SUBMANIFOLDS IN LOCALLY CONFORMAL ALMOST COSYMPLECTIC MANIFOLDS

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SUBMANIFOLDS IN LOCALLY CONFORMAL ALMOST COSYMPLECTIC MANIFOLDS

DAE WON YOON Received 23 June 2004

We establish inequalities between the Ricci curvature and the squared mean curvature, and also between thek-Ricci curvature and the scalar curvature for a slant, semi-slant, and bi-slant submanifold in a locally conformal almost cosymplectic manifold with arbitrary codimension.

1. Preliminaries

LetMbe a (2m+ 1)-dimensional almost contact manifold with almost contact structure (ϕ,ξ,η), that is, a global vector fieldξ, a (1, 1) tensor fieldϕ, and a 1-formηonMsuch that ϕ²X= −X+η(X)ξ,η(ξ)=1 for any vector fieldXonM. We consider a product manifold M×R, whereRdenotes a real line. Then a vector field onM×Ris given by (X,f(d/dt)), whereX is a vector field tangent toM, tthe coordinate ofR, and f a function onM× R. We define a linear mapJ on the tangent space ofM×RbyJ(X,f(d/dt))=(ϕX− f ξ,η(X)(d/dt)). Then we haveJ²= −I, and henceJ is an almost complex structure on M×R. The manifoldMis said to benormal(see [6]) if the almost complex structureJ is integrable (i.e.,J arises from a complex structure onM×R). Letg be a Riemannian metric onMcompatible with (ϕ,ξ,η), that is,g(ϕX,ϕY)=g(X,Y)−η(X)η(Y) for any vector fieldsX and Y tangent to M. Thus, the manifold Mis almost contact metric, and (ϕ,ξ,η,g) is its almost contact metric structure. Clearly, we haveη(X)₌g(X,ξ) for any vector fieldXtangent toM. LetΦdenote the fundamental 2-form ofMdefined by Φ(X,Y)=g(ϕX,Y) for any vector fieldsXandYtangent toM. The manifold Mis said to bealmost cosymplecticif the formsηandΦare closed. That is,dη=0 anddΦ=0, whered is the operator of exterior diﬀerentiation. IfMis almost cosymplectic and normal, then it is calledcosymplectic(see[1]). It is well known that the almost contact metric manifold is cosymplectic if and only if∇ϕvanishes identically, where∇ is the Levi-Civita connection onM. An almost contact metric manifold Mis a locally conformal almost cosymplectic manifold if and only if there exists a 1-formωsuch thatdΦ=2ω∧Φ,dη=ω∧η, and dω=0.

On the other hand, it is wellknown that the Riemannian curvature tensor ˜Ron a locally conformal almost cosymplectic manifoldM(m≥2) of pointwise constantϕ-sectional

International Journal of Mathematics and Mathematical Sciences 2005:10 (2005) 1621–1632 DOI:10.1155/IJMMS.2005.1621

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curvaturecsatisfies (see[6]) gR(X,Y˜ )Z,W

=c−3f² 4

g(X,W)g(Y,Z)−g(X,Z)g(Y,W) +c+ f²

4

g(X,ϕW)g(Y,ϕZ)−g(X,ϕZ)g(Y,ϕW)−2g(X,ϕY)g(Z,ϕW)

− c+f²

4 +f

g(X,W)η(Y)η(Z)−g(X,Z)η(Y)η(W) +g(Y,Z)η(X)η(W)

−g(Y,W)η(X)η(Z), X,Y,Z,W∈TpM,

(1.1) where f is the function such thatω= f η, f=ξ f.

In [5], Lotta has introduced the following notion of slant submanifolds into almost contact metric manifolds. A submanifoldM tangent toξ in locally conformal almost cosymplectic manifoldMis said to beslantif for anyp∈Mand anyX∈TpM, linearly independent ofξ, the angle betweenϕXandTpM is a constantθ∈[0,π/2], called the slant angleofMinM. Invariant and anti-invariant submanifolds of Mare slant submanifolds with slant anglesθ=0 andθ=π/2, respectively.

We say that a submanifoldMtangent toξis abi-slantsubmanifold inMif there exist two orthogonal distributionsᏰ1andᏰ2onMsuch that

(1)TMadmits the orthogonal direct decompositionTM=Ᏸ1⊕Ᏸ2⊕ {ξ}; (2) for anyi=1, 2,Ᏸiis slant distribution with slant angleθi.

