for each unit vectorX∈Ᏸ^⊥p, we have Ric(X)≤1

4

(n−1)(c−3)−(c+1)+n²H²

. (3.11)

4. k-Ricci curvature. In this section, we prove a relationship between thek-Ricci curvature and the squared mean curvature for submanifolds tangent toξin a Ken- motsu space form.

Theorem4.1. LetM(c)˜ be a Kenmotsu space form andM ann-dimensional sub- manifold tangent toξ. Then we have

H²≥ 2τ

n(n−1)−c−3 4 −

3P X²−2(n−1) n(n−1)

c+1

4 . (4.1)

Proof. We choose an orthonormal basis{e1, . . . , en, en+1, . . . , e2m+1=ξ}atpsuch thate_n+1is parallel to the mean curvature vectorH(p), ande1, . . . , endiagonalize the shape operatorAn+1. Then the shape operators take the forms

A_n+1=







a1 0 0 ··· 0 0 a2 0 ··· 0 ... ... ... ··· ... 0 0 0 ··· an







Ar= h^r_ij

, i, j=1, . . . , n;r=n+2, . . . ,2m, traceAr= n i=1

h^r_ii=0.

(4.2)

From (3.2), we get

n²H²=2τ+ n i=1

a²_i+

2m

r=n+2

n i,j=1

h^r_ij2

−n(n−1)c−3 4 −

3P X²−2(n−1) c+1 4 .

(4.3)

On the other hand, since 0≤

i<j

ai−aj

2=(n−1)

i

a²_i−2

i<j

aiaj, (4.4)

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we obtain

n²H²=



ⁿ

i=1

ai





2

= n i=1

a²_i+2

i<j

aiaj≤n n i=1

a²_i, (4.5) which implies that

n i=1

a²_i≥nH². (4.6)

We have from (4.3)

n²H²≥2τ+nH²−n(n−1)c+3 4 −

3P X²−2(n−1) c+1

4 , (4.7) which is equivalent to (4.1).

UsingTheorem 4.1, we obtain the following.

Theorem4.2. LetM(c)˜ be a Kenmotsu space form andM ann-dimensional sub- manifold tangent toξ. Then, for any integerk,2≤k≤n, and any pointp∈M, we have

H²(p)≥Θk(p)−c−3 4 −

3P²−2n+2 (c+1)

4n(n−1) . (4.8)

Proof. Let{e1, . . . , en}be an orthonormal basis ofTpM. Denote byLi₁···i_k thek- plane section spanned byei₁, . . . , ei_k. It follows from (2.9) and (2.10) that

τ

Li₁···i_k =1 2

i∈{i1,...,i_k}

RicL_i 1···ik

ei ,

τ(p)= 1 C_n−2^k−2

1≤i1<···<ik≤n

τ Li₁···ik .

(4.9)

Combining (2.11) and (4.9), we ﬁnd

τ(p)≥n(n−1)

2 Θk(p). (4.10)

From (4.1) and (4.10), we obtain (4.8).

In particular, we obtain the following.

Corollary4.3. LetMbe ann-dimensional invariant submanifold tangent toξin a Kenmotsu space formM(c). Then, for any integer˜ k,2≤k≤n, and any pointp∈M,

Θk(p)≤c−3 4 +c+1

4n . (4.11)

Corollary 4.4. LetM be an n-dimensional anti-invariant submanifold tangent toξin a Kenmotsu space formM(c). Then, for any integer˜ k,2≤k≤n, and any point p∈M,

H²(p)≥Θk(p)−c−3 4 +c+1

2n . (4.12)

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Corollary4.5. LetMbe ann-dimensional contactCR-submanifold of a Kenmotsu space formM(c). Then, for any integer˜ k,2≤k≤n, and any pointp∈M,

H²(p)≥Θk(p)−c−3

4 −(3h−n+1)(c+1)

2n(n−1) , (4.13)

where2h=dimᏰ.

Acknowledgments. The authors are very indebted to the referee for valuable suggestions. This paper is prepared during the third named author’s visit to the Uludag University, Bursa, Turkey in July 2000. The third author is supported by the Scientiﬁc and Technical Research Council of Turkey (TÜBITAK) for NATO-PC Advanced Fellowships Programme.

References

[1] D. E. Blair and B.-Y. Chen,On CR-submanifolds of Hermitian manifolds, Israel J. Math.34 (1979), no. 4, 353–363.

[2] B.-Y. Chen,CR-submanifolds of a Kaehler manifold. I, J. Diﬀerential Geom.16(1981), no. 2, 305–322.

[3] ,Relations between Ricci curvature and shape operator for submanifolds with arbi- trary codimensions, Glasgow Math. J.41(1999), no. 1, 33–41.

[4] B.-Y. Chen and K. Ogiue,On totally real submanifolds, Trans. Amer. Math. Soc.193(1974), 257–266.

[5] K. Kenmotsu,A class of almost contact Riemannian manifolds, Tôhoku Math. J. (2)24 (1972), 93–103.

[6] K. Matsumoto, I. Mihai, and R. Ro¸sca,A certain locally conformal almost cosymplectic man- ifold and its submanifolds, Tensor (N.S.)51(1992), no. 1, 91–102.

[7] S. Tanno, The automorphism groups of almost contact Riemannian manifolds, Tôhoku Math. J. (2)21(1969), 21–38.

[8] K. Yano and S. Ishihara,Thef-structure induced on submanifolds of complex and almost complex spaces, K¯odai Math. Sem. Rep.18(1966), 120–160.

[9] K. Yano and M. Kon,Structures on Manifolds, Series in Pure Mathematics, vol. 3, World Scientiﬁc Publishing, Singapore, 1984.

Kadri Arslan: Department of Mathematics, Faculty of Arts and Sciences, Uludag University, Görükle16059, Bursa, Turkey

E-mail address:[email protected]

Ridvan Ezentas, Cengizhan Murathan, and Cihan Özgür: Department of Mathemat- ics, Faculty of Arts and Sciences, Uludag University, Görükle16059, Bursa, Turkey

Ion Mihai: Faculty of Mathematics, University of Bucharest, Str. Academiei14, 70109Bucharest, Romania

E-mail address:[email protected]