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 arbi- trary 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
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:10 (2005) 1621–1632 DOI:10.1155/IJMMS.2005.1621
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 subman- ifolds 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
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)ni=1h(ei,ei).
We put
hri 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 iden- tically 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 cur- vature ofMassociated with a plane sectionπ⊂TpM,p∈M. For an orthonormal basis {e1,. . .,en}of the tangent spaceTpM, the scalar curvatureτis defined by
τ=
i< j
Ki j, (1.6)
whereKi jdenotes the sectional curvature of the 2-plane section spanned byeiandej. 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)
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 inTpMandXruns 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∈Np.
(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,. . .,en=ξ,en+1,. . .,e2m+1, such thate1,. . .,enare 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)
From (2.2), we get n2 H 2=2τ+
2m+1 r=n+1
hr112+hr22+···+hrnn2+ 2
1≤i< j≤n
hri j2
−2
2m+1 r=n+12≤i< j≤n
hriihrj 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
hr11+hr22+···+hrnn2+hr11−hr22− ··· −hrnn2
+ 2
2m+1 r=n+11≤i< j≤n
hri j2−2
2m+1 r=n+12≤i< j≤n
hriihrj 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
hriihrj j−
hri j2+(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 hr12= ··· =hr1n=0,
hr11=hr22+···+hrnn, r∈ {n+ 1,. . ., 2m+ 1}. (2.6) Thenhr1j=0 for all j∈ {1,. . .,n},r∈ {n+ 1,. . ., 2m+ 1}, that is,X∈Np.
(3) Then equality case of (2.1) holds for all unit tangent vectors orthogonal toξatpif and only if
hri j=0, i=j,r∈ {n+ 1,. . ., 2m+ 1},
hr11+···+hrnn−2hrii=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.
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∈TpMorthogonal 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∈TpM 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
hr112+hr22+···+hrnn2+ 2
1≤i< j≤n
hri j2
−2
2m+1 r=n+12≤i< j≤n
hriihrj j−n(n−1)c−3f2 4
−6c+f2 4
d1cos2θ1+d2cos2θ2
+ 2(n−1) c+f2
4 +f
=2τ+1 2
2m+1 r=n+1
hr11+hr22+···+hrnn2+hr11−hr22− ··· −hrnn2
+ 2
2m+1 r=n+11≤i< j≤n
hri j2−2
2m+1 r=n+12≤i< j≤n
hriihrj j−n(n−1)c−3f2 4
−6c+f2 4
d1cos2θ1+d2cos2θ2
+ 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
hriihrj j−
hri j2+(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
Ki j=
2m+1 r=n+12≤i< j≤n
hriihrj j−
hri j2+(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 hr12= ··· =hr1n=0,
hr11=hr22+···+hrnn, r∈ {n+ 1,. . ., 2m+ 1}. (2.16) Thenhr1j=0 for all j∈ {1,. . .,n},r∈ {n+ 1,. . ., 2m+ 1}, that is,X∈Np.
(3) Then equality case of (2.8) and (2.9) holds for all unit tangent vectors orthogonal toξatpif and only if
hri j=0, i=j,r∈ {n+ 1,. . ., 2m+ 1},
hr11+···+hrnn−2hrii=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∈TpMorthogonal 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∈Np.
(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∈TpMorthogonal 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∈Np.
(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,. . .,en=ξ} for the tangent spaceTpM and{en+1,. . .,e2m+1}for the normal spaceTp⊥M at p such that the normal vectoren+1is in the direction of the mean curvature vector ande1,e2,. . .,en
diagonalize the shape operatorAn+1. Then, we have
An+1=
a1 0 0 . . . 0 0 a2 0 . . . 0 0 0 a3 . . . 0 ... ... ... . .. ...
0 0 0 . . . an
,
Ar= hri j,
n i=1
hrii=0, n+ 2≤r≤2m+ 1.
(3.2)
From the equation of Gauss,
n2 H 2=2τ+
n i=1
a2i+
2m+1 r=n+2
n i,j=1
hri j2−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
ai−aj2=(n−1)
n i=1
a2i−2
i< j
aiaj. (3.4)
Therefore, from the above equation, we have n2 H 2=
n
i=1
ai
2
=
n i=1
a2i+ 2
i< j
aiaj≤n
n i=1
a2i. (3.5)
Combining (3.3) and (3.5), n(n−1) H 2≥2τ+
2m+1 r=n+2
n i,j=1
hri j2−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≥Θk(p)− 1 4n
nc−3f2+ 3c+f2cos2θ−8 c+f2
4 + f
. (3.9)
Proof. Let{e1,. . .,en} be an orthonormal basis ofTpM. Denote byLi1···ik the k-plane section spanned byei1,. . .,eik. It follows from (1.7) and (1.8) that
τLi1···ik
=1 2i∈{i
1,...,ik}
RicLi1···ik
ei , τ(p)=1
n−2 k−2
1≤i1<···<ik≤n
τLi1···ik.
(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≥Θk(p)− 1 4n
nc−3f2+ 3c+f2−8 c+f2
4 +f
. (3.14)
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≥Θk(p)− 1 4n(n−1)
n(n−1)c−3f2+ 6d1
c+f2−8(n−1) c+f2
4 + f
. (3.16) References
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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]