On the other hand,CR-submanifolds ofMare bi-slant submanifolds withθ1=0,θ2= π/2.

Let 2d1=dimᏰ1and 2d2=dimᏰ2.

Remark 1.1. If eitherd1ord2vanishes, the bi-slant submanifold is a slant submanifold.

Thus, slant submanifolds are particular cases of bi-slant submanifolds.

A submanifoldMtangent toξ is called asemi-slantsubmanifold inMif there exist two orthogonal distributionsᏰ1andᏰ2onMsuch that

(1)TMadmits the orthogonal direct decompositionTM=Ᏸ1⊕Ᏸ2⊕ {ξ}; (2) the distributionᏰ1is an invariant distribution, that is,ϕ(Ᏸ1)=Ᏸ1; (3) the distributionᏰ2is slant with angleθ=0.

Remark 1.2. The invariant distribution of a semi-slant submanifold is a slant distribution with zero angle. Thus, it is obvious that, in fact, semi-slant submanifolds are particular cases of bi-slant submanifolds.

(1) Ifd2=0, thenMis an invariant submanifold.

(2) Ifd1=0 andθ=π/2, thenMis an anti-invariant submanifold.

For the other properties and examples of slant, bi-slant, and semi-slant submanifolds in an almost contact metric manifold, we refer to [2,3].

LetMbe ann-dimensional submanifold of a locally conformal almost cosymplectic manifoldMequipped with a Riemannian metricg. The Gauss and Weingarten formulas

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are given, respectively, by

∇˜XY= ∇XY+h(X,Y), _∇˜_XN= −ANX+∇^⊥_XN, (1.2) for allX,Y ∈TMand N∈T^⊥M, where ˜∇,∇, and∇^⊥ are the Riemannian, induced Riemannian, and induced normal connections inM, M, and the normal bundleT^⊥Mof M, respectively, andhis the second fundamental form related to the shape operatorAby g(h(X,Y),N)=g(ANX,Y). Also, letRbe the Riemannian curvature tensor ofM. Then the equation of Gauss is given by

R(X,˜ Y,Z,W)=R(X,Y,Z,W) +gh(X,W),h(Y,Z)−gh(X,Z),h(Y,W), (1.3) for any vectorsX,Y,Z,Wtangent toM.

For any vectorX tangent toM, we putϕX=PX+FX, wherePX andFX are the tangential and the normal components ofϕX, respectively. Given an orthonormal basis {e1,. . .,en}ofM, we define the squared norm ofPby

P ²=

n i,j=1

g²Pei,ej

(1.4)

and the mean curvature vectorH(p) atp∈Mis given byH=(1/n)ⁿ_i₌₁h(ei,ei).

We put

h^r_{i j}=ghei,ej

,er

, h ²=

n i,j=1

ghei,ej

,hei,ej

, (1.5)

where{en+1,. . .,e2m+1}is an orthonormal basis ofT^⊥_pMandr=n+ 1,. . ., 2m+ 1. A sub- manifoldMinMis calledtotally geodesicif the second fundamental form vanishes identically andtotally umbilicalif there is a real numberλsuch thath(X,Y)=λg(X,Y)Hfor any tangent vectorsX,YonM.

For ann-dimensional Riemannian manifoldM, we denote byK(π) the sectional curvature ofMassociated with a plane sectionπ⊂T_pM,p∈M. For an orthonormal basis {e1,. . .,e_n}of the tangent spaceT_pM, the scalar curvatureτis defined by

τ=

i< j

K_{i j}, (1.6)

whereK_{i j}denotes the sectional curvature of the 2-plane section spanned bye_iande_j. Suppose thatLis ak-plane section ofTpM andX a unit vector inL. We choose an orthonormal basis{e1,. . .,ek}ofLsuch thate1=X. Define the Ricci curvature RicLofL atXby

RicL(X)=K12+···+K1k. (1.7)

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We simply called such a curvature ak-Ricci curvature.The scalar curvatureτ of thek- plane sectionLis given by

τ(L)=

1≤i< j≤k

Ki j. (1.8)

For each integerk, 2≤k≤n, the Riemannain invariantΘk on an n-dimensional Rie- mannian manifoldMis defined by

Θk(p)= 1 k−1inf

L,XRicL(X), p∈M, (1.9)

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

Recall that for a submanifoldMin a Riemannain manifold, the relative null space of Mat a pointp∈Mis defined by

Np=

X∈TpM|h(X,Y)=0∀Y∈TpM. (1.10) 2. 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 [4]). We prove similar inequalities for slant, bi-slant, and semi-slant submanifolds in a locally conformal almost cosymplectic manifoldM. We consider submanifolds Mtangent toξ.

Theorem2.1. LetMbe ann-dimensionalθ-slant submanifold tangent toξinto a(2m+ 1)- dimensional locally conformal almost cosymplectic manifoldM. Then, the following hold.

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

4

(n−1)c−3f²+3 2

c+f²cos²θ−4 c+f²

4 +f

+n² H ²

. (2.1) (2)IfH(p)=0, then a unit tangent vectorXorthogonal toξat psatisfies the equality

case of (2.1) if and only ifX∈N_p.

(3)The equality case of (2.1) holds identically for all unit tangent vectors orthogonal toξ atpif and only ifpis a totally geodesic point.

Proof. (1) LetX ∈TpM be a unit tangent vector at porthogonal toξ. We choose an orthonormal basise1,. . .,e_n₌ξ,e_n+1,. . .,e_2m+1, such thate1,. . .,e_nare tangent toM at p withe1=X. Then, from the equation of Gauss, we have

n² H ²=2τ+ h ²−n(n₋1)c₋3f² 4

−3(n−1)c+ f²

4 cos²θ+ 2(n−1) c+ f²

4 +f

.

(2.2)

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From (2.2), we get n² H ²=2τ+

2m+1 r=n+1

h^r₁₁²+h^r₂₂+···+h^r_nn²+ 2

1_≤i< j≤n

h^r_{i j}²

−2

2m+1 r=n+12_≤i< j≤n

h^r_iih^r_{j j}−n(n−1)c−3f² 4

−3(n−1)c+f²

4 cos²θ+ 2(n−1) c+f²

4 +f

=2τ+1 2

2m+1 r=n+1

h^r₁₁+h^r₂₂+···+h^r_nn²+h^r₁₁−h^r₂₂− ··· −h^r_nn²

+ 2

2m+1 r=n+11≤i< j≤n

h^r_{i j}²−2

2m+1 r=n+12≤i< j≤n

h^r_iih^r_{j j}

−n(n−1)c−3f²

4 ⁻

3(n−1)c+f²

4 cos²θ+ 2(n−1) c+f²

4 + f

. (2.3) By using the equation of Gauss, we have

2≤i< j≤n

Ki j=

2m+1 r=n+12≤i< j≤n

h^r_iih^r_{j j}−

h^r_{i j}²+(n−1)(n−2)c−3f² 8

+3(n−2)c+f²

8 cos²θ+1 2

c+f² 4 +f

(−2n+ 4).

(2.4)

Substituting (2.4) in (2.3), we get 1

2n² H ²≥2 Ric(X)−(n−1)c−3f²

2 ⁻

3c+f²

4 cos²θ+ 2 c+f²

4 +f

, (2.5) or equivalently (2.1).

(2) Assume thatH(P)=0. Equality holds in (2.1) if and only if h^r₁₂= ··· =h^r_1n=0,

h^r₁₁=h^r₂₂+···+h^r_nn, r∈ {n+ 1,. . ., 2m+ 1}. (2.6) Thenh^r_1j=0 for all j∈ {1,. . .,n},r∈ {n+ 1,. . ., 2m+ 1}, that is,X∈N_p.

(3) Then equality case of (2.1) holds for all unit tangent vectors orthogonal toξatpif and only if

h^r_{i j}=0, i=j,r∈ {n+ 1,. . ., 2m+ 1},

h^r₁₁+···+h^r_nn−2h^r_ii=0, i∈ {1,. . .,n},r∈ {n+ 1,. . ., 2m+ 1}. (2.7) In this case, it follows thatpis a totally geodesic point. The converse is trivial.

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Theorem2.2. LetM be ann-dimensional bi-slant submanifold satisfyingg(X,ϕY)=0, for anyX∈Ᏸ1and anyY∈Ᏸ2, tangent toξin a(2m+ 1)-dimensional locally conformal almost cosymplectic manifoldM. Then, the following hold.

(1)For each unit vectorX∈T_pMorthogonal toξand if (i)Xis tangent toᏰ1,

Ric(X)≤1 4

(n−1)c−3f²+3 2

c+ f²cos²θ1−4 c+f²

4 +f

+n² H ²

, (2.8) and if

(ii)Xis tangent toᏰ2,

Ric(X)≤1 4

(n−1)c−3f²+3 2

c+ f²cos²θ2−4 c+f²

4 + f

+n² H ²

. (2.9)

(2)IfH(p)=0, then a unit tangent vectorX orthogonal toξat psatisfies the equality case of (2.8) and (2.9) if and only ifX∈Np.

(3)The equality case of (2.8) and (2.9) holds identically for all unit tangent vectors or- thogonal toξatpif and only ifpis a totally geodesic point.

Proof. (1) LetX∈T_pM be a unit tangent vector at p orthogonal toξ. We choose an othonormal basise1,. . .,en=ξ,en+1,. . .,e2m+1 such thate1,. . .,en are tangent toM at p withe1=X. Then, from the equation of Gauss, we have

n² H ²=2τ+ h ²−n(n−1)c−3f² 4

−6c+f² 4

d1cos²θ1+d2cos²θ2

+ 2(n−1) c+ f²

4 +f

,

(2.10)

where 2d1=dimᏰ1and 2d2=dimᏰ2. From (2.10), we get

n² H ²=2τ+

2m+1 r=n+1

h^r₁₁²+h^r₂₂+···+h^r_nn²+ 2

1≤i< j≤n

h^r_{i j}²

−2

2m+1 r=n+12≤i< j≤n

h^r_iih^r_{j j}−n(n−1)c−3f² 4

−6c+f² 4

+ 2(n−1) c+f²

4 +f

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=2τ+1 2

2m+1 r=n+1

h^r₁₁+h^r₂₂+···+h^r_nn²+h^r₁₁−h^r₂₂− ··· −h^r_nn²

+ 2

2m+1 r=n+11≤i< j≤n

h^r_{i j}²−2

2m+1 r=n+12≤i< j≤n

h^r_iih^r_{j j}−n(n−1)c−3f² 4

−6c+f² 4

+ 2(n−1) c+f²

4 +f

.

(2.11) We distinguish two cases.

(i) IfXis tangent toᏰ1, then we have

2_≤i< j≤n

Ki j=

2m+1 r=n+12_≤i< j≤n

h^r_iih^r_{j j}−

h^r_{i j}²+(n−1)(n−2)c−3f² 8

+c+ f² 8

6d1cos²θ1+d2cos²θ2

−3 cos²θ1

+1 2

c+f² 4 +f

(−2n+ 4).

(2.12) Substituting (2.12) in (2.11), one gets

1

2n² H ²≥2 Ric(X)−(n−1)c−3f²

2 ⁻

3c+f²

4 cos²θ1+ 2 c+f²

4 +f

, (2.13) which is equivalent to (2.8).

(ii) IfXis tangent toᏰ2, then we have

2_≤i< j≤n

K_{i j}=

2m+1 r=n+12_≤i< j≤n

h^r_iih^r_{j j}−

h^r_{i j}²+(n−1)(n−2)c−3f² 8

+c+ f² 8

6d1cos²θ1+d2cos²θ2

−3 cos²θ2 +1

2 c+f²

4 +f

(−2n+ 4).

(2.14) Substituting (2.14) in (2.11), one gets

1

2n² H ²≥2 Ric(X)−(n−1)c−3f²

2 ⁻

3c+f²

4 cos²θ2+ 2 c+f²

4 +f

, (2.15) which is equivalent to (2.9).

(2) Assume thatH(p)=0. Equality holds in (2.8) and (2.9) if and only if h^r₁₂= ··· =h^r_1n=0,

h^r₁₁=h^r₂₂+···+h^r_nn, r∈ {n+ 1,. . ., 2m+ 1}. (2.16) Thenh^r_1j=0 for all j∈ {1,. . .,n},r∈ {n+ 1,. . ., 2m+ 1}, that is,X∈Np.

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(3) Then equality case of (2.8) and (2.9) holds for all unit tangent vectors orthogonal toξatpif and only if

h^r_{i j}=0, i=j,r∈ {n+ 1,. . ., 2m+ 1},

h^r₁₁+···+h^r_nn−2h^r_ii=0, i∈ {1,. . .,n},r∈ {n+ 1,. . ., 2m+ 1}. (2.17) In this case, it follows thatpis a totally geodesic point. The converse is trivial.

Corollary 2.3. Let M be an n-dimensional semi-slant submanifold in a (2m+ 1)- dimensional locally conformal almost cosymplectic manifoldM. Then, the following hold.

(1)For each unit vectorX∈TpMorthogonal toξand if (i)Xis tangent toᏰ1,

Ric(X)≤1 4

(n−1)c−3f²−4 c+f²

4 + f

+n² H ²

, (2.18)

and if

(ii)Xis tangent toᏰ2, Ric(X)≤1

4

(n−1)c−3f²+3 2

c+f²cos²θ−4 c+f²

4 +f

+n² H ²

. (2.19) (2)IfH(p)=0, then a unit tangent vectorX orthogonal toξat psatisfies the equality case of (2.18) and (2.19) if and only ifX∈Np.

(3)The equality case of (2.18) and (2.19) holds identically for all unit tangent vectors orthogonal toξatpif and only ifpis a totally geodesic point.

Corollary 2.4. Let M be an n-dimensional invariant submanifold in a (2m+ 1)- dimensional cosymplectic space formM(c). Then, the following hold.˜

(1)For each unit vectorX∈T_pMorthogonal toξ, Ric(X)≤1

4

(n−1)c−3f²+3 2

c+f²−4 c+ f²

4 +f

+n² H ²

. (2.20) (2)IfH(p)=0, then a unit tangent vectorXorthogonal toξat psatisfies the equality

case of (2.20) if and only ifX∈N_p.

(3)The equality case of (2.20) holds identically for all unit tangent vectors orthogonal to ξatpif and only ifpis a totally geodesic point.

Corollary2.5. Let M be ann-dimensional anti-invariant submanifold in a (2m+ 1)- dimensional cosymplectic space formM(c). Then, the following hold.˜

(1)For each unit vectorX∈T_pMorthogonal toξ, Ric(X)≤1

4

(n−1)c−3f²−4 c+f²

4 + f

+n² H ²

. (2.21)

(2)IfH(p)=0, then a unit tangent vectorXorthogonal toξat psatisfies the equality case of (2.21) if and only ifX∈N_p.

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(3)The equality case of (2.21) holds identically for all unit tangent vectors orthogonal to ξatpif and only ifpis a totally geodesic point.

3.k-Ricci curvature and squared mean curvature

In this section, we prove relationship between thek-Ricci curvature and the squared mean curvature for slant, bi-slant, and semi-slant submanifolds in a locally conformal almost cosymplectic manifoldM. We state an inequality between the scalar curvature and the squared mean curvature for submanifoldsMtangent to the vector fieldξ.

Theorem3.1. LetMbe ann-dimensionalθ-slant submanifold tangent toξinto a(2m+ 1)- dimensional locally conformal almost cosymplectic manifoldM. Then,

H ²≥ 2τ n(n−1)⁻

1 4n

nc−3f²+ 3c+f²cos²θ−8 c+ f²

4 +f

, (3.1)

equality holding at a pointp∈Mif and only ifpis a totally umbilical point.

Proof. Let p be a point ofM. We choose an orthonormal basis {e1,e2,. . .,e_n=ξ} for the tangent spaceTpM and{en+1,. . .,e2m+1}for the normal spaceT_p^⊥M at p such that the normal vectoren+1is in the direction of the mean curvature vector ande1,e2,. . .,en

diagonalize the shape operatorA_n+1. Then, we have

An+1=







a1 0 0 . . . 0 0 a2 0 . . . 0 0 0 a3 . . . 0 ... ... ... . .. ...

0 0 0 . . . a_n





 ,

Ar= h^r_{i j},

n i=1

h^r_ii=0, n+ 2≤r≤2m+ 1.

(3.2)

From the equation of Gauss,

n² H ²=2τ+

n i=1

a²_i+

2m+1 r=n+2

n i,j=1

h^r_{i j}²−n(n−1)c−3f² 4

−3(n−1)c+ f²

4 cos²θ+ 2(n−1) c+ f²

4 +f

.

(3.3)

On the other hand,

i< j

a_i−a_j²=(n−1)

n i=1

a²_i−2

i< j

a_ia_j. (3.4)

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Therefore, from the above equation, we have n² H ²=

_n

i=1

ai

2

=

n i=1

a²_i+ 2

i< j

aiaj≤n

n i=1

a²_i. (3.5)

Combining (3.3) and (3.5), n(n−1) H ²≥2τ+

2m+1 r=n+2

n i,j=1

h^r_{i j}²−n(n−1)c−3f² 4

−3(n−1)c+f²

4 cos²θ+ 2(n−1) c+f²

4 +f

,

(3.6)

which implies inequality (3.1). If the equality sign of (3.1) holds at a pointp∈M, then from (3.4) and (3.6) we get Ar=0 (r=n+ 2,. . ., 2m+ 1) and a1= ··· =an. Conse- quently,pis a totally umbilical point. The converse is trivial.

Theorem3.2. LetMbe ann-dimensional bi-slant submanifold satisfyingg(X,ϕY)=0, for anyX∈Ᏸ1and anyY∈Ᏸ2, tangent toξinto a(2m+ 1)-dimensional locally conformal almost cosymplectic manifoldM. Then,

H ²≥ 2τ n(n−1)⁻

1 4n(n−1)

n(n−1)c−3f²+ 6d1cos²θ1+d2cos²θ2

c+ f²

−8(n−1) c+ f²

4 +f

,

(3.7) where2d1=dimᏰ1and2d2=dimᏰ2.

Theorem3.3. LetMbe ann-dimensional semi-slant submanifold tangent toξinto a(2m+ 1)-dimensional locally conformal almost cosymplectic manifoldM. Then,

H ²≥ 2τ n(n−1)⁻

1 4n(n−1)

n(n−1)c−3f²+ 6d1+d2cos²θc+f²

−8(n−1) c+f²

4 +f

,

(3.8)

where2d1=dimᏰ1and2d2=dimᏰ2.

Theorem3.4. LetMbe ann-dimensionalθ-slant submanifold tangent toξinto a(2m+ 1)- dimensional locally conformal almost cosymplectic manifoldM. Then, for any integerk(2≤ k≤n)and any pointp∈M,

H ²≥Θ_k(p)− 1 4n

nc−3f²+ 3c+f²cos²θ−8 c+f²

4 + f

. (3.9)

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Proof. Let{e1,. . .,e_n} be an orthonormal basis ofT_pM. Denote byL_i₁_···_i_k the k-plane section spanned bye_i1,. . .,e_i_k. It follows from (1.7) and (1.8) that

τLi1···ik

=1 2_i_∈{_i

1,...,ik}

RicLi1···ik

ei , τ(p)=1

n−2 k−2

1≤i1<···<ik≤n

τL_i₁_···_i_k.

(3.10)

Combining (1.9) and (3.10), we obtain

τ(p)≥n(n−1)

2 Θk(p). (3.11)

Therefore, by using (3.1) and (3.11), we can obtain the inequality inTheorem 3.4.

Theorem3.5. LetMbe ann-dimensional bi-slant submanifold tangent toξinto a(2m+ 1)-dimensional locally conformal almost cosymplectic manifoldM. Then, for any integer k(2≤k≤n)and any pointp∈M,

H ²≥Θk(p)− 1 4n(n−1)

n(n−1)c−3f²+ 6d1cos²θ1+d2cos²θ2

c+f²

−8(n−1) c+f²

4 +f

,

(3.12) where2d1=dimᏰ1and2d2=dimᏰ2.

Theorem3.6. LetMbe ann-dimensional semi-slant submanifold tangent toξinto a(2m+ 1)-dimensional locally conformal almost cosymplectic manifoldM. Then, for any integer k(2≤k≤n)and any pointp∈M,

H ²≥Θk(p)− 1 4n(n−1)

n(n−1)c−3f²+ 6d1+d2cos²θc+f²

−8(n−1) c+f²

4 +f

,

(3.13)

where2d1=dimᏰ1and2d2=dimᏰ2.

Corollary 3.7. Let M be an n-dimensional invariant submanifold tangent to ξ into a (2m+ 1)-dimensional locally conformal almost cosymplectic manifoldM. Then, for any in- tegerk(2≤k≤n)and any pointp∈M,

H ²≥Θ_k(p)− 1 4n

nc−3f²+ 3c+f²−8 c+f²

4 +f

. (3.14)

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Corollary3.8. LetMbe ann-dimensional anti-invariant submanifold tangent toξinto a(2m+ 1)-dimensional locally conformal almost cosymplectic manifoldM. Then, for any integerk(2≤k≤n)and any pointp∈M,

H ²≥Θk(p)− 1 4n

nc−3f²−8 c+f²

4 +f

. (3.15)

Corollary 3.9. Let M be ann-dimensional contact CR-submanifold tangent to ξ into a(2m+ 1)-dimensional locally conformal almost cosymplectic manifoldM. Then, for any integerk(2≤k≤n)and any pointp∈M,

H ²≥Θ_k(p)− 1 4n(n−1)

n(n−1)c−3f²+ 6d1

c+f²−8(n−1) c+f²

4 + f

. (3.16) References

[1] D. E. Blair,Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, vol. 509, Springer, Berlin, 1976.

[2] J. L. Cabrerizo, A. Carriazo, L. M. Fern´andez, and M. Fern´andez,Semi-slant submanifolds of a Sasakian manifold, Geom. Dedicata78(1999), no. 2, 183–199.

[3] ,Slant submanifolds in Sasakian manifolds, Glasg. Math. J.42(2000), no. 1, 125–138.

[4] B.-Y. Chen,Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasg. Math. J.41(1999), no. 1, 33–41.

[5] A. Lotta,Slant submanifolds in contact geometry, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 39(1996), no. 1-4, 183–198.

[6] Z. Olszak,Locally conformal almost cosymplectic manifolds, Colloq. Math.57(1989), no. 1, 73–

87.

Dae Won Yoon: Department of Mathematics Education and Research Institute of Natural Science (RINS), Gyeongsang National University, 900 Gazwa-dong, Jinju 660-701, South Korea

E-mail address:[email protected]

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Special Issue on

Decision Support for Intermodal Transport

Call for Papers

Intermodal transport refers to the movement of goods in a single loading unit which uses successive various modes of transport (road, rail, water) without handling the goods during mode transfers. Intermodal transport has become an important policy issue, mainly because it is considered to be one of the means to lower the congestion caused by single-mode road transport and to be more environmentally friendly than the single-mode road transport. Both consider- ations have been followed by an increase in attention toward intermodal freight transportation research.

Various intermodal freight transport decision problems are in demand of mathematical models of supporting them.

As the intermodal transport system is more complex than a single-mode system, this fact oﬀers interesting and challeng- ing opportunities to modelers in applied mathematics. This special issue aims to fill in some gaps in the research agenda of decision-making in intermodal transport.

The mathematical models may be of the optimization type or of the evaluation type to gain an insight in intermodal operations. The mathematical models aim to support deci- sions on the strategic, tactical, and operational levels. The decision-makers belong to the various players in the intermodal transport world, namely, drayage operators, terminal operators, network operators, or intermodal operators.

Topics of relevance to this type of decision-making both in time horizon as in terms of operators are:

• Intermodal terminal design

• Infrastructure network configuration

• Location of terminals

• Cooperation between drayage companies

• Allocation of shippers/receivers to a terminal

• Pricing strategies

• Capacity levels of equipment and labour

• Operational routines and lay-out structure

• Redistribution of load units, railcars, barges, and so forth

• Scheduling of trips or jobs

• Allocation of capacity to jobs

• Loading orders

• Selection of routing and service

Before submission authors should carefully read over the journal’s Author Guidelines, which are located athttp://www .hindawi.com/journals/jamds/guidelines.html. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sys- tem athttp://mts.hindawi.com/, according to the following timetable:

Manuscript Due June 1, 2009 First Round of Reviews September 1, 2009 Publication Date December 1, 2009

Lead Guest Editor

Gerrit K. Janssens,Transportation Research Institute (IMOB), Hasselt University, Agoralaan, Building D, 3590 Diepenbeek (Hasselt), Belgium;[email protected]

Guest Editor

Cathy Macharis,Department of Mathematics, Operational Research, Statistics and Information for Systems (MOSI), Transport and Logistics Research Group, Management School, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium;[email protected]

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